AP Physics 1 review of Centripetal Forces

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What does
period and frequency mean?

00:00:02.601 --> 00:00:04.316
The period is the number of seconds

00:00:04.316 --> 00:00:06.111
it takes for a process to complete

00:00:06.111 --> 00:00:09.127
an entire cycle, circle, or revolution.

00:00:09.127 --> 00:00:10.844
So, if there's some repeating process,

00:00:10.844 --> 00:00:14.788
the time it takes that process
to reset is the period,

00:00:14.788 --> 00:00:16.298
and it's measured in seconds.

00:00:16.298 --> 00:00:18.457
The frequency is the number of cycles,

00:00:18.457 --> 00:00:22.012
or circles, or revolutions
completed in one second.

00:00:22.012 --> 00:00:24.117
So, if there's some
process that's repeating,

00:00:24.117 --> 00:00:26.287
the number of times the process repeats

00:00:26.287 --> 00:00:28.906
in one second would be the frequency.

00:00:28.906 --> 00:00:31.596
This means it has units
of one over second,

00:00:31.596 --> 00:00:33.755
which is just called, the hertz.

00:00:33.755 --> 00:00:35.959
And because the period
and frequency are defined

00:00:35.959 --> 00:00:38.677
in this inverse way as seconds per cycle

00:00:38.677 --> 00:00:40.714
or cycles per second, each one

00:00:40.714 --> 00:00:42.474
is just the inverse of the other.

00:00:42.474 --> 00:00:44.354
In other words, the period is

00:00:44.354 --> 00:00:45.929
just one over the frequency, and

00:00:45.929 --> 00:00:48.527
the frequency is equal
to one over the period.

00:00:48.527 --> 00:00:50.540
One example of a repeating process

00:00:50.540 --> 00:00:53.456
is an object going in a
circle at a constant speed.

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If this is the case, you can relate

00:00:55.433 --> 00:00:57.745
the speed, the radius of the circle,

00:00:57.745 --> 00:00:59.837
and the period of the motion since speed

00:00:59.837 --> 00:01:02.730
is just distance per
time, and the distance

00:01:02.730 --> 00:01:06.118
the object travels in
one cycle is two pi R

00:01:06.118 --> 00:01:08.087
the circumference, the speed would

00:01:08.087 --> 00:01:11.146
just be two pi R per the period,

00:01:11.146 --> 00:01:13.603
or since one over the
period is the frequency,

00:01:13.603 --> 00:01:16.933
you could write the speed as
two pi R times the frequency.

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Since time is not a
vector these quantities

00:01:18.858 --> 00:01:21.676
are not vectors and
they cannot be negative.

00:01:21.676 --> 00:01:23.193
So, what's an example involving period

00:01:23.193 --> 00:01:24.619
and frequency look like?

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Let's say moon travels around a planet

00:01:26.428 --> 00:01:30.234
in a circular orbit of radius
R at a constant speed S.

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And we wanna know what the
period and frequency are

00:01:32.349 --> 00:01:34.229
in terms of given quantities and

00:01:34.229 --> 00:01:36.394
fundamental constants, so
we'll use the relationship

00:01:36.394 --> 00:01:38.811
between the speed, the
period, and the frequency.

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We know that for object in circular motion

00:01:40.717 --> 00:01:43.183
the speed is two pi R over the period.

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And that means the period here

00:01:44.506 --> 00:01:47.277
would be equal to two pi R over the speed.

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And since frequency is
one over the period,

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if we take one over this quantity

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we just flip the top and bottom

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and we get that this is
the speed over two pi R.

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But we can't leave our
answer in terms of V.

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We had to express this in
terms of given quantities.

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We were given S, so our
answer for the period

00:02:02.576 --> 00:02:05.689
has to be two pi R over
S, and for frequency

00:02:05.689 --> 00:02:08.957
it would be S over two pi R, which is C.

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What is centripetal acceleration?

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The centripetal acceleration of an object

00:02:12.493 --> 00:02:14.446
is the acceleration that's causing

00:02:14.446 --> 00:02:16.499
that object to go in a circle.

00:02:16.499 --> 00:02:17.749
And it's important to note that

00:02:17.749 --> 00:02:20.015
this centripetal
acceleration always points

00:02:20.015 --> 00:02:22.014
toward the center of the circle.

00:02:22.014 --> 00:02:24.305
The formula to find the
centripetal acceleration

00:02:24.305 --> 00:02:27.153
is speed squared divided by the radius

00:02:27.153 --> 00:02:29.914
of the circle the object is travelling in.

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Even though this has a
bit of an exotic formula

00:02:32.381 --> 00:02:35.091
for the acceleration, it's
still an acceleration,

00:02:35.091 --> 00:02:37.870
so it still has units of
meters per second squared,

00:02:37.870 --> 00:02:39.793
and it is a vector, which means it does

00:02:39.793 --> 00:02:43.091
have a direction, i.e toward
the center of the circle.

00:02:43.091 --> 00:02:44.556
But this centripetal acceleration

00:02:44.556 --> 00:02:47.957
does not cause the object
to speed up or slow down.

00:02:47.957 --> 00:02:50.424
This centripetal
acceleration is only changing

00:02:50.424 --> 00:02:52.402
the direction of the velocity.

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If the object going in the circle

00:02:54.533 --> 00:02:57.082
is also speeding up or slowing down,

00:02:57.082 --> 00:02:58.331
there's also gotta be a component

00:02:58.331 --> 00:03:01.221
of the acceleration that's
tangential to the circle,

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in other words, if the object

00:03:02.426 --> 00:03:04.189
is going in a circle and speeding up,

00:03:04.189 --> 00:03:05.766
there's gotta be a
component of acceleration

00:03:05.766 --> 00:03:07.798
in direction of the velocity,

00:03:07.798 --> 00:03:09.225
and if the object is slowing down,

00:03:09.225 --> 00:03:10.913
there's gotta be a
component of acceleration

00:03:10.913 --> 00:03:13.343
in the opposite direction to the velocity.

00:03:13.343 --> 00:03:15.618
So, centripetal acceleration
changes the direction

00:03:15.618 --> 00:03:18.488
of the velocity, and
tangential acceleration changes

00:03:18.488 --> 00:03:21.451
the magnitude or size of the velocity.

00:03:21.451 --> 00:03:23.805
But this formula of V squared over R

00:03:23.805 --> 00:03:25.283
is only giving you the magnitude

00:03:25.283 --> 00:03:27.254
of the centripetal acceleration.

00:03:27.254 --> 00:03:30.436
This does not account for
any tangential acceleration.

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So, what's an example problem

00:03:31.569 --> 00:03:34.129
involving centripetal
acceleration look like?

00:03:34.129 --> 00:03:36.271
Let's say particle A is
travelling in a circle

00:03:36.271 --> 00:03:38.816
with a constant speed S and a radius R.

00:03:38.816 --> 00:03:40.676
If particle B is travelling in a circle

00:03:40.676 --> 00:03:44.154
with twice the speed of A
and twice the radius of A,

00:03:44.154 --> 00:03:46.086
what's the ratio of the acceleration

00:03:46.086 --> 00:03:48.458
of particle A compared to particle B.

