Inertial Mass vs. Gravitational Mass

WEBVTT
Kind: captions
Language: en

00:00:00.180 --> 00:00:01.520
- [Instructor] Knowing
the mass of an object

00:00:01.520 --> 00:00:05.110
actually tells you two independent
things about that object.

00:00:05.110 --> 00:00:08.610
For instance, if you knew that
this truck had a large mass

00:00:08.610 --> 00:00:11.000
you'd know that it has a
large amount of inertia,

00:00:11.000 --> 00:00:11.890
that is to say,

00:00:11.890 --> 00:00:14.310
it'd be very reluctant
to being accelerated.

00:00:14.310 --> 00:00:16.120
It'd be difficult to speed up.

00:00:16.120 --> 00:00:17.570
And once you got it up to speed,

00:00:17.570 --> 00:00:19.160
it'd be very difficult to stop.

00:00:19.160 --> 00:00:20.880
It would take a large amount of force

00:00:20.880 --> 00:00:23.910
that's 'cause it has a large
amount of inertial mass.

00:00:23.910 --> 00:00:27.240
And this idea of inertial
mass is best exemplified

00:00:27.240 --> 00:00:29.180
with Newton's second law.

00:00:29.180 --> 00:00:34.170
So acceleration equals the
net force divided by M.

00:00:34.170 --> 00:00:36.830
This M right here, down
here in the denominator,

00:00:36.830 --> 00:00:39.670
this is the inertial mass,

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because it's telling you
how reluctant that thing is

00:00:41.920 --> 00:00:43.070
to being accelerated more.

00:00:43.070 --> 00:00:46.340
Inertial mass would give
you less acceleration,

00:00:46.340 --> 00:00:48.110
but mass also tells you something else.

00:00:48.110 --> 00:00:51.080
It tells you how much that
object is gonna interact

00:00:51.080 --> 00:00:52.020
via gravity.

00:00:52.020 --> 00:00:55.280
So if this truck has a large
mass that also tells you

00:00:55.280 --> 00:00:58.430
its force of gravity is
going to be very large.

00:00:58.430 --> 00:01:02.880
So the force of gravity FG
on this truck is equal to MG

00:01:02.880 --> 00:01:06.320
that this M right here
is not inertial mass.

00:01:06.320 --> 00:01:09.350
This M here, is telling you
how much this truck interacts

00:01:09.350 --> 00:01:11.430
via gravity with other objects.

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And that means this is
the gravitational mass.

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Now in our universe for a given object,

00:01:16.750 --> 00:01:19.860
these two values inertial
mass and gravitational mass

00:01:19.860 --> 00:01:20.810
are gonna be the same.

00:01:20.810 --> 00:01:24.410
So this trucks, inertial
mass measured in kilograms

00:01:24.410 --> 00:01:27.930
is gonna be the exact
same value as this trucks,

00:01:27.930 --> 00:01:30.920
gravitational mass measured in kilograms,

00:01:30.920 --> 00:01:32.300
but it didn't have to be that way.

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I mean, these two ideas
are conceptually different.

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One, the inertial mass
tells you how much inertia

00:01:37.530 --> 00:01:39.850
or reluctance to
acceleration something has,

00:01:39.850 --> 00:01:42.140
but the gravitational mass
tells you how much that object

00:01:42.140 --> 00:01:43.580
interacts via gravity.

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So you could imagine a universe

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or maybe there's no force gravity,

00:01:47.900 --> 00:01:49.620
but objects still have a reluctance

00:01:49.620 --> 00:01:52.240
to being accelerated by other forces.

00:01:52.240 --> 00:01:54.280
Or maybe you can imagine a universe

00:01:54.280 --> 00:01:56.200
where there is a force of gravity,

00:01:56.200 --> 00:01:57.510
but the number that tells you

00:01:57.510 --> 00:01:59.970
how much something interacts via gravity,

00:01:59.970 --> 00:02:02.150
could have been different
from the number that tells you

00:02:02.150 --> 00:02:04.890
how reluctant that object
is to being accelerated.

00:02:04.890 --> 00:02:07.920
But for our universe, these
two numbers are the same.

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I mean, scientists to this day

00:02:09.420 --> 00:02:11.750
are still doing very delicate experiments

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to try to decern any small differences

00:02:14.400 --> 00:02:15.770
between these two.

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But as far I can tell, to the
best experiments up to date,

00:02:19.110 --> 00:02:21.020
these two numbers are exactly the same

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even though they're
conceptually different.

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So this is good to keep in mind.

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If you're gonna do an experiment,

00:02:26.290 --> 00:02:28.390
you're gonna be measuring
either inertial mass,

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or gravitational mass
typically, how would you know

00:02:31.390 --> 00:02:34.010
in a given experiment if you
measured one or the other?

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Well, I mean, if you just
use a simple experiment,

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like take a spring scale,
measure the force you're exerting

00:02:39.700 --> 00:02:42.600
on a cart and then measure
the acceleration of that cart

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using meter sticks and
stopwatches or a motion sensor.

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And if you just plug this
into Newton's second law,

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so if you know that acceleration
from a motion detector,

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stopwatches and rulers,

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and you measure the force
with the spring scale

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and you solve for this M well,

00:02:58.760 --> 00:03:01.650
this is the denominator
of Newton's second law.

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That means you just
solve for inertial mass,

00:03:04.540 --> 00:03:08.030
'cause you solved in a formula
that contained inertial mass.

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How would you experimentally determine

00:03:09.940 --> 00:03:12.020
the gravitational mass of this cart?

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Well, it's even easier.

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All you have to do take a scale, you know,

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just a digital scale, take your cart,

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put your cart on the digital scale

00:03:19.900 --> 00:03:21.820
and just measure how much the scale reads

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because you know that the force of gravity

00:03:24.020 --> 00:03:25.870
is gonna be measured by the scale.

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That's the number you
get out of the scales

00:03:28.030 --> 00:03:30.500
telling you how much
weight this object has.

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So the scale would just read this

00:03:32.350 --> 00:03:33.560
and if you know what planet you're on,

00:03:33.560 --> 00:03:34.550
you know what G you've got.

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So if you know, G is 9.8
and you solve for this M,

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well, look at you solve
for the gravitational mass,

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how much this thing interacts via gravity.

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So whenever you put something on a scale,

00:03:44.230 --> 00:03:46.370
weigh it like that and get M,

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you're getting gravitational mass.

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If you do the other way
with Newton's second law,

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you're getting inertial mass.

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People get this mixed
up, but it's pretty easy.

00:03:53.710 --> 00:03:56.510
If you ever use a formula
that involves little G

00:03:56.510 --> 00:03:59.620
or like big G, gravitational
constant big G,

00:03:59.620 --> 00:04:02.440
that means you've solved M in that formula

00:04:02.440 --> 00:04:03.920
for gravitational mass.

00:04:03.920 --> 00:04:06.960
If there isn't a G, then you're
solving for inertial mass.

00:04:06.960 --> 00:04:08.490
So for instance, (mumbles)

00:04:08.490 --> 00:04:10.930
you do some experiment where
you try to very delicately

00:04:10.930 --> 00:04:13.940
measure the force of
gravity between two spheres.

00:04:13.940 --> 00:04:14.773
This would be hard.

00:04:14.773 --> 00:04:15.840
You probably wouldn't set it up like this.

00:04:15.840 --> 00:04:17.200
You'd have to be more sophisticated,

00:04:17.200 --> 00:04:19.890
but let's say you could just
measure the force of gravity.

00:04:19.890 --> 00:04:22.220
These two spheres exert on each other.

00:04:22.220 --> 00:04:24.850
The formula for that would be big G, M one

00:04:24.850 --> 00:04:27.330
times M of the other
divided by the distance

00:04:27.330 --> 00:04:28.720
between them squared.

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You'd have to know one of the masses,

00:04:30.280 --> 00:04:32.370
but the spring scale
could give you the force.

00:04:32.370 --> 00:04:35.090
You can measure the distance
between them with a ruler

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big G you know, it's a
constant of the universe.

00:04:37.720 --> 00:04:41.490
If you knew one of the other
masses and solved for this one,

00:04:41.490 --> 00:04:43.297
you'd be getting the gravitational mass

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you could use the
formula that's got big G.

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Any formula with big G or with little G

00:04:49.140 --> 00:04:52.060
like force of gravity is MG.

00:04:52.060 --> 00:04:55.010
These are all formulas that
tell you how much the object M

00:04:55.010 --> 00:04:57.380
is gonna interact via gravity.

00:04:57.380 --> 00:05:01.300
Or you could even imagine
gravitational field is big GM

00:05:02.630 --> 00:05:03.910
over R squared.

00:05:03.910 --> 00:05:05.440
All of these M's here,

00:05:05.440 --> 00:05:08.560
this M here, that M there, that M there,

00:05:08.560 --> 00:05:11.170
and this M here all gravitational mass,

00:05:11.170 --> 00:05:13.370
'cause there's either big G or little G

00:05:13.370 --> 00:05:16.240
involved in that fundamental equation.

00:05:16.240 --> 00:05:18.340
If there's a fundamental
equation that doesn't have big G

00:05:18.340 --> 00:05:19.270
or little G,

00:05:19.270 --> 00:05:21.530
you're not talking about how
something interacts by gravity,

00:05:21.530 --> 00:05:23.090
you're talking about it's inertia,

00:05:23.090 --> 00:05:24.310
and that would be a natural mass.

00:05:24.310 --> 00:05:26.350
So for instance, if you
did some other experiment

00:05:26.350 --> 00:05:28.190
maybe you slam two carts together

00:05:28.190 --> 00:05:31.240
and use conservation of
momentum to solve for M

00:05:31.240 --> 00:05:33.540
well, momentum is MV.

00:05:33.540 --> 00:05:36.250
This formula has nothing to
do with little G or big G,

00:05:36.250 --> 00:05:38.060
no gravitational constants here.

00:05:38.060 --> 00:05:40.340
So if you use this collision experiment

00:05:40.340 --> 00:05:42.150
and solve for the mass
of one of the carts,

00:05:42.150 --> 00:05:45.190
you've solved for the
inertial mass of the cart.

00:05:45.190 --> 00:05:47.500
Similarly, if you use kinetic energy,

00:05:47.500 --> 00:05:50.630
this formula has nothing
fundamentally to do with gravity.

00:05:50.630 --> 00:05:53.150
One half MV squared,
there's no big G or little G

00:05:53.150 --> 00:05:55.700
this M here would be inertial mass.

00:05:55.700 --> 00:05:58.190
If you did the period
of a mass on a spring,

00:05:58.190 --> 00:06:01.400
is two PI root M over K.

00:06:01.400 --> 00:06:03.810
There's no little G or
big G to be found in here.

00:06:03.810 --> 00:06:06.330
That means this is also inertial mass.

00:06:06.330 --> 00:06:07.980
So unless there's a little G or big G

00:06:07.980 --> 00:06:10.760
in your fundamental equation
here, your basic equation,

00:06:10.760 --> 00:06:12.540
that mass is gonna be inertial mass

00:06:12.540 --> 00:06:13.960
if there is a little G or big G

00:06:13.960 --> 00:06:15.920
you're talking about gravitational mass.

00:06:15.920 --> 00:06:16.900
Now, if you're clever,

00:06:16.900 --> 00:06:19.700
you could do a single
experiment with two phases

00:06:19.700 --> 00:06:22.820
and get both masses at once for instance,

00:06:22.820 --> 00:06:25.130
let's say you got a spring
of known spring constant

00:06:25.130 --> 00:06:26.470
and you hung a block on it

00:06:26.470 --> 00:06:28.930
and you lowered it gently until it hangs

00:06:28.930 --> 00:06:31.140
at a certain distance,
unless you measured.

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How much did this thing stretch?

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Well, if you measure that with a ruler,

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then you know it this
position, the spring force KX

00:06:38.740 --> 00:06:42.600
had better be equal to the
gravitational force, MG.

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And so KX would just equal MG

00:06:45.030 --> 00:06:46.730
if the spring constance known

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and you measured X with a ruler,

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and you know what planning you're on

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'cause G is 9.8 on earth.

00:06:51.090 --> 00:06:53.470
If you solve for this and
look at G is right here,

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you multiplied by the G and this formulate

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came from a gravitational formula,

00:06:57.300 --> 00:07:00.130
you would have sold
for gravitational mass.

00:07:00.130 --> 00:07:02.260
And now you know the
gravitational mass of the object,

00:07:02.260 --> 00:07:04.080
how could you get the inertial mass?

00:07:04.080 --> 00:07:06.380
Well, let's say you just
pull down a little extra.

00:07:06.380 --> 00:07:08.390
You pull this down a
little extra, you let go.

00:07:08.390 --> 00:07:10.830
And then it's gonna oscillate
at a certain period.

00:07:10.830 --> 00:07:12.990
Let's say you measure that
period with a stopwatch.

00:07:12.990 --> 00:07:16.090
You measure how long it takes
to go through one full cycle.

00:07:16.090 --> 00:07:19.690
That's got to equal two PI, root M over K.

00:07:19.690 --> 00:07:21.670
Now, there's no little G or big G here.

00:07:21.670 --> 00:07:23.000
This has nothing to do with gravity.

00:07:23.000 --> 00:07:24.890
So if you measure this
period with a stopwatch

00:07:24.890 --> 00:07:26.320
and you know the spring constant

00:07:26.320 --> 00:07:28.160
and you solve for this M,

00:07:28.160 --> 00:07:32.030
well, now you've solved for the
inertial mass of that block.

00:07:32.030 --> 00:07:33.010
And now you know both.

00:07:33.010 --> 00:07:35.300
One stage got us the gravitational mass,

00:07:35.300 --> 00:07:37.780
'cause it came from MG the M did.

00:07:37.780 --> 00:07:39.480
The second stage, got us the inertial mass

00:07:39.480 --> 00:07:42.020
'cause it comes from
two PI, root M over K,

00:07:42.020 --> 00:07:43.970
and this formula has
nothing to do with gravity.

00:07:43.970 --> 00:07:46.720
So this would be a way you
could find both masses at once.

00:07:46.720 --> 00:07:49.670
So recapping, inertial
mass and gravitational mass

00:07:49.670 --> 00:07:53.190
are identical numbers, but
different conceptually.

00:07:53.190 --> 00:07:55.870
One, tells you how reluctant an object is

00:07:55.870 --> 00:07:57.200
to being accelerated.

00:07:57.200 --> 00:07:59.630
And the other tells you how
much the object will interact

00:07:59.630 --> 00:08:00.700
via gravity.

00:08:00.700 --> 00:08:03.160
And if the mass shows
up in a basic formula

00:08:03.160 --> 00:08:05.540
that involves little G or big G

00:08:05.540 --> 00:08:07.890
that's gonna be the gravitational mass.

00:08:07.890 --> 00:08:10.773
Otherwise it's gonna be the inertial mass.


Work-Energy Principle Example

WEBVTT
Kind: captions
Language: en

00:00:00.180 --> 00:00:01.310
- [Instructor] So, the
work-energy principle

00:00:01.310 --> 00:00:03.700
states that the net work done on an object

00:00:03.700 --> 00:00:06.800
is gonna equal the changing
kinetic energy of that object.

00:00:06.800 --> 00:00:08.680
And this works for systems as well.

00:00:08.680 --> 00:00:10.949
So, the net work done
on a system of objects

00:00:10.949 --> 00:00:14.250
is gonna equal the change
in the total kinetic energy

00:00:14.250 --> 00:00:16.030
of the objects in that system.

00:00:16.030 --> 00:00:18.480
Now, that sounds really
complicated and technical,

00:00:18.480 --> 00:00:19.313
but I like to think

00:00:19.313 --> 00:00:21.420
about the work-energy
principle's a shortcut.

00:00:21.420 --> 00:00:23.500
This is a really nice
shortcut that lets me

00:00:23.500 --> 00:00:26.500
determine the change in kinetic
energy without having to do

00:00:26.500 --> 00:00:29.720
a bunch of complicated
conservation of energy equations

00:00:29.720 --> 00:00:31.530
or kinematic formulas.

00:00:31.530 --> 00:00:33.040
The catch is that I need to know

00:00:33.040 --> 00:00:35.410
how to figure out what the net work is.

00:00:35.410 --> 00:00:36.960
So, how do you figure out net work?

00:00:36.960 --> 00:00:40.010
Well, the formula for work
done is F d cosine theta.

00:00:40.010 --> 00:00:42.020
Since this formula, work-energy principle,

00:00:42.020 --> 00:00:43.400
relies on net work,

00:00:43.400 --> 00:00:45.840
this has to be magnitude of the net force

00:00:45.840 --> 00:00:48.000
times magnitude of the distance traveled

00:00:48.000 --> 00:00:49.470
times cosine of theta.

00:00:49.470 --> 00:00:53.500
Remember this theta has to be
angle between, not any angle,

00:00:53.500 --> 00:00:55.810
angle between the net force direction

00:00:55.810 --> 00:00:57.286
and the direction of motion.

00:00:57.286 --> 00:00:59.190
And so, let's try this out.

00:00:59.190 --> 00:01:00.330
How do you use this thing?

00:01:00.330 --> 00:01:01.860
Let's kick the tires.

00:01:01.860 --> 00:01:03.050
Let's say there's a satellite.

00:01:03.050 --> 00:01:04.320
It's moving to the right

00:01:04.320 --> 00:01:06.690
and there's a net force on this satellite.

00:01:06.690 --> 00:01:08.940
Now, this net force could
go in any direction.

00:01:08.940 --> 00:01:11.909
If the net force has a component
in the direction of motion,

00:01:11.909 --> 00:01:14.560
then, the net work is gonna be positive.

00:01:14.560 --> 00:01:17.570
And if so, anything here
from like negative 90,

00:01:17.570 --> 00:01:20.120
well, like it's negative 89.9,

00:01:20.120 --> 00:01:21.670
because 90 would be
perpendicular for many,

00:01:21.670 --> 00:01:25.640
for like 89.9 negative to positive 89.9,

00:01:25.640 --> 00:01:28.440
you've got a component in
the direction of motion.

00:01:28.440 --> 00:01:30.857
That means you're gonna be
doing positive net work.

00:01:30.857 --> 00:01:32.660
And that means the
change of kinetic energy

00:01:32.660 --> 00:01:35.180
will be positive because
it just equals that number.

00:01:35.180 --> 00:01:37.290
That means kinetic energy increases.

00:01:37.290 --> 00:01:38.300
You're gonna be speeding up

00:01:38.300 --> 00:01:40.420
And that kind of, it
makes sense intuitively.

00:01:40.420 --> 00:01:42.570
If your force is in the
direction of motion,

00:01:42.570 --> 00:01:43.780
you're speeding up.

00:01:43.780 --> 00:01:44.760
What about the other case?

00:01:44.760 --> 00:01:45.900
What if your net force points

00:01:45.900 --> 00:01:47.430
in the opposite direction of motion?

00:01:47.430 --> 00:01:49.540
Well, now, the net work
is gonna be negative.

00:01:49.540 --> 00:01:52.490
You'll have a negative
change in kinetic energy.

00:01:52.490 --> 00:01:54.500
In other words, you're gonna slow down

00:01:54.500 --> 00:01:56.900
and if the net force points perpendicular,

00:01:56.900 --> 00:01:58.490
well, then, you're not doing any work

00:01:58.490 --> 00:02:01.430
because cosine of 90 is gonna be zero.

00:02:01.430 --> 00:02:03.040
No net work would be done.

00:02:03.040 --> 00:02:04.920
There's gonna be no
change in kinetic energy.

00:02:04.920 --> 00:02:05.890
That doesn't mean you stop.

00:02:05.890 --> 00:02:08.860
It just means you're not going
to speed up or slow down.

00:02:08.860 --> 00:02:09.693
This does something.

00:02:09.693 --> 00:02:10.650
You might be like,
"Don't you do something?"

00:02:10.650 --> 00:02:11.810
Yeah, you're gonna drift upward.

00:02:11.810 --> 00:02:13.740
You're gonna start
changing your direction,

00:02:13.740 --> 00:02:16.090
but this is not gonna
be doing any work on you

00:02:16.950 --> 00:02:18.890
at that moment.

00:02:18.890 --> 00:02:20.280
And so, just to be clear, I mean,

00:02:20.280 --> 00:02:21.920
let's just try a complicated one here.

00:02:21.920 --> 00:02:24.280
Let's say this force
goes in some direction.

00:02:24.280 --> 00:02:25.790
Let's say your velocity even goes down.

00:02:25.790 --> 00:02:28.400
So, maybe your satellite's
heading downward

00:02:28.400 --> 00:02:30.810
and your force is gonna
go in any direction.

00:02:30.810 --> 00:02:32.580
Well, if it goes this way,

00:02:32.580 --> 00:02:35.370
exactly backwards, it's gonna be 180.

00:02:35.370 --> 00:02:37.040
You're gonna be doing negative work.

00:02:37.040 --> 00:02:39.250
You're gonna be slowing down,
decreasing kinetic energy.

00:02:39.250 --> 00:02:41.130
And you're not gonna change direction.

00:02:41.130 --> 00:02:44.120
If you're like this, you
have a component opposite.

00:02:44.120 --> 00:02:47.271
So, you're gonna be slowing
down and changing direction.

00:02:47.271 --> 00:02:49.360
This will just be changing direction.

00:02:49.360 --> 00:02:51.676
You're not speeding up or
slowing down at that moment.

00:02:51.676 --> 00:02:54.890
This will speed you up
and change your direction.

00:02:54.890 --> 00:02:57.320
And finally, this will
just be speeding you up

00:02:57.320 --> 00:02:59.430
and you will not be
changing your direction.

00:02:59.430 --> 00:03:01.670
So, the work-energy principle's convenient

00:03:01.670 --> 00:03:04.120
to just get a conceptual
or qualitative idea

00:03:04.120 --> 00:03:05.050
of what's going on.

00:03:05.050 --> 00:03:07.510
And it can obviously also give you an idea

00:03:07.510 --> 00:03:09.380
of how to calculate things.

00:03:09.380 --> 00:03:11.970
So, let's try one where you
actually have to get a number.

00:03:11.970 --> 00:03:14.000
So, let's say there's a hot air balloon

00:03:14.000 --> 00:03:16.350
and it's a 300 kilogram hot air balloon.

00:03:16.350 --> 00:03:17.230
Drifting to the left,

00:03:17.230 --> 00:03:19.970
it had an initial speed of
seven meters per second,

00:03:19.970 --> 00:03:22.170
and it's traveling a total of 50 meters

00:03:22.170 --> 00:03:24.150
to the left during this journey.

00:03:24.150 --> 00:03:25.930
Now, there's gonna be forces
on this hot air balloon.

00:03:25.930 --> 00:03:29.010
Obviously, there's gonna be
gravity and some buoyant force,

00:03:29.010 --> 00:03:31.250
but because these are perpendicular

00:03:31.250 --> 00:03:32.510
to the direction of motion,

00:03:32.510 --> 00:03:33.420
they do no work.

00:03:33.420 --> 00:03:35.650
And so, when we're gonna use
this work-energy principle,

00:03:35.650 --> 00:03:36.690
they're not even gonna factor in.

00:03:36.690 --> 00:03:38.120
We don't even have to know these

00:03:38.120 --> 00:03:40.160
since they were perpendicular
and did no work.

00:03:40.160 --> 00:03:42.727
You only consider the forces
in the direction of motion.

00:03:42.727 --> 00:03:44.730
So, let's say there was a wind gust

00:03:44.730 --> 00:03:47.330
helping you to the left
here of 200 newtons,

00:03:47.330 --> 00:03:50.150
but this is a big, bulky
balloon, not that aerodynamic.

00:03:50.150 --> 00:03:51.890
And so, there was a drag force

00:03:51.890 --> 00:03:55.100
from air resistance of
104 newtons to the right.

00:03:55.100 --> 00:03:57.530
And what we wanna know
is we wanna determine

00:03:57.530 --> 00:03:59.150
the final speed of the hot air balloon

00:03:59.150 --> 00:04:02.170
after it travels 50 meters
directly to the left

00:04:02.170 --> 00:04:03.560
with the forces shown.

00:04:03.560 --> 00:04:05.490
Now, there's lots of
different ways to do this.

00:04:05.490 --> 00:04:06.820
You know, Newton's laws,

00:04:06.820 --> 00:04:10.330
you can do a kinematic formula,
there's all kinds of stuff,

00:04:10.330 --> 00:04:12.170
even momentum, technically, impulse,

00:04:12.170 --> 00:04:14.399
but the easiest, I'm pretty
sure the easiest way to do this

00:04:14.399 --> 00:04:17.310
is just gonna be the
work-energy principle,

00:04:17.310 --> 00:04:19.900
which states that the net work done

00:04:19.900 --> 00:04:22.690
is gonna equal the
change in kinetic energy.

00:04:22.690 --> 00:04:23.620
So, let's go ahead and do it.

00:04:23.620 --> 00:04:25.430
So, we know that net work is equal

00:04:25.430 --> 00:04:27.360
to the magnitude of the net force

00:04:27.360 --> 00:04:29.580
times the magnitude of
the distance traveled

00:04:29.580 --> 00:04:31.490
times cosine of the angle between them.

00:04:31.490 --> 00:04:33.050
What's change in kinetic energy mean?

00:04:33.050 --> 00:04:35.970
Well, change in anything
is final minus initial.

00:04:35.970 --> 00:04:39.530
So, since kinetic energy
is 1/2 m v squared,

00:04:39.530 --> 00:04:42.530
this is just gonna be
1/2 m v final squared,

00:04:42.530 --> 00:04:46.710
minus 1/2 m v initial squared.

00:04:46.710 --> 00:04:48.470
So, that's gonna be the
change in kinetic energy,

00:04:48.470 --> 00:04:49.760
final minus initial.

00:04:49.760 --> 00:04:50.910
All right, let's plug in numbers here.

00:04:50.910 --> 00:04:52.660
So, net force, how do we get that?

00:04:52.660 --> 00:04:53.580
Vertical pieces don't matter.

00:04:53.580 --> 00:04:54.860
We're just looking horizontally here.

00:04:54.860 --> 00:04:56.340
Those were the only ones
that are gonna affect it.

00:04:56.340 --> 00:04:58.040
These vertical ones just cancel.

00:04:58.040 --> 00:05:01.180
So, we have 200 to the
left, 104 to the right.

00:05:01.180 --> 00:05:02.320
So, we're gonna have to subtract those

00:05:02.320 --> 00:05:04.270
to get 96 to the left.

00:05:04.270 --> 00:05:05.370
And we just want magnitude.

00:05:05.370 --> 00:05:09.280
So, I'm gonna get 96 newtons to the left,

00:05:09.280 --> 00:05:11.770
not negative or anything,
I'm just taking magnitude,

00:05:11.770 --> 00:05:13.020
times the distance traveled.

00:05:13.020 --> 00:05:14.207
We know that's 50.

00:05:14.207 --> 00:05:18.720
So, we get times 50 meters,
cosine of the angle.

00:05:18.720 --> 00:05:19.553
Now, this is careful.