00:03:48.458 --> 00:03:51.137
So, particle A is gonna have
a centripetal acceleration

00:03:51.137 --> 00:03:53.683
of the speed squared over the radius,

00:03:53.683 --> 00:03:55.761
and particle B is also
gonna have an acceleration

00:03:55.761 --> 00:03:57.182
of the speed squared, but this speed

00:03:57.182 --> 00:04:00.216
is twice as much of the
speed of particle A,

00:04:00.216 --> 00:04:01.639
and is travelling in a circle

00:04:01.639 --> 00:04:03.716
with twice the radius of particle A.

00:04:03.716 --> 00:04:06.012
When we square the two
we'll get four over two,

00:04:06.012 --> 00:04:08.379
gives us a factor of two times the speed

00:04:08.379 --> 00:04:11.042
of A squared over the radius of A.

00:04:11.042 --> 00:04:13.212
So, the ratio of the
acceleration of particle A

00:04:13.212 --> 00:04:15.857
compared to particle
B is gonna be one half

00:04:15.857 --> 00:04:17.846
since the acceleration of particle A

00:04:17.846 --> 00:04:20.485
is half the acceleration of particle B.

00:04:20.485 --> 00:04:23.606
Centripetal forces are
not a new type of force,

00:04:23.606 --> 00:04:25.107
centripetal forces are just one

00:04:25.107 --> 00:04:27.387
of the any other forces
that we've already met

00:04:27.387 --> 00:04:29.618
that happen to be pointing
towards the center

00:04:29.618 --> 00:04:32.511
of the circle making an
object travel in a circle.

00:04:32.511 --> 00:04:34.152
So, for a moon going around the Earth,

00:04:34.152 --> 00:04:35.894
gravity is the centripetal force.

00:04:35.894 --> 00:04:37.932
For a yo-yo going around on a string,

00:04:37.932 --> 00:04:39.893
the tension is the centripetal force.

00:04:39.893 --> 00:04:42.080
For a skateboarder doing a loopty loop,

00:04:42.080 --> 00:04:44.218
the normal force is the centripetal force.

00:04:44.218 --> 00:04:46.148
And for a car going around a roundabout,

00:04:46.148 --> 00:04:49.226
the static frictional force
is the centripetal force.

00:04:49.226 --> 00:04:51.852
And these forces still
follow Newton's second law,

00:04:51.852 --> 00:04:53.729
but using centripetal forces means you're

00:04:53.729 --> 00:04:55.450
also going to have to use the expression

00:04:55.450 --> 00:04:57.185
for the centripetal acceleration.

00:04:57.185 --> 00:04:59.451
Now, if a force is
directed radially inward

00:04:59.451 --> 00:05:00.847
toward the center of the circle,

00:05:00.847 --> 00:05:02.510
you would count that force as positive

00:05:02.510 --> 00:05:04.167
since it points in the same direction

00:05:04.167 --> 00:05:05.741
as the centripetal acceleration.

00:05:05.741 --> 00:05:07.579
And if a force points radially out

00:05:07.579 --> 00:05:08.926
from the center of the circle,

00:05:08.926 --> 00:05:10.601
you would count that as a negative force.

00:05:10.601 --> 00:05:12.867
And if force is directed
tangential to the circle,

00:05:12.867 --> 00:05:15.433
you wouldn't include it in
this calculation at all.

00:05:15.433 --> 00:05:17.225
You can include those forces in their own

00:05:17.225 --> 00:05:19.032
Newton's second law equation,

00:05:19.032 --> 00:05:21.193
but you wouldn't be using V squared over R

00:05:21.193 --> 00:05:22.785
for that acceleration.

00:05:22.785 --> 00:05:26.016
Those tangential forces change
the speed of the object,

00:05:26.016 --> 00:05:27.518
but the centripetal force changes

00:05:27.518 --> 00:05:29.296
the direction of the object.

00:05:29.296 --> 00:05:30.979
So, what's an example problem involving

00:05:30.979 --> 00:05:32.671
centripetal forces look like?

00:05:32.671 --> 00:05:34.866
Imagine a ball of mass M rolling over

00:05:34.866 --> 00:05:37.280
the top of the hill of
radius R at a speed S.

00:05:37.280 --> 00:05:39.723
And we wanna know, at a top of the hill,

00:05:39.723 --> 00:05:41.731
what's the magnitude of the normal force

00:05:41.731 --> 00:05:44.044
exerted on the ball by the road.

00:05:44.044 --> 00:05:45.558
So, we'll draw our force diagram.

00:05:45.558 --> 00:05:47.184
There's gonna be an upward normal force

00:05:47.184 --> 00:05:49.020
on the ball from the road, and there's

00:05:49.020 --> 00:05:50.630
gonna be a downward force of gravity

00:05:50.630 --> 00:05:53.070
on the ball from the
Earth, and these two forces

00:05:53.070 --> 00:05:55.234
are not going to be equal and opposite.

00:05:55.234 --> 00:05:56.383
If they were equal and opposite

00:05:56.383 --> 00:05:58.135
they would balance, and
if the forces are balanced

00:05:58.135 --> 00:06:00.291
the object would maintain its velocity

00:06:00.291 --> 00:06:02.104
and keep travelling in a straight line.

00:06:02.104 --> 00:06:03.875
But this ball doesn't
travel in a straight line,

00:06:03.875 --> 00:06:05.617
it starts accelerating downward.

00:06:05.617 --> 00:06:07.094
So, this normal force
is gonna have to be less

00:06:07.094 --> 00:06:08.847
than the force of gravity.

00:06:08.847 --> 00:06:10.285
To figure out how much less, we can use

00:06:10.285 --> 00:06:12.032
Newton's second law with the formula

00:06:12.032 --> 00:06:13.668
for centripetal acceleration.

00:06:13.668 --> 00:06:15.848
The speed is S, the radius is R,

00:06:15.848 --> 00:06:18.279
the force of gravity
is gonna be a positive

00:06:18.279 --> 00:06:20.374
centripetal force since
it's directed toward

00:06:20.374 --> 00:06:21.803
the center of the circle.

00:06:21.803 --> 00:06:24.116
The normal force is gonna be
a negative centripetal force

00:06:24.116 --> 00:06:25.682
since it's directed radially away from

00:06:25.682 --> 00:06:27.078
the center of the circle.

00:06:27.078 --> 00:06:28.553
Then we divide by the mass, which,

00:06:28.553 --> 00:06:30.138
if you solve this for normal force,

00:06:30.138 --> 00:06:32.097
gives you the force of gravity

00:06:32.097 --> 00:06:35.319
minus M, S squared over
R, which makes sense

00:06:35.319 --> 00:06:37.637
'cause this normal force has to be less

00:06:37.637 --> 00:06:39.253
than the force of gravity.

00:06:39.253 --> 00:06:41.637
Newton's universal law
of gravity states that

00:06:41.637 --> 00:06:44.626
all masses in the
universe pull, i.e attract

00:06:44.626 --> 00:06:46.435
every other mass in the universe

00:06:46.435 --> 00:06:48.207
with gravitational force.

00:06:48.207 --> 00:06:49.588
And this force is proportional

00:06:49.588 --> 00:06:52.113
to each mass, and inversely proportional

00:06:52.113 --> 00:06:54.911
to the square of the
center to center distance

00:06:54.911 --> 00:06:56.593
between the two masses.

00:06:56.593 --> 00:06:58.270
In mathematical form, it just says

00:06:58.270 --> 00:07:00.780
that the force of gravity
is equal to big G,

00:07:00.780 --> 00:07:04.496
a constant, which is 6.67
times 10 to the negative 11th

00:07:04.496 --> 00:07:07.168
multiplied by each mass in kilograms,

00:07:07.168 --> 00:07:10.049
and then divided by the
center to center distance

00:07:10.049 --> 00:07:11.830
between the two masses, in other words,

00:07:11.830 --> 00:07:14.107
not the surface to surface distance,

00:07:14.107 --> 00:07:16.664
but the center to center distance.

00:07:16.664 --> 00:07:18.188
And even if these two
objects have different

00:07:18.188 --> 00:07:19.789
masses, the magnitude of the force

00:07:19.789 --> 00:07:22.330
they exert on each other
is gonna be the same.

00:07:22.330 --> 00:07:24.011
This is illustrated by the formula

00:07:24.011 --> 00:07:25.414
since you could swap these two masses

00:07:25.414 --> 00:07:26.802
and you get the same number.

00:07:26.802 --> 00:07:29.204
And it's also something we
know from Newton's third law.

00:07:29.204 --> 00:07:31.047
This force of gravity is a vector

00:07:31.047 --> 00:07:33.126
and it has a direction, the
direction is always such

00:07:33.126 --> 00:07:34.890
that it attracts every other mass,

00:07:34.890 --> 00:07:37.680
and since this is a force,
the unit is in Newtons.

00:07:37.680 --> 00:07:39.057
So, what's an example problem involving

00:07:39.057 --> 00:07:41.495
Newton's universal law
of gravity look like?

00:07:41.495 --> 00:07:43.403
Let's say two masses, both of mass M,

00:07:43.403 --> 00:07:45.861
exert a gravitational
force F on each other.

00:07:45.861 --> 00:07:49.144
If one of the masses is
exchanged for a mass 3M

00:07:49.144 --> 00:07:50.965
and the center to center distance between

00:07:50.965 --> 00:07:52.637
the masses is tripled, what would

00:07:52.637 --> 00:07:54.616
the new gravitational force be?

00:07:54.616 --> 00:07:55.983
We know the gravitational force

00:07:55.983 --> 00:07:58.918
is always big G times one
of the masses multiplied

00:07:58.918 --> 00:08:01.480
by the other mass divided
by the center to center

00:08:01.480 --> 00:08:02.598
distance squared.

00:08:02.598 --> 00:08:04.470
So, the initial force
between the two masses

00:08:04.470 --> 00:08:07.812
would be big G M times M over R squared,

00:08:07.812 --> 00:08:10.637
but the new force with the
exchanged values would be

00:08:10.637 --> 00:08:13.137
big G times 3M times M divided

00:08:14.046 --> 00:08:16.477
by three times the radius squared.

00:08:16.477 --> 00:08:18.177
The factor of three squared on the bottom

00:08:18.177 --> 00:08:19.879
gives nine, and three divided by nine

00:08:19.879 --> 00:08:23.838
is one over three times
big G M M over R squared.

00:08:23.838 --> 00:08:26.133
So, we can see that the
force with the new values

00:08:26.133 --> 00:08:29.345
1/3 of the force with the old values.

00:08:29.345 --> 00:08:31.268
What's gravitational field mean?

00:08:31.268 --> 00:08:33.189
The gravitational field
is just another word

00:08:33.189 --> 00:08:36.446
for the acceleration due
to gravity near an object.

00:08:36.446 --> 00:08:38.543
You can visualize a gravitational field

00:08:38.543 --> 00:08:42.286
as vectors pointing
radially in toward a mass.

00:08:42.286 --> 00:08:44.534
All masses create a
gravitational field that points

00:08:44.534 --> 00:08:47.193
radially in toward them and dies off like

00:08:47.193 --> 00:08:50.536
one over R squared the farther
you get away from them.

00:08:50.536 --> 00:08:53.301
So, the formula for the
gravitational field little G

00:08:53.301 --> 00:08:56.963
created by a mass M is big G times

00:08:56.963 --> 00:09:00.479
the mass creating the field
divided by the distance

00:09:00.479 --> 00:09:02.779
from the center of the
mass to the point where

00:09:02.779 --> 00:09:05.656
you're trying to determine
the value of the field.

00:09:05.656 --> 00:09:07.914
And again, this value for
the gravitational field

00:09:07.914 --> 00:09:09.805
is going to be equal to the value

00:09:09.805 --> 00:09:11.496
for the acceleration due to gravity

00:09:11.496 --> 00:09:13.845
of an object placed at that point.

00:09:13.845 --> 00:09:15.645
The gravitational field is a vector

00:09:15.645 --> 00:09:18.461
since it has a direction,
i.e. toward the center

00:09:18.461 --> 00:09:19.943
of the object creating it.

00:09:19.943 --> 00:09:21.751
And since gravitational
field is equivalent

00:09:21.751 --> 00:09:23.434
to acceleration due to gravity,

00:09:23.434 --> 00:09:25.981
the units are meters per second squared,

00:09:25.981 --> 00:09:29.104
but you could also write
that as newtons per kilogram,

00:09:29.104 --> 00:09:30.786
which is another way
of thinking about what

00:09:30.786 --> 00:09:32.435
gravitational field means.

00:09:32.435 --> 00:09:34.971
Not only is it the
acceleration due to gravity

00:09:34.971 --> 00:09:36.798
of an object placed at that point,

00:09:36.798 --> 00:09:38.832
but it's the amount of
the gravitational force

00:09:38.832 --> 00:09:42.028
exerted on a mass M placed at that point.

00:09:42.028 --> 00:09:44.123
So, you could think of
the gravitational field

00:09:44.123 --> 00:09:46.429
as measuring the amount
of gravitational force

00:09:46.429 --> 00:09:49.124
per kilogram at a point in space,

00:09:49.124 --> 00:09:51.753
which when rearranged gives
you the familiar formula

00:09:51.753 --> 00:09:54.535
that the force of gravity
is just M times G.

00:09:54.535 --> 00:09:55.956
So, what's an example problem involving

00:09:55.956 --> 00:09:57.891
gravitational field look like?

00:09:57.891 --> 00:10:00.403
Let's say a hypothetical
planet X had three times

00:10:00.403 --> 00:10:03.274
the mass of Earth and
half the radius of Earth.

00:10:03.274 --> 00:10:05.432
What would be the
acceleration due to gravity

00:10:05.432 --> 00:10:08.068
on planet X, i.e. the gravitational field

00:10:08.068 --> 00:10:10.649
on planet X, in terms of the acceleration

00:10:10.649 --> 00:10:13.195
due to gravity on Earth, which is GE.