00:05:19.553 --> 00:05:20.860
You might be like, "Oh 180 here."

00:05:20.860 --> 00:05:23.330
But no, the net force points to the left.

00:05:23.330 --> 00:05:25.060
This 200 is winning here.

00:05:25.060 --> 00:05:27.570
So, leftward net force and
the leftward direction,

00:05:27.570 --> 00:05:29.468
the angle between those two is zero,

00:05:29.468 --> 00:05:31.820
cosine of zero is just one.

00:05:31.820 --> 00:05:33.010
We're maxed out here.

00:05:33.010 --> 00:05:36.200
So, net force points in
the direction of motion.

00:05:36.200 --> 00:05:38.797
So, equals, let's plug
in the rest of this,

00:05:38.797 --> 00:05:42.410
1/2, the mass is 300 kilograms,

00:05:42.410 --> 00:05:44.040
times v final squared,

00:05:44.040 --> 00:05:45.570
that's what we wanna determine,

00:05:45.570 --> 00:05:49.883
minus 1/2, 300 kilograms,

00:05:51.630 --> 00:05:54.890
times the initial speed is
seven meters per second.

00:05:54.890 --> 00:05:58.359
So, we got seven meters per squared.

00:05:58.359 --> 00:06:01.467
Well, 96 times 50 is
gonna come out to 4,800.

00:06:02.590 --> 00:06:04.960
That means that's the net work done.

00:06:04.960 --> 00:06:06.370
Notice that's joules.

00:06:06.370 --> 00:06:08.210
That's how much energy we've added.

00:06:08.210 --> 00:06:09.840
That's the change in kinetic energy.

00:06:09.840 --> 00:06:11.790
So, we know the net work is
change in kinetic energy.

00:06:11.790 --> 00:06:14.864
We've added 4,800 joules
of kinetic energy.

00:06:14.864 --> 00:06:19.864
That's gonna have to equal,
half of 300 is 150 kilograms,

00:06:21.160 --> 00:06:23.630
times v final squared,

00:06:23.630 --> 00:06:28.630
minus, if you take a half
of 300 times seven squared,

00:06:28.700 --> 00:06:31.710
you're gonna get 7,350.

00:06:31.710 --> 00:06:32.890
So, this is how much energy

00:06:32.890 --> 00:06:35.410
the hot air balloon
started with initially.

00:06:35.410 --> 00:06:36.650
So, after we moved that to the left,

00:06:36.650 --> 00:06:38.070
we add those together.

00:06:38.070 --> 00:06:41.760
I'm gonna get 12,150 joules

00:06:41.760 --> 00:06:45.050
is how much kinetic energy
the balloon ends with.

00:06:45.050 --> 00:06:48.740
And that's gotta equal 150 kilograms

00:06:48.740 --> 00:06:50.800
times v final squared.

00:06:50.800 --> 00:06:55.800
I could divide 12,150 by
150 and you get exactly 81

00:06:56.743 --> 00:06:59.530
and that's gonna equal v final squared.

00:06:59.530 --> 00:07:01.050
And if you take a square root of that,

00:07:01.050 --> 00:07:02.440
you get exactly nine.

00:07:02.440 --> 00:07:04.664
So, the final velocity
of this hot air balloon

00:07:04.664 --> 00:07:06.970
is nine meters per second.

00:07:06.970 --> 00:07:07.894
It sped up.

00:07:07.894 --> 00:07:09.630
That's not surprising.

00:07:09.630 --> 00:07:11.850
This net force was directed leftward

00:07:11.850 --> 00:07:13.890
and the object was moving leftward.

00:07:13.890 --> 00:07:14.930
So, we're doing positive work.

00:07:14.930 --> 00:07:16.640
We're increasing the kinetic energy.

00:07:16.640 --> 00:07:17.928
We started with seven meters per second.

00:07:17.928 --> 00:07:20.210
We ended up with nine meters per second.

00:07:20.210 --> 00:07:21.043
And this is an example

00:07:21.043 --> 00:07:23.180
of how you use the work-energy principle.

00:07:23.180 --> 00:07:24.160
So, to recap,

00:07:24.160 --> 00:07:25.470
the work-energy principle states,

00:07:25.470 --> 00:07:28.110
the net work is equal to the
change in kinetic energy.

00:07:28.110 --> 00:07:29.520
This can help you just conceptually

00:07:29.520 --> 00:07:31.350
or qualitatively determine

00:07:31.350 --> 00:07:33.200
whether something is
gonna speed up, slow down,

00:07:33.200 --> 00:07:34.960
or change direction or both.

00:07:34.960 --> 00:07:36.010
And then, quantitatively,

00:07:36.010 --> 00:07:37.930
you can use this to specifically solve

00:07:37.930 --> 00:07:40.140
for the change in kinetic energy,

00:07:40.140 --> 00:07:42.830
as well as the final or initial speed

00:07:42.830 --> 00:07:44.053
something might've had.


Changes in Momentum Worked Examples

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.430
- [Instructor] So, here's a pink ball

00:00:01.430 --> 00:00:03.610
rolling toward a green cube

00:00:03.610 --> 00:00:05.878
that's sitting at rest on
a frictionless surface.

00:00:05.878 --> 00:00:09.920
When the pink ball hits and
slams into the green cube,

00:00:09.920 --> 00:00:12.490
it's gonna exert a force to
the right on the green cube

00:00:12.490 --> 00:00:14.270
and the green cube's gonna speed up.

00:00:14.270 --> 00:00:16.250
But because of Newton's third law,

00:00:16.250 --> 00:00:17.970
whatever force the pink ball

00:00:17.970 --> 00:00:19.860
exerts on the green cube to the right

00:00:19.860 --> 00:00:22.090
has to be equal and opposite to the force

00:00:22.090 --> 00:00:25.600
the green cube exerts
backward on the pink ball.

00:00:25.600 --> 00:00:27.860
So, the green cube's
gonna gain some momentum,

00:00:27.860 --> 00:00:29.800
but the pink ball is
gonna lose some momentum.

00:00:29.800 --> 00:00:31.520
And here's the cool thing.

00:00:31.520 --> 00:00:34.390
Whatever momentum is
gained by this green cube

00:00:34.390 --> 00:00:36.310
has to be the same amount of momentum

00:00:36.310 --> 00:00:38.410
that's lost by the pink ball.

00:00:38.410 --> 00:00:40.510
So, if the green cube gained

00:00:40.510 --> 00:00:42.560
three units of momentum to the right,

00:00:42.560 --> 00:00:44.210
and then, the pink ball has to lose

00:00:44.210 --> 00:00:45.870
three units of momentum to the right.

00:00:45.870 --> 00:00:46.960
And this is true,

00:00:46.960 --> 00:00:49.540
whether these are the same
mass or different masses,

00:00:49.540 --> 00:00:50.760
or whether the green cube

00:00:50.760 --> 00:00:52.890
was moving left to
start or right to start.

00:00:52.890 --> 00:00:53.840
However they collide,

00:00:53.840 --> 00:00:55.340
if two objects collide,

00:00:55.340 --> 00:00:57.670
the change in momentum
between the two objects

00:00:57.670 --> 00:00:59.760
has to be equal and opposite.

00:00:59.760 --> 00:01:01.300
And this is why we love momentum.

00:01:01.300 --> 00:01:02.750
Whatever momentum is gained by one

00:01:02.750 --> 00:01:04.230
has to be lost by the other.

00:01:04.230 --> 00:01:05.840
And so, how is that possible?

00:01:05.840 --> 00:01:07.000
You might be like, "How can that work?"

00:01:07.000 --> 00:01:08.250
Well, let's go up here.

00:01:08.250 --> 00:01:09.620
Think about these forces.

00:01:09.620 --> 00:01:12.150
The forces on each object
are equal and opposite.

00:01:12.150 --> 00:01:13.300
Here's the cool part though.

00:01:13.300 --> 00:01:17.150
The time they're in contact
also has to be equal

00:01:17.150 --> 00:01:19.786
because as soon as one of them
loses contact with the other,

00:01:19.786 --> 00:01:22.580
the other loses contact with the one.

00:01:22.580 --> 00:01:26.050
And in physics, force times
time is called the impulse.

00:01:26.050 --> 00:01:28.260
Now, this is the change in momentum.

00:01:28.260 --> 00:01:29.870
This will equal the change in momentum.

00:01:29.870 --> 00:01:31.100
So, if you wanna know how much momentum

00:01:31.100 --> 00:01:33.508
was gained by this green cube,

00:01:33.508 --> 00:01:36.490
multiply the force
exerted on it by the time,

00:01:36.490 --> 00:01:38.630
you'll get the change in
momentum of the green cube.

00:01:38.630 --> 00:01:39.780
That's true for this pink ball.

00:01:39.780 --> 00:01:41.710
And look, the force is backwards.

00:01:41.710 --> 00:01:44.368
It's just the negative
of this other impulse.

00:01:44.368 --> 00:01:46.900
So, this is why the pink ball,

00:01:46.900 --> 00:01:49.170
no matter what it's mass
or how fast it's going,

00:01:49.170 --> 00:01:50.244
same with green cube,

00:01:50.244 --> 00:01:54.300
these changes in momentum
must be equal and opposite

00:01:54.300 --> 00:01:56.980
simply due to Newton's third law.

00:01:56.980 --> 00:01:58.520
So, to be clear, what I'm saying is this.

00:01:58.520 --> 00:02:00.490
If you are to graph the momentum,

00:02:00.490 --> 00:02:03.340
this pink ball would have
started off with some momentum.

00:02:03.340 --> 00:02:05.600
And that would have been
constant before the collision

00:02:05.600 --> 00:02:07.520
if there's no friction or resistance.

00:02:07.520 --> 00:02:08.840
But then, it lost some momentum.

00:02:08.840 --> 00:02:11.520
So, this goes down and
maybe end up right here.

00:02:11.520 --> 00:02:13.160
And then, afterward, it'll maintain

00:02:13.160 --> 00:02:14.910
a constant amount of momentum.

00:02:14.910 --> 00:02:17.220
The green box, the green cube here,

00:02:17.220 --> 00:02:19.188
started with zero
momentum, it was at rest.

00:02:19.188 --> 00:02:21.980
During this collision,
it gained some momentum.

00:02:21.980 --> 00:02:24.110
So, it's gonna jump up maybe to here.

00:02:24.110 --> 00:02:27.430
What I'm saying is that, and
afterward, it stays constant,

00:02:27.430 --> 00:02:29.759
what I'm saying is that
these two jumps are the same.

00:02:29.759 --> 00:02:31.780
You know, if this thing lost,

00:02:31.780 --> 00:02:34.900
let's say it lost four units of momentum,

00:02:34.900 --> 00:02:37.140
well then, this green cube has to gain

00:02:37.140 --> 00:02:38.357
four units of momentum.

00:02:38.357 --> 00:02:40.010
And you might be like,
"Okay, I don't care."

00:02:40.010 --> 00:02:41.190
But here's why you should care.

00:02:41.190 --> 00:02:44.260
If you are to graph the total momentum,

00:02:44.260 --> 00:02:47.010
what this means is that
the total momentum,

00:02:47.010 --> 00:02:48.930
which was just the pink ball initially,

00:02:48.930 --> 00:02:51.150
is gonna remain constant the whole time.

00:02:51.150 --> 00:02:55.450
It's as if the momentum never
realized a collision occurred.

00:02:55.450 --> 00:02:58.660
This total momentum just remains constant.

00:02:58.660 --> 00:03:01.400
And this is why conservation
and momentum is a thing.

00:03:01.400 --> 00:03:02.750
It is why it's really useful.

00:03:02.750 --> 00:03:05.600
The total initial momentum in a system,

00:03:05.600 --> 00:03:09.000
even if a collision occurs
between objects in that system,

00:03:09.000 --> 00:03:11.460
the total initial momentum must be equal

00:03:11.460 --> 00:03:13.690
to the total final momentum.

00:03:13.690 --> 00:03:16.330
As long as the forces are only internal,

00:03:16.330 --> 00:03:18.277
that is to say between
objects in the system,

00:03:18.277 --> 00:03:20.280
and there's no external forces,

00:03:20.280 --> 00:03:21.820
then, this will always be true.

00:03:21.820 --> 00:03:24.160
So, this is a super powerful tool

00:03:24.160 --> 00:03:26.110
we can use to problem-solve.

00:03:26.110 --> 00:03:28.070
This saves us a lot of time and trouble.

00:03:28.070 --> 00:03:29.240
If a collision is occurring,

00:03:29.240 --> 00:03:32.060
this is one of our best
methods to solve for things

00:03:32.060 --> 00:03:33.825
is conservation of momentum.

00:03:33.825 --> 00:03:35.170
Now, keep in mind.

00:03:35.170 --> 00:03:37.350
Even though the total momentum in a system

00:03:37.350 --> 00:03:38.461
has to stay the same,

00:03:38.461 --> 00:03:40.730
the momentum of an individual object

00:03:40.730 --> 00:03:42.220
does not have to stay the same.

00:03:42.220 --> 00:03:44.679
These objects can exchange
momentum, but again,

00:03:44.679 --> 00:03:47.010
the reason this is gonna be conserved

00:03:47.010 --> 00:03:48.500
is that they do so equally.

00:03:48.500 --> 00:03:51.540
If one gains five, the
other loses five and so on,

00:03:51.540 --> 00:03:53.550
and the total amount stays the same.

00:03:53.550 --> 00:03:55.230
So, let me show you how
this works real quick.

00:03:55.230 --> 00:03:57.200
So, to give you an idea numerically here.

00:03:57.200 --> 00:03:59.450
So, let's say the pink
ball was two kilograms

00:03:59.450 --> 00:04:01.565
and it was going five
meters per second start.