00:10:13.195 --> 00:10:15.122
So, we know that the
gravitational field on Earth

00:10:15.122 --> 00:10:17.643
has to be big G times mass of the Earth

00:10:17.643 --> 00:10:19.442
over the radius of Earth squared,

00:10:19.442 --> 00:10:21.565
which we're calling G sub E, and

00:10:21.565 --> 00:10:24.317
the gravitational field
on planet X would be

00:10:24.317 --> 00:10:27.696
big G times three times
the mass of the Earth

00:10:27.696 --> 00:10:30.762
divided by half the radius
of the Earth squared,

00:10:30.762 --> 00:10:32.677
and when we square this factor of a half

00:10:32.677 --> 00:10:34.757
we'll get 1/4, which
is in the denominator,

00:10:34.757 --> 00:10:38.168
so three divided by a
1/4 is 12 times big G

00:10:38.168 --> 00:10:40.966
mass of the Earth over
radius of the Earth squared,

00:10:40.966 --> 00:10:43.425
and since this entire term
here is the acceleration

00:10:43.425 --> 00:10:45.877
due to gravity on Earth,
the acceleration due

00:10:45.877 --> 00:10:48.340
to gravity on planet
X is gonna be 12 times

00:10:48.340 --> 00:10:51.154
the acceleration due to gravity on Earth.

00:10:51.154 --> 00:10:53.152
Sometimes when you're solving
gravitational problems,

00:10:53.152 --> 00:10:55.625
you'll be given the density
instead of the mass.

00:10:55.625 --> 00:10:58.327
The density is the
amount of mass per volume

00:10:58.327 --> 00:10:59.606
for a given material.

00:10:59.606 --> 00:11:02.171
The symbol for density
is the Greek letter rho,

00:11:02.171 --> 00:11:03.899
and you can find it by taking the mass

00:11:03.899 --> 00:11:05.609
divided by the volume.

00:11:05.609 --> 00:11:08.922
So, the units of density are
kilograms per meter queued.

00:11:08.922 --> 00:11:11.054
And it's not a vector
since it has no direction,

00:11:11.054 --> 00:11:12.810
but it does let you solve for mass.

00:11:12.810 --> 00:11:14.393
If you know the density you could say

00:11:14.393 --> 00:11:17.408
that the mass is the
density times the volume.

00:11:17.408 --> 00:11:20.169
So, what's an example problem
involving density look like?

00:11:20.169 --> 00:11:22.518
Let's try the hypothetical
planet problem again,

00:11:22.518 --> 00:11:24.293
but this time instead of being told

00:11:24.293 --> 00:11:26.696
that planet X has three
times the mass of Earth,

00:11:26.696 --> 00:11:28.411
let's say that planet X has three times

00:11:28.411 --> 00:11:31.681
the density of Earth, and
again, half the radius of Earth.

00:11:31.681 --> 00:11:33.373
What would be the
acceleration due to gravity

00:11:33.373 --> 00:11:35.259
on planet X in terms of the acceleration

00:11:35.259 --> 00:11:37.501
due to gravity on Earth GE.

00:11:37.501 --> 00:11:38.961
We could write down the formula

00:11:38.961 --> 00:11:41.666
for gravitational acceleration
or gravitational field,

00:11:41.666 --> 00:11:44.111
which is big G M over R squared,

00:11:44.111 --> 00:11:45.924
but this time we don't know the mass,

00:11:45.924 --> 00:11:47.861
we just know the density,
so we want to rewrite

00:11:47.861 --> 00:11:49.744
this formula in terms of density,

00:11:49.744 --> 00:11:53.789
which we can do by rewriting
the M as rho times V,

00:11:53.789 --> 00:11:55.895
since density is mass per volume,

00:11:55.895 --> 00:11:58.388
and mass is density times volume.

00:11:58.388 --> 00:11:59.892
But we don't know the
volume of this planet,

00:11:59.892 --> 00:12:02.177
we just know the radius, so
we need to rewrite volume

00:12:02.177 --> 00:12:04.093
in terms of radius, which we could do

00:12:04.093 --> 00:12:06.955
since planets are spherical
and the volume of a sphere

00:12:06.955 --> 00:12:09.566
is four thirds pi R cubed.

00:12:09.566 --> 00:12:11.391
We can substitute this expression in

00:12:11.391 --> 00:12:13.766
for the volume and
finally get an expression

00:12:13.766 --> 00:12:16.282
for the acceleration
due to gravity of big G

00:12:16.282 --> 00:12:20.373
times rho four thirds pi R
cubed divided by R squared.

00:12:20.373 --> 00:12:23.234
And we can cancel an R squared
on the top and the bottom,

00:12:23.234 --> 00:12:26.016
which leaves this little
G as equaling big G

00:12:26.016 --> 00:12:27.766
rho four thirds pi R.

00:12:28.808 --> 00:12:30.685
So, the gravitational
acceleration on Earth

00:12:30.685 --> 00:12:33.494
would be big G rho of Earth four thirds pi

00:12:33.494 --> 00:12:34.990
times the radius of Earth.

00:12:34.990 --> 00:12:36.862
And the gravitational acceleration

00:12:36.862 --> 00:12:38.824
on planet X would be big G times

00:12:38.824 --> 00:12:41.374
the density of planet
X, which is three times

00:12:41.374 --> 00:12:44.314
the density of Earth times four thirds pi

00:12:44.314 --> 00:12:47.018
times the radius of planet
X, which is one half

00:12:47.018 --> 00:12:48.932
the radius of Earth,
which when we pull out

00:12:48.932 --> 00:12:50.838
the three and factor of a half,

00:12:50.838 --> 00:12:53.196
gives us three halves times the expression

00:12:53.196 --> 00:12:55.607
for the acceleration
due to gravity on Earth.

00:12:55.607 --> 00:12:57.166
So, the gravitational acceleration

00:12:57.166 --> 00:12:59.378
on planet X is gonna be three halves

00:12:59.378 --> 00:13:02.195
the gravitational
acceleration on planet Earth.

00:13:02.195 --> 00:13:04.319
Gravitational orbits
are just a special case

00:13:04.319 --> 00:13:06.640
of centripetal acceleration
where some object

00:13:06.640 --> 00:13:08.225
is orbiting another object due

00:13:08.225 --> 00:13:09.862
to the gravitational force.

00:13:09.862 --> 00:13:11.284
And if that orbit is a circle,

00:13:11.284 --> 00:13:14.072
we can relate the speed,
the radius of the orbit,

00:13:14.072 --> 00:13:15.721
and the larger mass to each other

00:13:15.721 --> 00:13:19.186
using Newton's second law
and centripetal acceleration.

00:13:19.186 --> 00:13:20.616
You just plugin the acceleration

00:13:20.616 --> 00:13:23.243
as the centripetal
acceleration, V squared over R,

00:13:23.243 --> 00:13:26.294
and since the centripetal
force is the force of gravity,

00:13:26.294 --> 00:13:27.583
you can plugin the expression for

00:13:27.583 --> 00:13:30.208
the force of gravity as
the centripetal force,

00:13:30.208 --> 00:13:33.637
which is big G M M over the
distance between them squared.

00:13:33.637 --> 00:13:37.056
And since the mass of the
orbiting object cancels

00:13:37.056 --> 00:13:39.010
we get an expression
that relates to the speed

00:13:39.010 --> 00:13:40.644
of the orbiting object, the larger mass

00:13:40.644 --> 00:13:42.479
that's pulling that object in,

00:13:42.479 --> 00:13:45.653
and the center to center
distance between the objects,

00:13:45.653 --> 00:13:47.243
which, if we solve this for V,

00:13:47.243 --> 00:13:50.595
gives us the square root
of big G times the mass

00:13:50.595 --> 00:13:52.631
pulling in the object divided by

00:13:52.631 --> 00:13:55.123
the center to center
distance between the objects.

00:13:55.123 --> 00:13:57.318
Note that this formula
does not depend on the mass

00:13:57.318 --> 00:14:00.037
that's in orbit since
that mass canceled out

00:14:00.037 --> 00:14:01.223
in the calculation.

00:14:01.223 --> 00:14:02.608
So, what's an example problem involving

00:14:02.608 --> 00:14:04.489
gravitational orbits look like?

00:14:04.489 --> 00:14:07.161
Well, imagine a space station of mass MS

00:14:07.161 --> 00:14:09.646
is orbiting at an altitude of 3R

00:14:09.646 --> 00:14:13.183
above a planet of mass MP and radius R

00:14:13.183 --> 00:14:15.734
as seen in this diagram, and then imagine

00:14:15.734 --> 00:14:18.208
a different space station of mass 3MS

00:14:18.208 --> 00:14:21.472
is orbiting at an altitude
of 2R above a planet

00:14:21.472 --> 00:14:26.447
of mass 4MP and a radius of
2R as seen in this diagram,

00:14:26.447 --> 00:14:27.879
and we wanna know, if the speed

00:14:27.879 --> 00:14:30.695
of the space station of mass MS is V,

00:14:30.695 --> 00:14:32.523
then in terms of V, what's the speed

00:14:32.523 --> 00:14:34.951
of the space station of mass 3MS?

00:14:34.951 --> 00:14:36.636
Well, we just showed that the speed

00:14:36.636 --> 00:14:38.473
of an orbiting object is gonna be equal

00:14:38.473 --> 00:14:40.822
to the square root of big G times

00:14:40.822 --> 00:14:43.149
the mass of the larger object pulling in

00:14:43.149 --> 00:14:44.933
the smaller object divided by the

00:14:44.933 --> 00:14:48.081
center to center distance
between the objects.

00:14:48.081 --> 00:14:49.926
And since this formula
doesn't involve the mass

00:14:49.926 --> 00:14:51.762
of the orbiting object, it doesn't matter

00:14:51.762 --> 00:14:53.861
that the objects have different masses,

00:14:53.861 --> 00:14:56.082
but the mass of the planet
can make a difference.

00:14:56.082 --> 00:14:58.438
So, to get the speed of
the space station MS,

00:14:58.438 --> 00:15:00.851
we could say that it's
the square root of Big G

00:15:00.851 --> 00:15:04.139
the mass of planet P over the
center to center distance,

00:15:04.139 --> 00:15:05.764
which is not gonna be the radius

00:15:05.764 --> 00:15:07.844
of the planet or the altitude,

00:15:07.844 --> 00:15:11.036
it's gonna be the radius of
the planet plus the altitude

00:15:11.036 --> 00:15:14.241
since this has to be the
center to center distance,

00:15:14.241 --> 00:15:17.371
which in this case will
be 3R plus R, which is 4R.

00:15:17.371 --> 00:15:19.157
And now to get the speed
of the space station

00:15:19.157 --> 00:15:21.656
of mass 3MS, we'll use the same formula,

00:15:21.656 --> 00:15:23.690
which is big G mass of the planet, which

00:15:23.690 --> 00:15:26.203
in this case is 4MP divided by

00:15:26.203 --> 00:15:27.608
the center to center distance,

00:15:27.608 --> 00:15:30.701
which in this case would be 2R plus 2R,

00:15:30.701 --> 00:15:33.003
and again that's 4R, and if we compare,

00:15:33.003 --> 00:15:35.028
the only difference
between these expressions

00:15:35.028 --> 00:15:37.218
is that there's an extra
factor of four within

00:15:37.218 --> 00:15:38.502
this square root.

00:15:38.502 --> 00:15:40.533
So, if we take that factor
out, the square root

00:15:40.533 --> 00:15:43.533
of four is two, we'd get
two times the expression

00:15:43.533 --> 00:15:46.373
for the speed of the space station MS.

00:15:46.373 --> 00:15:49.930
So, the space station 3MS
is travelling two times

00:15:49.930 --> 00:15:52.763
the speed of the space station MS.


Super hot tension

WEBVTT
Kind: captions
Language: en

00:00:00.709 --> 00:00:01.616
- [Voiceover] Oh, it's time.

00:00:01.616 --> 00:00:03.391
It's time for the super
hot tension problem.

00:00:03.391 --> 00:00:04.837
We're about to do this right here.

00:00:04.837 --> 00:00:07.032
We've got our super hot can of red peppers

00:00:07.032 --> 00:00:08.821
hanging from these strings.

00:00:08.821 --> 00:00:11.761
We want to know what the
tension is in these ropes.

00:00:11.761 --> 00:00:15.077
This is for real now, this
is a real tension problem.

00:00:15.077 --> 00:00:16.309
And here's the deal.

00:00:16.309 --> 00:00:18.377
You might look at this,
you might get frightened.

00:00:18.377 --> 00:00:20.232
You might think, I've gotta come up with

00:00:20.232 --> 00:00:22.551
a completely new strategy to tackle this.

00:00:22.551 --> 00:00:24.439
I've gotta throw away
everything I've learned

00:00:24.439 --> 00:00:25.902
and just try something new.

00:00:25.902 --> 00:00:27.058
And that's a lie.

00:00:27.058 --> 00:00:28.499
You should not lie to yourself.

00:00:28.499 --> 00:00:30.439
Use the same process.

00:00:30.439 --> 00:00:32.290
We're gonna use the same process we used

00:00:32.290 --> 00:00:33.408
for the easy tension problems,

00:00:33.408 --> 00:00:35.134
because it's gonna lead
us to the answer again.

00:00:35.134 --> 00:00:39.170
Be careful. Don't stray
from the strategy here.

00:00:39.170 --> 00:00:40.109
The strategy works.

00:00:40.109 --> 00:00:41.650
So we're gonna draw our
force diagram first.

00:00:41.650 --> 00:00:43.178
That's what we always do.

00:00:43.178 --> 00:00:45.031
We're gonna say that the forces are

00:00:45.031 --> 00:00:48.170
force of gravity on
this can of red peppers,

00:00:48.170 --> 00:00:50.544
which is MG, and if it's 3 kilograms,

00:00:50.544 --> 00:00:52.962
we know 3 kilograms times about 10,

00:00:52.962 --> 00:00:55.708
we're gonna say, let's
approximate G as 10 again

00:00:55.708 --> 00:00:57.417
to make the numbers come out nice.