00:04:01.565 --> 00:04:05.330
And the green cube had a
mass, m, and afterward,

00:04:05.330 --> 00:04:08.110
let's say the pink ball's
going four meters per seconds.

00:04:08.110 --> 00:04:09.040
So, it's slowed down.

00:04:09.040 --> 00:04:10.650
And the green cubes speeds up.

00:04:10.650 --> 00:04:12.620
Let's say, it's going
eight meters per second.

00:04:12.620 --> 00:04:14.640
You might think, "Wait,
this isn't conserved.

00:04:14.640 --> 00:04:18.310
The pink ball only lost one, five to four,

00:04:18.310 --> 00:04:20.130
but the green cube gained eight."

00:04:20.130 --> 00:04:22.040
But, remember, we're
not conserving velocity.

00:04:22.040 --> 00:04:24.130
We're conserving momentum.

00:04:24.130 --> 00:04:26.690
So, momentum is m times v,

00:04:26.690 --> 00:04:28.740
momentum is mv,

00:04:28.740 --> 00:04:30.060
and it's a vector, you gotta be careful.

00:04:30.060 --> 00:04:31.430
It has a direction.

00:04:31.430 --> 00:04:33.620
So, I'm not saying the amount
of velocity is conserved

00:04:33.620 --> 00:04:34.453
or anything like that.

00:04:34.453 --> 00:04:35.520
I'm saying momentum is conserved.

00:04:35.520 --> 00:04:38.270
So, what I'm saying is
that p initial total

00:04:38.270 --> 00:04:41.040
has to equal p final total.

00:04:41.040 --> 00:04:43.560
So, only the pink ball
had momentum initially.

00:04:43.560 --> 00:04:46.370
So, it had two kilograms

00:04:46.370 --> 00:04:49.920
times five meters per second of momentum.

00:04:49.920 --> 00:04:52.520
To start with, the green cube had none.

00:04:52.520 --> 00:04:53.978
So, this has gotta equal.

00:04:53.978 --> 00:04:57.390
Afterward, the pink ball has two kilograms

00:04:57.390 --> 00:05:00.170
times four meters per second.

00:05:00.170 --> 00:05:02.320
And the green cube does
have momentum afterward.

00:05:02.320 --> 00:05:03.153
Now, we gotta add it up.

00:05:03.153 --> 00:05:04.130
There's gotta be the total momentum.

00:05:04.130 --> 00:05:06.800
So, plus mass of the cube

00:05:06.800 --> 00:05:09.570
times its eight meters per second.

00:05:09.570 --> 00:05:10.640
Well, so this isn't too hard.

00:05:10.640 --> 00:05:11.473
The math here is easy.

00:05:11.473 --> 00:05:13.080
So, this is gonna be 10 units of momentum,

00:05:13.080 --> 00:05:15.410
10 kilogram meters per second

00:05:15.410 --> 00:05:17.160
is what the ball started with.

00:05:17.160 --> 00:05:18.020
And then, the ball is gonna end

00:05:18.020 --> 00:05:21.220
with eight kilogram meters per second.

00:05:21.220 --> 00:05:22.053
So, we can see right here,

00:05:22.053 --> 00:05:24.330
the ball lost two units of momentum.

00:05:24.330 --> 00:05:26.500
That means the green cube better gain

00:05:26.500 --> 00:05:27.333
two units of momentum.

00:05:27.333 --> 00:05:29.980
So, it's gonna be plus m times eight.

00:05:29.980 --> 00:05:30.813
And so, this is what I mean

00:05:30.813 --> 00:05:32.200
when this can help you solve problems,

00:05:32.200 --> 00:05:33.360
we can just solve for the mass now.

00:05:33.360 --> 00:05:35.530
Now, we can know what the
mass of the cube had to be.

00:05:35.530 --> 00:05:38.600
So, 10 minus eight,
it's gonna be two units.

00:05:38.600 --> 00:05:41.976
It's how much momentum this
green cube has to gain.

00:05:41.976 --> 00:05:44.530
And if we divide two by eight,

00:05:44.530 --> 00:05:46.740
we get that the mass of the green cube

00:05:46.740 --> 00:05:50.610
had to be .25 kilograms.

00:05:50.610 --> 00:05:55.436
So, indeed, if we took the
.25 kilograms times the eight

00:05:55.436 --> 00:05:58.130
that this cube ended with, .25 times eight

00:05:58.130 --> 00:06:00.140
really does give us two, positive two,

00:06:00.140 --> 00:06:03.270
it gained two units of momentum.

00:06:03.270 --> 00:06:05.660
This is why momentum is conserved.

00:06:05.660 --> 00:06:07.434
Whatever a gain in
momentum one thing gets,

00:06:07.434 --> 00:06:09.709
there's a corresponding loss in the other.

00:06:09.709 --> 00:06:12.190
And so, this is gonna be equal.

00:06:12.190 --> 00:06:15.770
Now, how could you ever make
it so that this was not equal?

00:06:15.770 --> 00:06:17.580
Well, this will always be equal

00:06:17.580 --> 00:06:19.140
if the only force is being exerted

00:06:19.140 --> 00:06:20.480
are internal to your system.

00:06:20.480 --> 00:06:23.320
The only way you make this nonequal

00:06:23.320 --> 00:06:25.420
is to have external forces.

00:06:25.420 --> 00:06:26.530
What would that look like?

00:06:26.530 --> 00:06:30.300
Well, imagine the ball and
the cube were on a ramp now.

00:06:30.300 --> 00:06:32.220
So, now that this thing's inclined,

00:06:32.220 --> 00:06:35.370
gravity is gonna be
exerting an external force.

00:06:35.370 --> 00:06:37.150
Let's say it sit like 30 degrees.

00:06:37.150 --> 00:06:40.222
And if we graph the total
momentum of the ball and cube,

00:06:40.222 --> 00:06:42.740
it's not gonna be a straight line anymore.

00:06:42.740 --> 00:06:44.770
It is not gonna look like this.

00:06:44.770 --> 00:06:45.780
That's what it looked like before

00:06:45.780 --> 00:06:47.470
when there were no external forces.

00:06:47.470 --> 00:06:49.540
This time, it's gonna be angled up.

00:06:49.540 --> 00:06:51.430
Let's just call down the ramp positive.

00:06:51.430 --> 00:06:53.270
This time, it's gonna look like this.

00:06:53.270 --> 00:06:55.330
Gravity's pulling down the ramp.

00:06:55.330 --> 00:06:57.100
Let's call that the positive direction.

00:06:57.100 --> 00:06:58.860
If we're measuring the momentum

00:06:58.860 --> 00:07:00.993
in that parallel to the ramp direction,

00:07:00.993 --> 00:07:03.690
we're gonna see an increase
in momentum that direction,

00:07:03.690 --> 00:07:04.523
because that's the way

00:07:04.523 --> 00:07:07.856
that this external
force of gravity points.

00:07:07.856 --> 00:07:09.810
So, how much will this increase?

00:07:09.810 --> 00:07:11.840
How could you figure out what the change

00:07:11.840 --> 00:07:14.120
in the total momentum is gonna be?

00:07:14.120 --> 00:07:16.040
One way to do it is to find the impulse.

00:07:16.040 --> 00:07:19.770
So, remember, force times time
would give you the impulse.

00:07:19.770 --> 00:07:21.550
Let's just say we were looking at this

00:07:21.550 --> 00:07:23.730
for about one second of the time

00:07:23.730 --> 00:07:25.130
that these were on the ramp.

00:07:25.130 --> 00:07:26.820
And let's say they have
the values they did before.

00:07:26.820 --> 00:07:30.620
So, two kilograms and .25 kilograms.

00:07:30.620 --> 00:07:34.790
That would mean the force down
the ramp is mg sine theta.

00:07:34.790 --> 00:07:38.060
So, the force along this
parallel direction is m,

00:07:38.060 --> 00:07:42.440
which the total mass of our
system is 2.25 kilograms.

00:07:42.440 --> 00:07:47.380
So, m times g, 9.8, times sine of 30.

00:07:47.380 --> 00:07:48.940
So, you need to take this sine of 30

00:07:48.940 --> 00:07:50.640
if you wanna know the change in momentum

00:07:50.640 --> 00:07:52.090
along this ramp direction.

00:07:52.090 --> 00:07:57.030
So, the force of gravity
parallel is mg sine theta.

00:07:57.030 --> 00:07:58.220
And then, that's the force.

00:07:58.220 --> 00:07:59.670
So, that's the force parallel to the ramp.

00:07:59.670 --> 00:08:01.970
That would give us the
change in momentum parallel.

00:08:01.970 --> 00:08:04.803
We just have to multiply by
the time, which is one second.

00:08:04.803 --> 00:08:09.140
And that gives us about
11 units of momentum.

00:08:09.140 --> 00:08:12.690
So, this system would
gain 11 units of momentum.

00:08:12.690 --> 00:08:13.673
That doesn't tell you who's gonna get it.

00:08:13.673 --> 00:08:15.520
It's not like the two gets all of it,

00:08:15.520 --> 00:08:17.260
or the .25 gets all of it.

00:08:17.260 --> 00:08:18.470
But the total system,

00:08:18.470 --> 00:08:19.860
if you watched for a second,

00:08:19.860 --> 00:08:21.689
while it's on this
ramp, frictionless ramp,

00:08:21.689 --> 00:08:23.448
it would increase its momentum.

00:08:23.448 --> 00:08:25.397
It'd be changing its total momentum

00:08:25.397 --> 00:08:28.780
because there was an external force.

00:08:28.780 --> 00:08:32.090
So, recapping, if there are no
external forces on a system,

00:08:32.090 --> 00:08:33.534
then, the total momentum initial

00:08:33.534 --> 00:08:37.110
will equal the total momentum final.

00:08:37.110 --> 00:08:39.520
So, using p equals mv,

00:08:39.520 --> 00:08:41.380
you can add up these contributions

00:08:41.380 --> 00:08:43.070
and set them equal to problem-solve.

00:08:43.070 --> 00:08:46.020
And in a case where there
is an external force,

00:08:46.020 --> 00:08:48.040
you can find how much that system

00:08:48.040 --> 00:08:50.030
will gain or lose in momentum

00:08:50.030 --> 00:08:52.100
by taking that external force

00:08:52.100 --> 00:08:54.550
times the time that force was applied.

00:08:54.550 --> 00:08:57.140
That would give you the
total change in momentum

00:08:57.140 --> 00:08:59.153
of that system over that time.


Torque Basics

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.540
- [Instructor] Imagine
you've got a door here

00:00:01.540 --> 00:00:03.310
with a blue door knob.

00:00:03.310 --> 00:00:05.370
Any one of these 10 newton forces

00:00:05.370 --> 00:00:09.170
will cause the door to rotate
around the hinge or the axis,

00:00:09.170 --> 00:00:11.320
or sometimes this is
called the pivot point.

00:00:11.320 --> 00:00:13.930
Any one of these forces will
cause the door to rotate.

00:00:13.930 --> 00:00:16.090
My question is, if you
could insert one of these

00:00:16.090 --> 00:00:17.640
in one of these locations,

00:00:17.640 --> 00:00:19.650
which one of these forces, if any,

00:00:19.650 --> 00:00:23.310
would cause the most angular
acceleration of this door?

00:00:23.310 --> 00:00:26.070
And you might think, "Oh,
well, 10 newtons is 10 newtons.

00:00:26.070 --> 00:00:28.750
They'll all cause the same
amount," but that's not true.

00:00:28.750 --> 00:00:30.080
It turns out we put door knobs

00:00:30.080 --> 00:00:32.210
at the end of doors for a reason.

00:00:32.210 --> 00:00:34.280
This red 10 newtons at the outside edge

00:00:34.280 --> 00:00:36.840
will cause the most angular acceleration.

00:00:36.840 --> 00:00:39.570
It'll cause this door to
speed up most rapidly.

00:00:39.570 --> 00:00:40.630
And this used to bother me.

00:00:40.630 --> 00:00:43.230
I was like, "How come this
is getting an advantage?"

00:00:43.230 --> 00:00:45.010
I think the best way to
think about it is this.

00:00:45.010 --> 00:00:46.930
Even though these forces
are all going through

00:00:46.930 --> 00:00:49.597
the same angle, so they've
gone through 20 degrees,

00:00:49.597 --> 00:00:51.870
and now they've all
gone through 30 degrees,

00:00:51.870 --> 00:00:55.760
now they've all gone
through 45, 60 and 90.

00:00:55.760 --> 00:00:57.120
Even though these forces

00:00:57.120 --> 00:00:58.710
have all gone through the same angle,

00:00:58.710 --> 00:01:01.320
they have not gone
through the same distance.

00:01:01.320 --> 00:01:02.153
Some of these forces

00:01:02.153 --> 00:01:03.860
have been exerted through
a larger distance.

00:01:03.860 --> 00:01:04.760
So just look at it.

00:01:04.760 --> 00:01:06.830
If you imagine rotating this thing,

00:01:06.830 --> 00:01:09.770
that red force, this
outside pink force here

00:01:09.770 --> 00:01:12.230
goes through a much larger distance

00:01:12.230 --> 00:01:14.410
than that inner yellow force.

00:01:14.410 --> 00:01:17.190
This force has gone through
very little distance whatsoever.

00:01:17.190 --> 00:01:18.840
And you might think, well,
why does that matter?

00:01:18.840 --> 00:01:22.260
Well, it matters because
if you remember work done.

00:01:22.260 --> 00:01:24.670
Work done is proportional
to the amount of force,

00:01:24.670 --> 00:01:25.780
but these are all 10 newtons

00:01:25.780 --> 00:01:27.370
so that doesn't really matter here.