00:00:57.417 --> 00:01:00.816
So instead of using 9.8,
we'll say G is about 10,

00:01:00.816 --> 00:01:02.910
and so we'll say 3 kilograms

00:01:02.910 --> 00:01:07.077
times 10 meters per second
squared is gonna be 30 Newtons.

00:01:09.947 --> 00:01:12.728
And so the force of gravity
downward is 30 Newtons.

00:01:12.728 --> 00:01:14.078
What other forces do we have?

00:01:14.078 --> 00:01:17.330
We've got this T1, remember
tension does not push.

00:01:17.330 --> 00:01:20.197
Ropes can't push, ropes can only pull,

00:01:20.197 --> 00:01:21.880
so T1's gonna pull that way.

00:01:21.880 --> 00:01:23.961
So I'm gonna draw T1 coming this way.

00:01:23.961 --> 00:01:25.378
So here's our T1.

00:01:26.219 --> 00:01:28.447
And then we're gonna have
T2 pointing this way,

00:01:28.447 --> 00:01:29.630
so this is T2.

00:01:29.630 --> 00:01:31.731
Again, T2 pulls, just like all tension.

00:01:31.731 --> 00:01:34.018
Tension pulls, tension can't push.

00:01:34.018 --> 00:01:37.101
So I've got tension 2 going this way.

00:01:39.580 --> 00:01:41.656
That's it, that's our force diagram.

00:01:41.656 --> 00:01:42.584
There's no other forces.

00:01:42.584 --> 00:01:43.849
I don't draw a normal force,

00:01:43.849 --> 00:01:46.276
'cause this can isn't in
contact with another surface.

00:01:46.276 --> 00:01:48.956
So there's no normal force,
you've got these two tensions,

00:01:48.956 --> 00:01:50.436
the force of gravity.

00:01:50.436 --> 00:01:52.155
And now we do the same thing we always do.

00:01:52.155 --> 00:01:54.742
After our force diagram,
we use Newton's Second Law

00:01:54.742 --> 00:01:56.809
in one direction or another.

00:01:56.809 --> 00:01:57.863
So let's do it.

00:01:57.863 --> 00:02:00.687
Let's say that acceleration
is the net force

00:02:00.687 --> 00:02:03.779
in a given direction, divided by the mass.

00:02:03.779 --> 00:02:05.827
Which direction did we pick again?

00:02:05.827 --> 00:02:08.786
It's hard to say, we've
got forces vertical,

00:02:08.786 --> 00:02:10.137
we've got forces horizontal.

00:02:10.137 --> 00:02:13.258
There's only two directions to
pick, X or Y in this problem.

00:02:13.258 --> 00:02:15.222
We're gonna pick the vertical direction,

00:02:15.222 --> 00:02:17.094
even though it doesn't
really matter too much.

00:02:17.094 --> 00:02:19.416
But because we know one of
the forces in the vertical

00:02:19.416 --> 00:02:20.909
direction, we know the force of gravity.

00:02:20.909 --> 00:02:23.089
Force of gravity is 30 Newtons.

00:02:23.089 --> 00:02:25.267
Usually that's a guod
strategy, pick the direction

00:02:25.267 --> 00:02:27.770
that you know something about at least.

00:02:27.770 --> 00:02:29.526
So we're gonna do that here.

00:02:29.526 --> 00:02:31.392
We're gonna say that the
acceleration vertically

00:02:31.392 --> 00:02:33.896
equals to the net force
vertically over the mass.

00:02:33.896 --> 00:02:34.968
And so now we plug in.

00:02:34.968 --> 00:02:36.784
If this can is just sitting here,

00:02:36.784 --> 00:02:39.473
if there's no acceleration,
if this is in not an elevator

00:02:39.473 --> 00:02:42.557
transporting these peppers up or down,

00:02:42.557 --> 00:02:44.447
and it's not in a rocket,
if it's just sitting here

00:02:44.447 --> 00:02:47.466
with no acceleration, our
acceleration will be zero.

00:02:47.466 --> 00:02:50.995
That's gonna equal the net force
and the vertical direction.

00:02:50.995 --> 00:02:52.377
So what are we gonna have?

00:02:52.377 --> 00:02:54.197
So what are the forces in
the vertical direction here?

00:02:54.197 --> 00:02:57.222
One force is this 30
Newton force of gravity.

00:02:57.222 --> 00:03:00.571
This points down, we're gonna
assume upward is positive,

00:03:00.571 --> 00:03:02.194
that means down in a negative.

00:03:02.194 --> 00:03:03.608
So I'll just put -30 Newtons.

00:03:03.608 --> 00:03:05.916
I could have written -MG,

00:03:05.916 --> 00:03:07.462
but we already knew it was 30 Newtons,

00:03:07.462 --> 00:03:08.921
so I'll write -30 Newtons.

00:03:08.921 --> 00:03:10.583
Then we've got T1 and T2.

00:03:10.583 --> 00:03:12.255
Both of those point up.

00:03:12.255 --> 00:03:13.648
But they don't completely point up,

00:03:13.648 --> 00:03:15.004
they partially point up.

00:03:15.004 --> 00:03:17.326
So part of them points to the right,

00:03:17.326 --> 00:03:19.584
part of them points upward.

00:03:19.584 --> 00:03:22.656
Only this vertical
component, we'll call it T1Y,

00:03:22.656 --> 00:03:25.179
is gonna get included
into this calculation,

00:03:25.179 --> 00:03:28.212
'cause this calculation
only uses Y directed forces.

00:03:28.212 --> 00:03:30.868
And the reason is only Y directed forces,

00:03:30.868 --> 00:03:34.325
vertical forces, affect
the vertical acceleration.

00:03:34.325 --> 00:03:36.427
So this T1Y points upward,

00:03:36.427 --> 00:03:39.785
I'll do plus T1 in the Y direction.

00:03:39.785 --> 00:03:41.675
And similarly, this T2.

00:03:41.675 --> 00:03:44.113
It doesn't all point vertically,

00:03:44.113 --> 00:03:45.594
only part of it points vertically.

00:03:45.594 --> 00:03:49.138
So I'll write this as
T2 in the Y direction.

00:03:49.138 --> 00:03:50.437
And that's also upward,

00:03:50.437 --> 00:03:52.597
so since that's up, I'll count it

00:03:52.597 --> 00:03:55.314
as plus T2 in the Y direction.

00:03:55.314 --> 00:03:57.101
And that's it, that's all our forces.

00:03:57.101 --> 00:04:00.483
Notice we can't plug in
the total amount T2 in this

00:04:00.483 --> 00:04:02.870
formula, 'cause only part of it points up.

00:04:02.870 --> 00:04:05.340
Similarly, we have to plug in
only the vertical component

00:04:05.340 --> 00:04:09.031
of the T1 force because only
part of it points vertically.

00:04:09.031 --> 00:04:13.175
And then we divide by the
mass, the mass is 3 kilograms.

00:04:13.175 --> 00:04:15.875
But we're gonna multiply
both sides by 3 kilograms,

00:04:15.875 --> 00:04:19.668
and we're gonna get zero
equals all of this right here,

00:04:19.668 --> 00:04:21.066
so I'll just copy this right here.

00:04:21.066 --> 00:04:25.233
We use this over again,
that comes down right there.

00:04:26.489 --> 00:04:29.331
But now there's nothing
on the bottom here.

00:04:29.331 --> 00:04:30.227
So what do we do at this point?

00:04:30.227 --> 00:04:31.556
Now you might think we're stuck.

00:04:31.556 --> 00:04:33.525
I mean, we've got two unknowns in here.

00:04:33.525 --> 00:04:35.605
I can't solve for either one,

00:04:35.605 --> 00:04:36.938
I don't know either one of these.

00:04:36.938 --> 00:04:38.624
I know they have to add up to 30,

00:04:38.624 --> 00:04:40.920
so I'd do fine, if I
added 30 to both sides,

00:04:40.920 --> 00:04:43.129
I'd realize that these
two vertical components

00:04:43.129 --> 00:04:45.097
of these tension forces added up

00:04:45.097 --> 00:04:47.308
have to add up to 30,
and that makes sense.

00:04:47.308 --> 00:04:49.319
They have to balance the force downward.

00:04:49.319 --> 00:04:52.664
But I don't know either of
them, so how do I solve here?

00:04:52.664 --> 00:04:54.193
Well, let's do this.

00:04:54.193 --> 00:04:57.380
If you ever get stuck on
one of the force equations

00:04:57.380 --> 00:05:00.090
for a single direction, just
go to the next equation.

00:05:00.090 --> 00:05:01.889
Let's try A in the X direction.

00:05:01.889 --> 00:05:04.053
So for A in the X direction,
we have the net force

00:05:04.053 --> 00:05:06.198
in the X direction, over the mass,

00:05:06.198 --> 00:05:09.231
again, the acceleration is gonna be zero

00:05:09.231 --> 00:05:11.887
if these peppers are not
accelerating horizontally.

00:05:11.887 --> 00:05:14.270
So unless this thing's in
a train car or something,

00:05:14.270 --> 00:05:16.435
and the whole thing's accelerating,

00:05:16.435 --> 00:05:18.076
then you might have
horizontal acceleration.

00:05:18.076 --> 00:05:19.946
And if it did, it's
not that big of a deal,

00:05:19.946 --> 00:05:21.639
you just plug it in there.

00:05:21.639 --> 00:05:23.341
But assuming it's acceleration zero,

00:05:23.341 --> 00:05:24.725
because the peppers
are just sitting there,

00:05:24.725 --> 00:05:25.768
not changing their velocity,
we'll plug in zero.

00:05:25.768 --> 00:05:28.206
We'll plug in the forces
in the X direction.

00:05:28.206 --> 00:05:31.193
These are gonna be T1 in the X.

00:05:31.193 --> 00:05:33.927
So part of this T1 points
in the X direction.

00:05:33.927 --> 00:05:37.566
Similarly, part of T2
points in the X direction.

00:05:37.566 --> 00:05:38.706
We'll call this T2X.

00:05:38.706 --> 00:05:40.833
We use these as the magnitude.

00:05:40.833 --> 00:05:42.994
Let's say T2X is the
magnitude of the force

00:05:42.994 --> 00:05:45.140
that T2 pulls with to the left,

00:05:45.140 --> 00:05:49.307
and T1 is the magnitude that
T1 pulls with to the right.

00:05:50.723 --> 00:05:53.079
So to plug these in, we've got to decide

00:05:53.079 --> 00:05:55.585
whether they should be
positive or negative.

00:05:55.585 --> 00:05:57.988
So this T1X, since it pulls to the right,

00:05:57.988 --> 00:05:59.696
T1X will be positive.

00:05:59.696 --> 00:06:02.835
We're gonna consider rightward
to be the positive direction,

00:06:02.835 --> 00:06:05.252
'cause that's the typical
convention that we're gonna adopt.

00:06:05.252 --> 00:06:08.229
And T2X pulls to the left.

00:06:08.229 --> 00:06:10.033
That's gonna be a negative contribution,

00:06:10.033 --> 00:06:12.587
so minus T2 in the X direction.

00:06:12.587 --> 00:06:14.115
'Cause leftward would be negative.

00:06:14.115 --> 00:06:16.597
We divided by the mass,
the mass was 3 kilograms,

00:06:16.597 --> 00:06:19.314
but again, we'll multiply both sides by 3,

00:06:19.314 --> 00:06:23.313
we'll get zero equals,
and then we just get T,

00:06:23.313 --> 00:06:24.502
the same thing up here,

00:06:24.502 --> 00:06:28.669
so we'll just copy this
thing here, put it down here.

00:06:30.614 --> 00:06:31.673
And again, you might be concerned.

00:06:31.673 --> 00:06:33.526
I can't solve this either.

00:06:33.526 --> 00:06:37.132
I mean, I can solve for
T1X, but look at what I get.

00:06:37.132 --> 00:06:40.870
If I just multi, or if I
added T2X to both sides,

00:06:40.870 --> 00:06:42.432
I'm just gonna get T1 in the X direction

00:06:42.432 --> 00:06:45.349
has to equal T2 in the X direction.

00:06:46.224 --> 00:06:47.684
And that makes sense.

00:06:47.684 --> 00:06:49.472
These two forces have to
be equal and opposite,

00:06:49.472 --> 00:06:52.137
because they have to cancel so
that you have no acceleration

00:06:52.137 --> 00:06:53.144
in the X direction.

00:06:53.144 --> 00:06:55.373
And this was not drawn
proportionately, sorry,

00:06:55.373 --> 00:06:57.452
this should be the exact
same size as this force

00:06:57.452 --> 00:06:59.423
because they have to cancel,

00:06:59.423 --> 00:07:01.517
since there's no horizontal acceleration.

00:07:01.517 --> 00:07:02.817
But what do we do?

00:07:02.817 --> 00:07:05.187
We can't solve this equation
we got from X direction.

00:07:05.187 --> 00:07:08.635
We can't solve this equation
we got from the Y direction.

00:07:08.635 --> 00:07:11.223
Whenever this happens,
when you get two equations,

00:07:11.223 --> 00:07:12.761
and you can't solve either

00:07:12.761 --> 00:07:13.995
because there's too many unknowns,

00:07:13.995 --> 00:07:16.666
you're gonna have to end up
plugging one into the other.

00:07:16.666 --> 00:07:18.407
But I can't even do that yet.

00:07:18.407 --> 00:07:20.358
I've got four different variables here.

00:07:20.358 --> 00:07:22.275
T1X, T2X, T1Y, and T2Y,

00:07:24.305 --> 00:07:25.720
these are all four different variables,

00:07:25.720 --> 00:07:28.387
I've only got two equations,
I can't solve this.

00:07:28.387 --> 00:07:30.383
So the trick, the trick we're gonna use

00:07:30.383 --> 00:07:31.717
that a lot of people don't like doing

00:07:31.717 --> 00:07:33.147
because it's a little more sophisticated,

00:07:33.147 --> 00:07:37.144
now we've gotta put these
all in terms of T1 and T2

00:07:37.144 --> 00:07:38.595
so that we can solve.