00:01:27.370 --> 00:01:29.560
And it's also proportional
to the amount of distance

00:01:29.560 --> 00:01:31.280
through which that force is applied.

00:01:31.280 --> 00:01:33.360
And because this outside force

00:01:33.360 --> 00:01:35.590
has gone through so much more distance

00:01:35.590 --> 00:01:37.440
than these inner forces,

00:01:37.440 --> 00:01:40.240
it's done more work over the same angle.

00:01:40.240 --> 00:01:42.180
And if you do more work,

00:01:42.180 --> 00:01:44.930
you input more kinetic
energy into the door,

00:01:44.930 --> 00:01:48.560
it's gonna be moving
faster for the same angle

00:01:48.560 --> 00:01:51.610
compared to what's caused
by these other forces here.

00:01:51.610 --> 00:01:53.550
And this is why in angular mechanics

00:01:53.550 --> 00:01:55.370
you can't just think about forces,

00:01:55.370 --> 00:01:58.160
you have to think about
something called torque.

00:01:58.160 --> 00:02:00.880
The symbol for torque is this fancy T.

00:02:00.880 --> 00:02:02.820
It's the Greek letter tau.

00:02:02.820 --> 00:02:05.210
And the amount of torque
caused by a force,

00:02:05.210 --> 00:02:07.380
so you need a force to cause a torque,

00:02:07.380 --> 00:02:08.740
but it's more than just force.

00:02:08.740 --> 00:02:10.620
You have to multiply r,

00:02:10.620 --> 00:02:13.190
the distance from the access to the force

00:02:13.190 --> 00:02:15.170
by the amount of force

00:02:15.170 --> 00:02:17.790
in order to find how much
torque is being exerted

00:02:17.790 --> 00:02:19.350
by a given force.

00:02:19.350 --> 00:02:21.220
The more torque that's exerted,

00:02:21.220 --> 00:02:23.900
the more angular acceleration you'd get,

00:02:23.900 --> 00:02:25.960
the faster you'd get
something to speed up.

00:02:25.960 --> 00:02:26.793
Now you might wonder like,

00:02:26.793 --> 00:02:29.210
"Okay, I get that more
force gives me more torque,

00:02:29.210 --> 00:02:31.690
how come this is just r
and not like r squared?

00:02:31.690 --> 00:02:34.100
It seems kind of random, maybe
its like square root of r."

00:02:34.100 --> 00:02:37.240
Well, if you remember arc
lengths from back in the day,

00:02:37.240 --> 00:02:39.550
arc length is r times theta.

00:02:39.550 --> 00:02:42.650
So if I'm twice as far away from an axis,

00:02:42.650 --> 00:02:44.400
I get twice the arc length.

00:02:44.400 --> 00:02:47.170
If I get twice the arc length,
I get twice the work done.

00:02:47.170 --> 00:02:48.180
You get twice the work done,

00:02:48.180 --> 00:02:50.350
you get twice the input kinetic energy

00:02:50.350 --> 00:02:52.470
and it turns out twice the kinetic energy

00:02:52.470 --> 00:02:54.940
will give you twice the
angular acceleration.

00:02:54.940 --> 00:02:57.680
This is why everything
is just proportional to r

00:02:57.680 --> 00:03:00.510
in terms of torque, it's
not like r squared here.

00:03:00.510 --> 00:03:03.830
So for example, let's just say
the distance from the axis,

00:03:03.830 --> 00:03:04.663
because that's what matters,

00:03:04.663 --> 00:03:07.080
to this 10 newtons here was one meter.

00:03:07.080 --> 00:03:10.210
And from the access to the
purple force was two meters.

00:03:10.210 --> 00:03:14.180
And from the access to this
doorknob force was three meters.

00:03:14.180 --> 00:03:16.070
What this torque formula
means is that even though

00:03:16.070 --> 00:03:17.330
these are all 10 newtons,

00:03:17.330 --> 00:03:19.170
they'd all be exerting
different amounts of torque.

00:03:19.170 --> 00:03:21.680
I'd have to take the one
meter times 10 newtons

00:03:21.680 --> 00:03:24.160
would give me 10 newton meters.

00:03:24.160 --> 00:03:27.710
So the unit for torque
is meters times newtons,

00:03:27.710 --> 00:03:29.970
but we usually write it as newton meters.

00:03:29.970 --> 00:03:31.240
If you buy a torque wrench,

00:03:31.240 --> 00:03:34.170
you could set it in newton
meters or in foot pounds,

00:03:34.170 --> 00:03:36.470
if you're doing the US system.

00:03:36.470 --> 00:03:38.420
And then this purple force,
even though it's 10 newtons,

00:03:38.420 --> 00:03:41.360
you'd have to take two
meters times 10 newtons.

00:03:41.360 --> 00:03:45.290
This would exert a torque
of 20 newton meters

00:03:45.290 --> 00:03:47.280
and the doorknob force wins the battle

00:03:47.280 --> 00:03:51.100
because it would have three
times 10 would exert a torque

00:03:51.100 --> 00:03:53.140
of 30 newton meters.

00:03:53.140 --> 00:03:56.070
So the same size force can
exert a different amount

00:03:56.070 --> 00:03:59.070
of torque, depending on how
far away it is from the axis.

00:03:59.070 --> 00:04:01.020
So one area you have to
be careful of this torque

00:04:01.020 --> 00:04:02.480
technically is a vector.

00:04:02.480 --> 00:04:03.313
It has a direction,

00:04:03.313 --> 00:04:04.710
it could be positive or negative.

00:04:04.710 --> 00:04:07.690
If you're doing full-blown
engineering 3D physics,

00:04:07.690 --> 00:04:10.450
technically these torques would
point out of the screen here

00:04:10.450 --> 00:04:13.350
out of the page, but for
intro algebra-based physics,

00:04:13.350 --> 00:04:15.700
and for most problems, you
can usually get away with

00:04:15.700 --> 00:04:19.730
just considering
counterclockwise or clockwise

00:04:19.730 --> 00:04:21.810
as being the direction of the torque.

00:04:21.810 --> 00:04:25.410
That is to say these forces
were making this object rotate

00:04:25.410 --> 00:04:27.840
in the counterclockwise direction,

00:04:27.840 --> 00:04:29.630
so they were all at the same sign.

00:04:29.630 --> 00:04:32.590
The convention is to call
counterclockwise positive.

00:04:32.590 --> 00:04:34.230
So we'd call these all positive.

00:04:34.230 --> 00:04:37.440
If there were any forces that
tried to rotate the system

00:04:37.440 --> 00:04:40.140
clockwise, you'd call
those torques negative.

00:04:40.140 --> 00:04:42.600
You can do it either way, as
long as you're consistent.

00:04:42.600 --> 00:04:44.740
Most books pick this as
the convention though,

00:04:44.740 --> 00:04:45.890
so you should be aware of that.

00:04:45.890 --> 00:04:47.560
And then the last little
bit to be careful about,

00:04:47.560 --> 00:04:50.180
I'm drawing all these forces
nice and perpendicular

00:04:50.180 --> 00:04:53.960
to the r, and if that's the
case, you just do r times F.

00:04:53.960 --> 00:04:56.200
If your force has different components,

00:04:56.200 --> 00:04:58.080
you need to make sure
that the only component

00:04:58.080 --> 00:05:00.360
you plug in here is the
perpendicular piece.

00:05:00.360 --> 00:05:02.360
So if this had some weird angle here,

00:05:02.360 --> 00:05:05.420
you'd only want the piece
that was directly into

00:05:05.420 --> 00:05:10.060
this perpendicular lever arm
here at a perpendicular angle.

00:05:10.060 --> 00:05:11.480
We'll talk about that more later.

00:05:11.480 --> 00:05:13.980
For now, let's just try some
problems to kick the tires

00:05:13.980 --> 00:05:15.560
and get used to this formula.

00:05:15.560 --> 00:05:16.850
So imagine this example here

00:05:16.850 --> 00:05:18.900
where you've got the fancy door, you know,

00:05:18.900 --> 00:05:20.520
with a fancy hotel or restaurant

00:05:20.520 --> 00:05:22.930
that's a rotating circle and you can go in

00:05:22.930 --> 00:05:23.840
from either direction.

00:05:23.840 --> 00:05:25.470
This would be a bird's-eye view.

00:05:25.470 --> 00:05:27.550
Now, imagine you go into the hotel,

00:05:27.550 --> 00:05:28.383
you're pushing over here,

00:05:28.383 --> 00:05:29.670
you took physics, you know what to do.

00:05:29.670 --> 00:05:31.410
So you exert this 20 newtons over here.

00:05:31.410 --> 00:05:33.310
Let's say someone else comes
in from the other edge.

00:05:33.310 --> 00:05:35.680
It's all awkward and they're
trying to go in the other way

00:05:35.680 --> 00:05:36.740
and it's a stalemate.

00:05:36.740 --> 00:05:39.920
You're both pushing with
forces, but nothing's happening.

00:05:39.920 --> 00:05:41.810
And that doesn't mean
the two forces are equal.

00:05:41.810 --> 00:05:44.650
If you're in a stalemate here
in terms of angular motion,

00:05:44.650 --> 00:05:47.660
that means your torques
are equal and opposite.

00:05:47.660 --> 00:05:49.890
They're opposed, they
have the same magnitude,

00:05:49.890 --> 00:05:51.630
but they'll have opposite
directions of torque.

00:05:51.630 --> 00:05:53.180
So if you're locked in a stalemate here,

00:05:53.180 --> 00:05:55.950
that means the torque that
you exert with your 20 newtons

00:05:55.950 --> 00:05:59.650
has to be equal to the
torque from the other person.

00:05:59.650 --> 00:06:01.370
So let's try to figure out how much force

00:06:01.370 --> 00:06:02.860
would this person have to exert?

00:06:02.860 --> 00:06:04.600
It's not gonna be 20 newtons.

00:06:04.600 --> 00:06:06.480
They're pushing closer to the axis here

00:06:06.480 --> 00:06:08.110
so they're gonna have
to push with more force.

00:06:08.110 --> 00:06:09.230
How much more force?

00:06:09.230 --> 00:06:11.810
Well, we can use the formula
for torque to find it.

00:06:11.810 --> 00:06:13.060
The torques have to be equal.

00:06:13.060 --> 00:06:15.240
If there's no rotation
here, you're balanced out.

00:06:15.240 --> 00:06:16.630
If your force is 20 newtons,

00:06:16.630 --> 00:06:19.910
you're exerting a force three
meters away from the axis.

00:06:19.910 --> 00:06:22.170
That's your r, would be three meters

00:06:22.170 --> 00:06:25.770
times 20 newtons means you're
exerting 20 times three,

00:06:25.770 --> 00:06:27.970
so 60 newton meters of torque.

00:06:27.970 --> 00:06:29.940
That means the other
person has to be exerting

00:06:29.940 --> 00:06:33.000
60 newton meters of torque,
but there r isn't two.

00:06:33.000 --> 00:06:35.140
Be careful here, you always
have to measure from the axis,

00:06:35.140 --> 00:06:36.560
the point where you're rotating about.

00:06:36.560 --> 00:06:38.640
That'd be one meter.

00:06:38.640 --> 00:06:40.210
This door's all symmetric here.

00:06:40.210 --> 00:06:42.590
So it'd be one meter times F.

00:06:42.590 --> 00:06:45.410
And if you take this 60 newton meters

00:06:45.410 --> 00:06:47.260
and you divide by one meter,

00:06:47.260 --> 00:06:48.450
you're gonna get that this force here

00:06:48.450 --> 00:06:50.040
is gonna have to be 60 newtons.

00:06:50.040 --> 00:06:52.040
So this person is gonna
have to exert more force.

00:06:52.040 --> 00:06:55.280
In fact, they pushed three
times closer to the axis,

00:06:55.280 --> 00:06:57.010
so they're gonna have to exert three times

00:06:57.010 --> 00:06:58.310
the force that you do.

00:06:58.310 --> 00:07:02.030
You have a three times advantage
here in holding this door

00:07:02.030 --> 00:07:03.770
compared to the other person.

00:07:03.770 --> 00:07:05.070
All right, let's try one more

00:07:05.070 --> 00:07:06.590
just to make sure we understand it.

00:07:06.590 --> 00:07:08.190
Let's say it's now rush hour, you know,

00:07:08.190 --> 00:07:10.410
bird's eye view here, same circular door.

00:07:10.410 --> 00:07:12.240
Three people are trying
to go through at once.

00:07:12.240 --> 00:07:13.660
It's gonna be a mad house.

00:07:13.660 --> 00:07:16.020
This time I want to know,
it's not gonna be a stalemate.

00:07:16.020 --> 00:07:17.780
This door is gonna
rotate in some direction.

00:07:17.780 --> 00:07:20.130
I want to know what the net torque is.

00:07:20.130 --> 00:07:21.660
So just like you can find net force,

00:07:21.660 --> 00:07:24.120
you can find the net torque,
but you gotta be careful.

00:07:24.120 --> 00:07:25.290
These might have different signs,

00:07:25.290 --> 00:07:27.320
you gotta add or subtract accordingly.

00:07:27.320 --> 00:07:28.370
So start over here.

00:07:28.370 --> 00:07:31.020
How much torque would
be from this 10 newtons?

00:07:31.020 --> 00:07:34.600
Well, it's exerted three
meters away from the axis,

00:07:34.600 --> 00:07:35.990
so it's r is three meters.

00:07:35.990 --> 00:07:38.750
So the torque from that
force would be three meters

00:07:38.750 --> 00:07:42.010
times 10 newtons, and
since this is directed

00:07:42.010 --> 00:07:45.270
counterclockwise, I'm just
gonna call that positive

00:07:45.270 --> 00:07:47.110
and I'll have to be
consistent with that choice.

00:07:47.110 --> 00:07:48.700
So now let's consider this eight newtons.

00:07:48.700 --> 00:07:50.960
You might think it would
have an oppositely directed

00:07:50.960 --> 00:07:53.120
sign of torque from this 10 newtons

00:07:53.120 --> 00:07:54.880
'cause the eight is down, the 10 is up,

00:07:54.880 --> 00:07:57.440
but it's also trying to rotate this door

00:07:57.440 --> 00:07:59.270
in the counterclockwise direction.

00:07:59.270 --> 00:08:00.780
So in terms of forces,

00:08:00.780 --> 00:08:03.560
this 10 newton and eight
newton are oppositely directed,

00:08:03.560 --> 00:08:06.220
but in terms of torques,
they are the same direction.

00:08:06.220 --> 00:08:08.890
They're both causing
rotation counterclockwise.

00:08:08.890 --> 00:08:11.770
So if I called this torque
from the 10 newtons positive,

00:08:11.770 --> 00:08:14.620
I've gotta call the torque from
this eight newtons positive

00:08:14.620 --> 00:08:16.780
'cause it's trying to exert a
torque in the same direction.

00:08:16.780 --> 00:08:20.080
So I'd have one meter
is the r for the eight

00:08:20.080 --> 00:08:22.780
times eight newtons would be the torque

00:08:22.780 --> 00:08:24.250
from the eight newtons.

00:08:24.250 --> 00:08:25.500
And then I have one more force here.

00:08:25.500 --> 00:08:28.960
This five newton is trying
to rotate clockwise.

00:08:28.960 --> 00:08:31.130
Since I called counterclockwise positive,

00:08:31.130 --> 00:08:32.760
I'm gonna have to make
this a negative torque,

00:08:32.760 --> 00:08:36.470
so minus three meters,
the r from the access

00:08:36.470 --> 00:08:39.180
to this five newtons is three meters

00:08:39.180 --> 00:08:41.250
multiplied by five newtons.

00:08:41.250 --> 00:08:44.360
And if you take 30 plus eight minus 15,

00:08:44.360 --> 00:08:45.230
you're gonna get a total

00:08:45.230 --> 00:08:48.760
of positive 23 newton meters of torque,

00:08:48.760 --> 00:08:49.930
so this is not a stalemate.

00:08:49.930 --> 00:08:52.920
There will be an amount
of angular acceleration

00:08:52.920 --> 00:08:55.120
caused by this net torque.

00:08:55.120 --> 00:08:58.410
So to recap, just like net forces

00:08:58.410 --> 00:09:01.010
can cause regular acceleration,

00:09:01.010 --> 00:09:04.140
net torques can cause
angular acceleration.

00:09:04.140 --> 00:09:05.400
If there is no net torque,

00:09:05.400 --> 00:09:07.950
that means there is no
angular acceleration.

00:09:07.950 --> 00:09:10.820
The way you find the
torque from a given force

00:09:10.820 --> 00:09:13.880
is you take r, the distance from the axis

00:09:13.880 --> 00:09:16.360
to where that force is
applied and you multiply

00:09:16.360 --> 00:09:19.970
by the amount of force, as long
as it's that amount of force

00:09:19.970 --> 00:09:23.170
that runs perpendicular to this lever arm

00:09:23.170 --> 00:09:25.240
or this r direction.

00:09:25.240 --> 00:09:27.240
Be careful that torque is a vector.

00:09:27.240 --> 00:09:30.840
We typically count
counterclockwise as positive

00:09:30.840 --> 00:09:34.610
and clockwise as negative,
but if you're consistent,

00:09:34.610 --> 00:09:36.050
you can call whichever one of these

00:09:36.050 --> 00:09:37.450
you want to be positive

00:09:37.450 --> 00:09:39.600
as long as you call
the other one negative.


Period of a Pendulum

WEBVTT
Kind: captions
Language: en

00:00:00.150 --> 00:00:01.310
- [Instructor] So, a simple pendulum

00:00:01.310 --> 00:00:03.540
is just a mass hanging from a string.

00:00:03.540 --> 00:00:05.210
And if you are to pull this mass,

00:00:05.210 --> 00:00:07.004
sometimes it's called a pendulum bob,

00:00:07.004 --> 00:00:09.380
if you are to pull it
back and then let go,

00:00:09.380 --> 00:00:11.860
gravity would act as a restoring force

00:00:11.860 --> 00:00:15.310
and this mass would swing
back and forth over and over.

00:00:15.310 --> 00:00:17.220
And because this simple pendulum

00:00:17.220 --> 00:00:19.920
is a simple harmonic
oscillator, it's motion.

00:00:19.920 --> 00:00:21.552
Its angle as a function of time

00:00:21.552 --> 00:00:25.320
would be accurately described
by a sine or a cosine graph.

00:00:25.320 --> 00:00:26.153
So, in other words,

00:00:26.153 --> 00:00:30.120
if you are to pull this back
initially by 15 degrees,

00:00:30.120 --> 00:00:32.600
you might get a graph that
looks something like this.

00:00:32.600 --> 00:00:34.840
Now, every simple harmonic oscillator

00:00:34.840 --> 00:00:38.010
has a characteristic period of motion.

00:00:38.010 --> 00:00:40.670
Now, the period of motion
is the time it takes

00:00:40.670 --> 00:00:42.800
to complete one full cycle.

00:00:42.800 --> 00:00:45.210
So, the time it takes to swing from here

00:00:45.210 --> 00:00:46.043
all the way to there,

00:00:46.043 --> 00:00:47.330
all the way back to here,

00:00:47.330 --> 00:00:49.660
would be one full period.

00:00:49.660 --> 00:00:52.570
And in this graph, I've
written it as .5 seconds,

00:00:52.570 --> 00:00:54.690
but the period of every simple pendulum

00:00:54.690 --> 00:00:56.290
is not gonna be .5 seconds.

00:00:56.290 --> 00:00:57.930
The period of a simple pendulum

00:00:57.930 --> 00:01:00.687
depends on the characteristics
of that pendulum

00:01:00.687 --> 00:01:02.520
and the environment that it's in.

00:01:02.520 --> 00:01:05.760
So, to derive this formula
for the period of a pendulum,

00:01:05.760 --> 00:01:07.300
you would need calculus.

00:01:07.300 --> 00:01:08.420
So, I'm just gonna write it down

00:01:08.420 --> 00:01:10.390
and give you a quick tour and compare it

00:01:10.390 --> 00:01:12.560
with the period formula
for a mass on a spring,

00:01:12.560 --> 00:01:13.600
to the period of a pendulum

00:01:13.600 --> 00:01:18.600
is gonna be equal to two
pi times the square root

00:01:18.740 --> 00:01:21.870
of the ratio of the
length of the pendulum, L,

00:01:21.870 --> 00:01:25.070
so the length of that
string here, the length L,

00:01:25.070 --> 00:01:28.145
divided by g, the
gravitational acceleration

00:01:28.145 --> 00:01:31.530
of the planet that the
pendulum is being used on.

00:01:31.530 --> 00:01:32.510
Now, if you look at this

00:01:32.510 --> 00:01:33.830
and you've been paying
attention in physics,

00:01:33.830 --> 00:01:35.800
you might be like, "Wait,
that looks really similar

00:01:35.800 --> 00:01:38.620
to the formula for the period
of a mass on a spring."

00:01:38.620 --> 00:01:40.199
So, if you are to take a mass on a spring,

00:01:40.199 --> 00:01:42.130
displace it 15 centimeters,

00:01:42.130 --> 00:01:44.098
you'd get a similar graph

00:01:44.098 --> 00:01:47.880
and this would also have a
characteristic period of motion.

00:01:47.880 --> 00:01:51.020
And if you wrote down the
period for a mass on a spring,

00:01:51.020 --> 00:01:53.590
it looks like this, its
period for a mass on a spring

00:01:53.590 --> 00:01:56.470
is also two pi, so that's identical.

00:01:56.470 --> 00:01:59.120
And then, it's also the
square root of a ratio,

00:01:59.120 --> 00:02:02.294
but instead of L over g
for the mass on a spring,

00:02:02.294 --> 00:02:06.340
it's m, the mass of the block
connected to the spring,

00:02:06.340 --> 00:02:10.690
divided by k, the spring
constant of the spring.

00:02:10.690 --> 00:02:13.021
So, one obvious similarity
between these two formulas

00:02:13.021 --> 00:02:14.630
is just their format.

00:02:14.630 --> 00:02:18.110
They're both a two pi times
a square root of a ratio,

00:02:18.110 --> 00:02:20.810
but another important
similarity between these

00:02:20.810 --> 00:02:22.140
that might not be evident

00:02:22.140 --> 00:02:23.840
is that neither of these formulas

00:02:23.840 --> 00:02:26.090
depend on the amplitude of the motion.

00:02:26.090 --> 00:02:28.090
So, the period of a pendulum

00:02:28.090 --> 00:02:30.360
does not depend on the
amplitude of the motion

00:02:30.360 --> 00:02:32.000
and the period of a mass on a spring

00:02:32.000 --> 00:02:35.020
also does not depend on the
amplitude of the motion.

00:02:35.020 --> 00:02:35.853
What I mean by that

00:02:35.853 --> 00:02:37.850
is if you are to pull this pendulum back,

00:02:37.850 --> 00:02:38.920
instead of 15 degrees,

00:02:38.920 --> 00:02:41.370
pull it back 20 degrees and let go,

00:02:41.370 --> 00:02:43.100
it would have farther to swing.

00:02:43.100 --> 00:02:45.350
So, its motion might
look something like this,

00:02:45.350 --> 00:02:47.830
and we get down farther
and it would get back up,

00:02:47.830 --> 00:02:50.120
but it would take the
exact same amount of time.

00:02:50.120 --> 00:02:51.790
The period would not change

00:02:51.790 --> 00:02:54.320
if you pull this amplitude back farther.

00:02:54.320 --> 00:02:56.490
Same goes with this mass on a spring.

00:02:56.490 --> 00:02:58.300
Instead of pulling it 15 centimeters,

00:02:58.300 --> 00:03:00.630
let's say you pulled it
20 centimeters, again,

00:03:00.630 --> 00:03:02.060
this would start up higher.

00:03:02.060 --> 00:03:03.960
It would get down lower.

00:03:03.960 --> 00:03:06.900
But the time it takes to
complete one full cycle

00:03:06.900 --> 00:03:09.730
would not vary as you vary this amplitude.

00:03:09.730 --> 00:03:10.690
And that might seem weird.

00:03:10.690 --> 00:03:12.410
You're like, "Wait, don't these objects

00:03:12.410 --> 00:03:14.210
have farther to go now

00:03:14.210 --> 00:03:15.990
that you've pulled it back
to a larger amplitude?"

00:03:15.990 --> 00:03:18.780
That's true, they'll both
have farther to travel,

00:03:18.780 --> 00:03:20.550
but they'll be going faster now.

00:03:20.550 --> 00:03:24.632
And faster motion over a bigger
distance is gonna offset.

00:03:24.632 --> 00:03:27.499
And the amplitude does
not affect the period

00:03:27.499 --> 00:03:30.760
of the motion of a pendulum
or of a mass on a spring.

00:03:30.760 --> 00:03:33.220
As for the differences, well,

00:03:33.220 --> 00:03:35.370
the denominator here
for the mass on a spring

00:03:35.370 --> 00:03:38.690
depends on k, and that's
the spring constant k,

00:03:38.690 --> 00:03:39.710
and that should make sense.

00:03:39.710 --> 00:03:43.150
More spring constant means
you get more restoring force,

00:03:43.150 --> 00:03:45.400
the spring force is the
restoring force here.

00:03:45.400 --> 00:03:47.170
More restoring force means this mass

00:03:47.170 --> 00:03:48.800
is gonna be moving faster.

00:03:48.800 --> 00:03:51.280
That means it's gonna take less time

00:03:51.280 --> 00:03:52.940
to go through a full period.

00:03:52.940 --> 00:03:56.000
So, dividing by a bigger
number, bigger spring k,

00:03:56.000 --> 00:03:57.520
gives you less period.

00:03:57.520 --> 00:03:59.660
That's also true here but it's not k.

00:03:59.660 --> 00:04:02.270
The force that's the
restoring force for a pendulum

00:04:02.270 --> 00:04:04.980
isn't a spring, it's the force of gravity.

00:04:04.980 --> 00:04:07.220
So, mg depends on the g.

00:04:07.220 --> 00:04:10.810
So, a bigger g would give
you a bigger restoring force

00:04:10.810 --> 00:04:12.310
over here for the pendulum.

00:04:12.310 --> 00:04:14.700
That means the pendulum
could be moving faster.

00:04:14.700 --> 00:04:17.400
Moving faster means it's
gonna take less time

00:04:17.400 --> 00:04:18.920
to complete a full period.

00:04:18.920 --> 00:04:21.250
So, dividing by a bigger g,
if you took this pendulum

00:04:21.250 --> 00:04:23.570
to the surface of Jupiter or something,

00:04:23.570 --> 00:04:25.070
where the g is bigger,

00:04:25.070 --> 00:04:27.480
it would swing back and forth faster.

00:04:27.480 --> 00:04:29.560
It would take less time to complete it

00:04:29.560 --> 00:04:31.420
because that restoring force is bigger.

00:04:31.420 --> 00:04:33.890
So, even though these denominators
are different letters,

00:04:33.890 --> 00:04:35.760
they're arriving from the same source.

00:04:35.760 --> 00:04:38.880
They're both arriving
because the restoring force

00:04:38.880 --> 00:04:41.520
is larger when you increase
these denominators,

00:04:41.520 --> 00:04:43.460
which increases the speed of the object.

00:04:43.460 --> 00:04:45.320
Now, maybe the biggest difference here

00:04:45.320 --> 00:04:47.610
is that the numerator here

00:04:47.610 --> 00:04:49.620
for the mass on a spring depends on mass,

00:04:49.620 --> 00:04:51.730
but nowhere is mass to be found

00:04:51.730 --> 00:04:53.010
in this period of a pendulum.

00:04:53.010 --> 00:04:56.020
The period of the pendulum
does not depend on the mass.

00:04:56.020 --> 00:04:57.480
Now, this is kind of interesting.

00:04:57.480 --> 00:05:00.520
So, you know, really big,
heavy person gets on a swing,

00:05:00.520 --> 00:05:01.730
swings back and forth,

00:05:01.730 --> 00:05:04.240
very light child gets on the same swing,

00:05:04.240 --> 00:05:06.240
they will take the same amount of time

00:05:06.240 --> 00:05:07.730
to complete a full cycle.

00:05:07.730 --> 00:05:09.800
Their mass does not factor in here

00:05:09.800 --> 00:05:10.790
to the period of pendulum,

00:05:10.790 --> 00:05:13.840
but it does for the mass
on a spring, why is that?

00:05:13.840 --> 00:05:16.440
Well, bigger mass on a spring

00:05:16.440 --> 00:05:18.770
gives you more inertia in the system.

00:05:18.770 --> 00:05:20.430
If you have more inertia in the system,

00:05:20.430 --> 00:05:22.130
it's more sluggish to movement.

00:05:22.130 --> 00:05:23.510
It's gonna go slower.

00:05:23.510 --> 00:05:25.460
That means it's gonna take more time

00:05:25.460 --> 00:05:27.210
to complete a full cycle.

00:05:27.210 --> 00:05:28.490
Now, you might think, "Wait a minute.

00:05:28.490 --> 00:05:29.660
don't that hold true up here?

00:05:29.660 --> 00:05:33.470
Look, if we have a
bigger mass pendulum bob,

00:05:33.470 --> 00:05:35.982
that should increase
the rotational inertia.

00:05:35.982 --> 00:05:37.970
And so, that should make this take longer.

00:05:37.970 --> 00:05:39.560
It should be more sluggish to movement.

00:05:39.560 --> 00:05:41.680
It should take longer to
go through a full period."

00:05:41.680 --> 00:05:45.370
But look at the restoring
force is also, for a pendulum,

00:05:45.370 --> 00:05:46.970
proportional to mass.

00:05:46.970 --> 00:05:48.510
So, if you increase the
mass of this pendulum,

00:05:48.510 --> 00:05:49.810
you do get more inertia,

00:05:49.810 --> 00:05:51.370
but you're getting more restoring force

00:05:51.370 --> 00:05:53.120
because the restoring force is gravity.

00:05:53.120 --> 00:05:54.084
Those completely offset

00:05:54.084 --> 00:05:57.930
mass does not end up showing
up into this pendulum formula,

00:05:57.930 --> 00:05:59.730
even though it does down here.

00:05:59.730 --> 00:06:02.140
So, the spring force is
not proportional to mass.

00:06:02.140 --> 00:06:03.510
Spring force is kx.

00:06:03.510 --> 00:06:06.487
Increasing the mass of
this block on the end

00:06:06.487 --> 00:06:09.140
does not increase the
spring restoring force.

00:06:09.140 --> 00:06:11.020
So, this mass stays in the numerator here,

00:06:11.020 --> 00:06:13.640
but it does not affect
the period of a pendulum.

00:06:13.640 --> 00:06:15.590
So, why does this L show up then?

00:06:15.590 --> 00:06:18.210
Why is it length of the
pendulum in the numerator?

00:06:18.210 --> 00:06:20.880
Well, the rotational
inertia does get increased

00:06:20.880 --> 00:06:23.280
when you increase the
length of the pendulum.

00:06:23.280 --> 00:06:24.210
But increasing that length

00:06:24.210 --> 00:06:26.550
that does not increase
the force of gravity.

00:06:26.550 --> 00:06:27.940
If you wanna get technical,

00:06:27.940 --> 00:06:31.270
rotational inertia is
proportional to length squared,

00:06:31.270 --> 00:06:33.840
but the torque would only
be proportional to length.

00:06:33.840 --> 00:06:36.040
That's why only one L shows up here.

00:06:36.040 --> 00:06:39.920
Long story short, if you increase
the length of a pendulum,

00:06:39.920 --> 00:06:42.150
you're gonna increase the
inertia of that pendulum.

00:06:42.150 --> 00:06:43.380
That's gonna make it take longer

00:06:43.380 --> 00:06:44.730
to go through a full cycle.

00:06:44.730 --> 00:06:45.900
This is while I go into the park

00:06:45.900 --> 00:06:47.650
and finding the long swings.

00:06:47.650 --> 00:06:50.410
The longer the swing, the
longer it actually takes,

00:06:50.410 --> 00:06:52.670
the more time it actually
takes to swing back and forth.

00:06:52.670 --> 00:06:55.480
I think that's more fun
than the little short swings

00:06:55.480 --> 00:06:57.380
that go back and forth really quick.

00:06:57.380 --> 00:06:58.670
So, let's try a sample problem

00:06:58.670 --> 00:07:01.160
to see how this period formula works.

00:07:01.160 --> 00:07:02.660
Let's say you went to the park.

00:07:02.660 --> 00:07:03.900
You're 60 kilograms.

00:07:03.900 --> 00:07:04.990
You're swinging on a swing

00:07:04.990 --> 00:07:07.880
and your friend pulls you back 20 degrees

00:07:07.880 --> 00:07:10.370
on a swing that's one meter in length.

00:07:10.370 --> 00:07:12.370
Let's find the period of the motion.

00:07:12.370 --> 00:07:13.680
So, in other words, the time it takes

00:07:13.680 --> 00:07:15.150
to go all the way to here

00:07:15.150 --> 00:07:16.720
and then all the way back to there.

00:07:16.720 --> 00:07:19.710
We use the period formula for a pendulum.

00:07:19.710 --> 00:07:23.600
It's two pi, root L over g.

00:07:23.600 --> 00:07:27.670
And so, we would do two
pi times the square root,

00:07:27.670 --> 00:07:29.860
the length here is the
length of the string here.

00:07:29.860 --> 00:07:31.800
So, one meter, technically,

00:07:31.800 --> 00:07:34.040
it'd be to wherever
your center of mass is,

00:07:34.040 --> 00:07:35.820
but we're gonna assume our center mass

00:07:35.820 --> 00:07:37.250
is right here at the end,

00:07:37.250 --> 00:07:40.260
divided by g, well, we're
on earth, I'm assuming,

00:07:40.260 --> 00:07:43.420
so, 9.8 meters per second squared.

00:07:43.420 --> 00:07:44.359
If you solve all of that,

00:07:44.359 --> 00:07:48.070
you get a period of about
two seconds exactly.

00:07:48.070 --> 00:07:51.490
So, this swing would have a
period of about two seconds.

00:07:51.490 --> 00:07:54.480
Now, notice we did not
use this 20 degrees.

00:07:54.480 --> 00:07:55.670
That's the amplitude.

00:07:55.670 --> 00:07:58.020
This period does not
depend on the amplitude.

00:07:58.020 --> 00:08:01.200
We also did not use the fact
that I was 60 kilograms.

00:08:01.200 --> 00:08:03.410
The period of a pendulum
also does not depend

00:08:03.410 --> 00:08:06.250
on the mass of the bob at
the end of the pendulum.

00:08:06.250 --> 00:08:08.640
We only use the length
and if you're on earth,

00:08:08.640 --> 00:08:11.260
the denominator here
is always gonna be 9.8.

00:08:11.260 --> 00:08:14.310
Now, one thing I should be
clear about is that a pendulum

00:08:14.310 --> 00:08:17.100
is technically not a perfect
simple harmonic oscillator.

00:08:17.100 --> 00:08:19.760
It's only approximately a
simple harmonic oscillator.

00:08:19.760 --> 00:08:22.780
So, this formula here is
only approximately correct,

00:08:22.780 --> 00:08:25.400
but for small angles, it's almost perfect.

00:08:25.400 --> 00:08:26.560
So, at 20 degrees,

00:08:26.560 --> 00:08:28.610
this formula is only gonna be off.

00:08:28.610 --> 00:08:31.066
The error's only going to be about 1%,

00:08:31.066 --> 00:08:32.230
which isn't bad.

00:08:32.230 --> 00:08:34.420
If you get this back to like 70 degrees,

00:08:34.420 --> 00:08:37.920
even then, the error's only around 10%.

00:08:37.920 --> 00:08:41.190
So, this is a really,
really good approximation

00:08:41.190 --> 00:08:44.710
if you're in small angles,
like around 20 to 30 degrees.

00:08:44.710 --> 00:08:46.340
The bigger the angle gets though,

00:08:46.340 --> 00:08:47.960
the worst the approximation gets.

00:08:47.960 --> 00:08:51.310
For small angles, most
physicists just treat a pendulum

00:08:51.310 --> 00:08:53.656
as if it's a perfect
simple harmonic oscillator.

00:08:53.656 --> 00:08:55.560
As you get to those higher angles though,

00:08:55.560 --> 00:08:56.700
you gotta be careful.

00:08:56.700 --> 00:08:59.470
This can start deviating
more significantly

00:08:59.470 --> 00:09:00.870
from the actual value.

00:09:00.870 --> 00:09:02.860
So, to recap, the period of a pendulum

00:09:02.860 --> 00:09:04.930
depends on the length of the pendulum

00:09:04.930 --> 00:09:07.650
and the surface gravity of
the planet that you're on.

00:09:07.650 --> 00:09:10.110
It does not depend on the amplitude

00:09:10.110 --> 00:09:12.040
or the mass of the pendulum.

00:09:12.040 --> 00:09:13.940
And the form it takes is very similar

00:09:13.940 --> 00:09:16.070
to the period of a mass on a spring

00:09:16.070 --> 00:09:17.890
where the numerator increases

00:09:17.890 --> 00:09:19.530
due to increased inertia

00:09:19.530 --> 00:09:21.680
and the denominator increases

00:09:21.680 --> 00:09:24.253
due to increased restoring force.


Dihybrid cross and the Law of Independent Assortment

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.070
- [Instructor] In this
video, we're going to build

00:00:02.070 --> 00:00:03.977
on our understanding of Mendelian genetics

00:00:03.977 --> 00:00:05.390
and Punnett squares

00:00:05.390 --> 00:00:08.340
by starting to think
about two different genes.

00:00:08.340 --> 00:00:09.820
So we're going back to the pea plant,

00:00:09.820 --> 00:00:12.070
and we're gonna think about
the gene for pea color

00:00:12.070 --> 00:00:15.090
and the gene for pea shape.

00:00:15.090 --> 00:00:18.430
So let's say that in
the parental generation,

00:00:18.430 --> 00:00:21.640
you have one parent that
is homozygous dominant

00:00:21.640 --> 00:00:24.210
for both of these genes.

00:00:24.210 --> 00:00:28.500
So their genotype is capital Y, capital Y,

00:00:28.500 --> 00:00:32.260
and capital R, capital R.

00:00:32.260 --> 00:00:35.640
So their phenotype for sure
is going to be yellow round,

00:00:35.640 --> 00:00:36.840
and we also see the genotype.

00:00:36.840 --> 00:00:39.680
We see which alleles it has.

00:00:39.680 --> 00:00:41.270
Now, let's say that that is crossed

00:00:41.270 --> 00:00:44.100
with the homozygous recessive parent.

00:00:44.100 --> 00:00:47.710
So in this case, it's going
to be green-colored peas.

00:00:47.710 --> 00:00:49.950
It's counterintuitive
to write green with a y,

00:00:49.950 --> 00:00:52.350
a lowercase y in the color yellow,

00:00:52.350 --> 00:00:54.730
but the lowercase y represents green.

00:00:54.730 --> 00:00:57.930
And also these are wrinkled green peas.

00:00:57.930 --> 00:01:01.410
So I will write that with
the lowercase r here.

00:01:01.410 --> 00:01:03.370
And so the phenotype here is going

00:01:03.370 --> 00:01:06.900
to be green and wrinkled for sure.

00:01:06.900 --> 00:01:09.320
Now, what's going to
happen when they cross?

00:01:09.320 --> 00:01:12.110
What does the F1 generation look like?

00:01:12.110 --> 00:01:15.460
Well, we know from
Mendel's law of segregation

00:01:15.460 --> 00:01:19.180
for each of these genes, that
when a gamete is created,

00:01:19.180 --> 00:01:23.290
it randomly gets one copy
of each of these genes.

00:01:23.290 --> 00:01:24.123
So for this first one,

00:01:24.123 --> 00:01:27.150
it's going to randomly get
one of these capital Ys.

00:01:27.150 --> 00:01:29.430
So it's going to get a capital Y for sure

00:01:29.430 --> 00:01:30.620
from this first parent.

00:01:30.620 --> 00:01:34.460
And it's also going to randomly
get one of these capital Rs.

00:01:34.460 --> 00:01:36.860
So it's going to get a capital R for sure

00:01:36.860 --> 00:01:38.370
from that first parent.

00:01:38.370 --> 00:01:40.370
And then by the same logic,

00:01:40.370 --> 00:01:43.340
it's going to randomly get one
of these two lowercase y's.

00:01:43.340 --> 00:01:46.390
So it's going to get a
lowercase y for sure,

00:01:46.390 --> 00:01:47.900
and it's going to randomly get one

00:01:47.900 --> 00:01:49.650
of these two lowercase r's,

00:01:49.650 --> 00:01:52.470
so it's going to get a
lowercase r for sure.

00:01:52.470 --> 00:01:57.060
So this is the genotype for
all of the F1 generation.

00:01:57.060 --> 00:01:58.893
This is often known as a dihybrid.

00:01:59.976 --> 00:02:03.120
It is heterozygous in both genes.

00:02:03.120 --> 00:02:04.830
Now, what's the phenotype here?

00:02:04.830 --> 00:02:06.450
Well, we know yellow is dominant,

00:02:06.450 --> 00:02:08.050
and we know round is dominant.

00:02:08.050 --> 00:02:10.210
So if we looked at these
plants right over here,

00:02:10.210 --> 00:02:12.430
their peas would still be yellow round,

00:02:12.430 --> 00:02:15.380
just like this homozygous
parent over here.

00:02:15.380 --> 00:02:17.080
Now, what's interesting is when you do

00:02:17.080 --> 00:02:19.760
what's known as a dihybrid cross

00:02:19.760 --> 00:02:22.890
when you cross one of this F1 generation

00:02:22.890 --> 00:02:25.130
with itself or with each other.

00:02:25.130 --> 00:02:26.020
And to do that,

00:02:26.020 --> 00:02:30.440
I'm gonna create a four by
four Punnett square here.

00:02:30.440 --> 00:02:34.210
And so one parent here
is going to be hybrid in

00:02:34.210 --> 00:02:37.540
or heterozygous in the color gene

00:02:37.540 --> 00:02:40.470
and also heterozygous in the shape gene.

00:02:40.470 --> 00:02:42.900
And that's going to be true
of the other parent as well.

00:02:42.900 --> 00:02:46.380
Heterozygous or hybrid in the color gene

00:02:46.380 --> 00:02:49.890
and also heterozygous in the shape gene.

00:02:49.890 --> 00:02:52.160
And so that's why this is
called a dihybrid cross.

00:02:52.160 --> 00:02:53.300
You're crossing things

00:02:53.300 --> 00:02:56.330
that are hybrid in two different genes.

00:02:56.330 --> 00:02:59.300
Now, we've already talked
about the law of segregation.

00:02:59.300 --> 00:03:03.320
The gamete is randomly going
to get one copy of each gene.

00:03:03.320 --> 00:03:07.190
Now, Mendel also has the law
of independent assortment,

00:03:07.190 --> 00:03:08.580
which tells us the alleles

00:03:08.580 --> 00:03:12.100
of different genes
segregate independently.

00:03:12.100 --> 00:03:14.020
So for this parent here,

00:03:14.020 --> 00:03:17.230
whether it contributes a
capital Y or a lowercase y

00:03:17.230 --> 00:03:19.910
is independent of whether
it contribute a capital R

00:03:19.910 --> 00:03:21.330
or a lowercase r.

00:03:21.330 --> 00:03:22.920
Now, there is a little bit of a asterisk,

00:03:22.920 --> 00:03:24.490
a little caveat on there.

00:03:24.490 --> 00:03:27.020
We now know that genes sit on chromosomes.

00:03:27.020 --> 00:03:28.810
One chromosome will have many genes on it.

00:03:28.810 --> 00:03:31.630
And this law of independent
assortment only applies

00:03:31.630 --> 00:03:34.690
to genes that are actually
sitting on different chromosomes.

00:03:34.690 --> 00:03:36.710
If they sit on the same chromosome,

00:03:36.710 --> 00:03:40.230
they generally are not going
to assort independently.

00:03:40.230 --> 00:03:43.040
But let's just assume the
law of independent assortment

00:03:43.040 --> 00:03:45.380
'cause this is true for most genes.

00:03:45.380 --> 00:03:49.920
So this first parent can
contribute a capital Y

00:03:49.920 --> 00:03:51.530
out of this first gene

00:03:51.530 --> 00:03:54.900
and a capital R out of the second gene,

00:03:54.900 --> 00:03:57.380
or they could contribute
the lowercase copy

00:03:57.380 --> 00:03:58.600
of the first gene

00:03:58.600 --> 00:04:02.780
and the capital R copy of the second gene,

00:04:02.780 --> 00:04:06.000
this capital R, the round
allele of the second gene.

00:04:06.000 --> 00:04:08.210
And we could go through
every combination here.

00:04:08.210 --> 00:04:12.580
It could also contribute the yellow allele

00:04:12.580 --> 00:04:14.363
and the wrinkled allele.

00:04:15.490 --> 00:04:20.290
Or it could contribute the green allele

00:04:20.290 --> 00:04:22.543
and the wrinkled allele as well.

00:04:23.540 --> 00:04:26.000
And the same would be true
for this other dihybrid,

00:04:26.000 --> 00:04:28.130
this other parent right over here.

00:04:28.130 --> 00:04:29.510
So let me just write that down.

00:04:29.510 --> 00:04:33.440
They could contribute capital
Y in two of the scenarios.

00:04:33.440 --> 00:04:35.270
They could contribute a lowercase y

00:04:35.270 --> 00:04:37.590
or the green allele in
two of the scenarios.

00:04:37.590 --> 00:04:39.920
And they could contribute a capital R

00:04:39.920 --> 00:04:42.360
in two of the scenarios, a round allele,

00:04:42.360 --> 00:04:44.700
or a lowercase r in two of the scenarios,

00:04:44.700 --> 00:04:46.100
a wrinkled allele.

00:04:46.100 --> 00:04:48.050
So you have all of the
different combinations

00:04:48.050 --> 00:04:49.690
that each of them can contribute.

00:04:49.690 --> 00:04:51.730
Once again, whether you get the yellow

00:04:51.730 --> 00:04:53.110
or the green is independent

00:04:53.110 --> 00:04:55.260
of whether you get the
round or the wrinkled.

00:04:55.260 --> 00:04:58.083
So these are all equally
probably right over here.

00:04:59.100 --> 00:05:01.270
When the two gametes from
these two parents merge,

00:05:01.270 --> 00:05:03.000
we can then look at what the genotype

00:05:03.000 --> 00:05:04.350
of the offspring is going to be,

00:05:04.350 --> 00:05:06.900
really the genotype of the F2 generation

00:05:06.900 --> 00:05:10.470
'cause we're crossing two
members of the F1 generation.

00:05:10.470 --> 00:05:12.210
So I encourage you to pause this video

00:05:12.210 --> 00:05:13.720
and fill in this grid.

00:05:13.720 --> 00:05:15.820
See if you can figure
the different genotypes

00:05:15.820 --> 00:05:16.803
that will result.

00:05:18.140 --> 00:05:20.270
All right, now let's do this together.

00:05:20.270 --> 00:05:21.710
So this scenario right over here,

00:05:21.710 --> 00:05:25.060
you're getting a capital
Y from both parents,

00:05:25.060 --> 00:05:29.810
and you're getting a
capital R from both parents.

00:05:29.810 --> 00:05:30.730
This scenario over here,

00:05:30.730 --> 00:05:33.330
you're getting a capital
Y from this parent,

00:05:33.330 --> 00:05:35.840
lowercase y from that parent,

00:05:35.840 --> 00:05:38.663
and then you're getting
a capital R from both.

00:05:39.650 --> 00:05:43.940
This scenario over here,
capital Y from both parents.

00:05:43.940 --> 00:05:45.500
Capital Y, capital Y.

00:05:45.500 --> 00:05:49.940
And you're getting a
capital R from this parent

00:05:49.940 --> 00:05:52.480
and a lowercase r from that parent.

00:05:52.480 --> 00:05:54.460
And then this scenario over here,

00:05:54.460 --> 00:05:57.550
you're getting gonna capital
Y allele from this parent

00:05:57.550 --> 00:05:59.640
and lowercase y from that parent,

00:05:59.640 --> 00:06:03.130
and you're getting a
capital R from this parent

00:06:03.130 --> 00:06:05.630
and a lowercase r from that parent.

00:06:05.630 --> 00:06:07.810
And now I'm just going
to speed up the video

00:06:07.810 --> 00:06:10.923
and just fill in the rest of
these using the same logic.

00:06:14.440 --> 00:06:17.080
All right, now that we've
filled out this Punnett square,

00:06:17.080 --> 00:06:19.470
let's think about the
different phenotypes.

00:06:19.470 --> 00:06:20.530
How many of these plants are going

00:06:20.530 --> 00:06:22.560
to produce yellow round peas?

00:06:22.560 --> 00:06:24.310
Pause the video and think about it.

00:06:25.320 --> 00:06:27.310
Well, it's yellow and round.

00:06:27.310 --> 00:06:31.260
It has to have at least one
capital Y and one capital R.

00:06:31.260 --> 00:06:33.620
So that one's going to
be yellow and round.

00:06:33.620 --> 00:06:35.560
This is going to be yellow and round.

00:06:35.560 --> 00:06:37.970
This is going to be yellow and round.

00:06:37.970 --> 00:06:40.270
That's yellow and round as well.

00:06:40.270 --> 00:06:42.500
This is yellow and round.

00:06:42.500 --> 00:06:44.500
That's yellow and round.

00:06:44.500 --> 00:06:46.610
This is yellow and round.

00:06:46.610 --> 00:06:49.560
And that one is yellow and round.

00:06:49.560 --> 00:06:53.040
And then last but not least,
I think this is the last one

00:06:53.040 --> 00:06:55.430
that is both yellow and round.

00:06:55.430 --> 00:06:57.090
And actually let me make
a little color code here.

00:06:57.090 --> 00:07:00.240
Yellow plus round.

00:07:00.240 --> 00:07:01.850
And here we're talking
about the phenotype.

00:07:01.850 --> 00:07:03.840
You can see we have
different genotypes here,

00:07:03.840 --> 00:07:06.730
but because both yellow
and round are dominant,

00:07:06.730 --> 00:07:09.170
as long as you have at
least one Y and one R,

00:07:09.170 --> 00:07:10.340
you're going to have a yellow

00:07:10.340 --> 00:07:12.330
plus round phenotype over here.

00:07:12.330 --> 00:07:14.300
So you have one, two, three, four,

00:07:14.300 --> 00:07:16.540
five, six, seven, eight, nine.

00:07:16.540 --> 00:07:19.580
And I will say there's
nine of these over here.

00:07:19.580 --> 00:07:24.580
Now, how many of these are going
to be yellow plus wrinkled?

00:07:25.490 --> 00:07:28.820
Pause the video and think
about that, that phenotype.

00:07:28.820 --> 00:07:30.000
So yellow and wrinkled,

00:07:30.000 --> 00:07:32.300
you're going to have to have a capital Y

00:07:32.300 --> 00:07:35.610
and two lowercase r's
in order to be wrinkled.

00:07:35.610 --> 00:07:39.310
So you have at least one
capital Y and two lowercase r's,

00:07:39.310 --> 00:07:42.860
and least one capital Y
and two lowercase r's.

00:07:42.860 --> 00:07:47.170
Let's see, at least one capital
Y and two lowercase r's.

00:07:47.170 --> 00:07:50.390
It looks like you have
exactly three of them,

00:07:50.390 --> 00:07:51.550
that phenotype.

00:07:51.550 --> 00:07:54.390
And then what about the other way around?

00:07:54.390 --> 00:07:58.410
What if we are looking for,
I'll do it in this green color,

00:07:58.410 --> 00:08:03.410
green plus round?

00:08:03.840 --> 00:08:06.180
How many of them exhibit that phenotype?

00:08:06.180 --> 00:08:07.450
Well, to be green and round,

00:08:07.450 --> 00:08:09.590
you have to have two lowercase y's,

00:08:09.590 --> 00:08:12.370
and you have to have
at least one capital R.

00:08:12.370 --> 00:08:15.950
So this would be green and round.

00:08:15.950 --> 00:08:18.090
This would be green and round.

00:08:18.090 --> 00:08:20.270
And then this would be
green and round as well.

00:08:20.270 --> 00:08:21.800
So you have three of those.

00:08:21.800 --> 00:08:23.300
And then how many of them are going

00:08:23.300 --> 00:08:25.430
to be both green and wrinkled?

00:08:25.430 --> 00:08:28.290
Well, I think you see that
one scenario over here

00:08:28.290 --> 00:08:30.020
that is both green and wrinkled,

00:08:30.020 --> 00:08:33.450
having that homozygous
recessive phenotype.

00:08:33.450 --> 00:08:35.290
And so if you were to do this many times,

00:08:35.290 --> 00:08:37.870
you'd expect the ratios between
these various phenotypes

00:08:37.870 --> 00:08:41.400
to be nine to three to three to one.

00:08:41.400 --> 00:08:43.000
And when Mendel and many other people

00:08:43.000 --> 00:08:45.180
since Mendel have done
these types of experiments,

00:08:45.180 --> 00:08:47.060
they have seen that statistically,

00:08:47.060 --> 00:08:49.870
this is what you see
in that F2 generation.

00:08:49.870 --> 00:08:51.820
Now, you're unlikely to get exactly a nine

00:08:51.820 --> 00:08:53.590
to three to three to one ratio.

00:08:53.590 --> 00:08:54.950
It's all probabilistic.

00:08:54.950 --> 00:08:58.650
Every one of these 16
scenarios are equally likely,

00:08:58.650 --> 00:09:00.530
so you would expect this nine to three

00:09:00.530 --> 00:09:01.950
to three to one ratio,

00:09:01.950 --> 00:09:04.870
but you're not always going
to get that exact ratio.

00:09:04.870 --> 00:09:06.970
You'll probably get something close to it.