00:07:38.595 --> 00:07:41.859
If I put T1Y in terms of the total T1,

00:07:41.859 --> 00:07:44.706
and then sines of angles,
and cosines of angles,

00:07:44.706 --> 00:07:48.039
and I put T2Y in terms of T2 and angles,

00:07:49.582 --> 00:07:51.240
and I do the same thing for 1X and 2X,

00:07:51.240 --> 00:07:53.042
I'll have two equations,
and the only two unknowns

00:07:53.042 --> 00:07:57.501
will be T1 and T2, then
we can finally solve.

00:07:57.501 --> 00:07:59.435
If that didn't make any
sense, here's what I'm saying.

00:07:59.435 --> 00:08:02.751
I'm saying figure out what
T1Y is in terms of T1.

00:08:02.751 --> 00:08:05.629
So I know this angle here,
let's figure out these angles.

00:08:05.629 --> 00:08:08.666
So these angles here are, if this is 30,

00:08:08.666 --> 00:08:11.202
this angle down here has
to be 30 because these

00:08:11.202 --> 00:08:13.312
are alternate interior angles.

00:08:13.312 --> 00:08:14.696
And if you don't believe me,

00:08:14.696 --> 00:08:16.811
imagine this big triangle over here,

00:08:16.811 --> 00:08:18.692
where this is a right angle.

00:08:18.692 --> 00:08:20.401
So this triangle from here
to there, down to here,

00:08:20.401 --> 00:08:24.725
up to here, if this is 30,
that's 90, this has gotta be 60,

00:08:24.725 --> 00:08:27.212
'cause it all adds up
to 180 for a triangle.

00:08:27.212 --> 00:08:29.132
And if this right angle
is 90, and this side's 60,

00:08:29.132 --> 00:08:31.291
this side's gotta be 30.

00:08:31.291 --> 00:08:34.102
Similarly, this side's a right angle.

00:08:34.102 --> 00:08:36.132
Look at this triangle, 60, 90,

00:08:36.132 --> 00:08:37.841
that means this would have to be 30.

00:08:37.841 --> 00:08:40.751
And so if I come down here,
this angle would have to be 60.

00:08:40.751 --> 00:08:42.334
Just like this one,

00:08:43.189 --> 00:08:45.919
'cause it's an alternate
interior angle, so that's 60.

00:08:45.919 --> 00:08:49.788
So this angle here is 60,
this angle here is 30,

00:08:49.788 --> 00:08:51.934
we can figure out what
these components are

00:08:51.934 --> 00:08:53.462
in terms of the total vectors.

00:08:53.462 --> 00:08:54.957
Once we find those,

00:08:54.957 --> 00:08:56.502
we're gonna plug those
expressions into here,

00:08:56.502 --> 00:08:57.753
and that will let us solve.

00:08:57.753 --> 00:08:59.362
In other words, T1Y is gonna be,

00:08:59.362 --> 00:09:02.108
once you do this for awhile you realize,

00:09:02.108 --> 00:09:03.604
this is the opposite side.

00:09:03.604 --> 00:09:05.440
So this component here is going to be

00:09:05.440 --> 00:09:07.607
total T1 times sine of 30.

00:09:09.082 --> 00:09:11.132
Because it's the opposite side.

00:09:11.132 --> 00:09:13.779
And if that didn't make sense,
we'll derive it right here.

00:09:13.779 --> 00:09:15.957
So what we're saying is that sine of 30,

00:09:15.957 --> 00:09:19.207
sine of 30 is opposite over hypotenuse,

00:09:21.588 --> 00:09:25.150
and in this case, the
opposite side is T1Y.

00:09:25.150 --> 00:09:29.150
So T1Y over the total T1
is equal to sine of 30.

00:09:30.918 --> 00:09:33.275
And we can solve this for T1Y now,

00:09:33.275 --> 00:09:36.106
we can get the T1Y if I
multiply both sides by T1.

00:09:36.106 --> 00:09:39.273
I get that that's T1 times sine of 30.

00:09:40.764 --> 00:09:42.199
So that's what I said down here.

00:09:42.199 --> 00:09:45.074
T1 is just T, oh sorry, forgot the one.

00:09:45.074 --> 00:09:46.741
T1 times sine of 30.

00:09:48.048 --> 00:09:50.506
Similarly, if you do the
same thing with cosine 30,

00:09:50.506 --> 00:09:53.506
you'll get that T1X is T1 cosine 30,

00:09:55.852 --> 00:09:57.702
by the exact same process.

00:09:57.702 --> 00:10:01.147
Similarly over here, T2 is going to be,

00:10:01.147 --> 00:10:03.564
I'm sorry, T2X is gonna be 2.

00:10:05.004 --> 00:10:09.171
So T2 cosine 60, because
this is the adjacent side.

00:10:10.133 --> 00:10:12.966
And T2Y is gonna be T2 sine of 60.

00:10:15.068 --> 00:10:16.531
And if any of that doesn't make sense,

00:10:16.531 --> 00:10:19.712
just go back to the
definition of sine and cosine,

00:10:19.712 --> 00:10:22.152
write what the opposite side is,

00:10:22.152 --> 00:10:25.600
the total hypotenuse side,
solve for your expression,

00:10:25.600 --> 00:10:26.719
you'll get these.

00:10:26.719 --> 00:10:29.191
If you don't believe me on
those, try those out yourselves.

00:10:29.191 --> 00:10:31.494
But those are what these components are,

00:10:31.494 --> 00:10:35.966
in terms of T2 and the
angles T2, T1 and the angles.

00:10:35.966 --> 00:10:37.574
And why are we doing this?

00:10:37.574 --> 00:10:39.839
We're doing this so that
when plug in over here,

00:10:39.839 --> 00:10:41.251
we'll only have two variables.

00:10:41.251 --> 00:10:45.008
In other words, if I plug
T1Y, this expression here,

00:10:45.008 --> 00:10:48.758
T1 sine 30 in for T1Y,
similarly if I plug in

00:10:49.703 --> 00:10:53.870
T2Y is T2 sine 60 into
this expression right there

00:10:55.357 --> 00:10:57.408
for T2Y, look at what I'll get.

00:10:57.408 --> 00:10:58.987
I'll get zero equals.

00:10:58.987 --> 00:11:01.654
So I'll get negative 30 Newtons,

00:11:02.712 --> 00:11:06.795
and then I'll get plus
T1Y was T1 sine 30, so T1,

00:11:08.495 --> 00:11:11.145
and then sine 30, we can
clean this up a little bit.

00:11:11.145 --> 00:11:12.334
Sine 30 is just a half.

00:11:12.334 --> 00:11:15.938
So I'll just write T1 over 2, and then

00:11:15.938 --> 00:11:17.874
'cause sine 30 is just one half.

00:11:17.874 --> 00:11:20.874
And then T2Y is gonna be T2 sine 60,

00:11:23.726 --> 00:11:26.651
and sine 60 is just root 3 over 2.

00:11:26.651 --> 00:11:30.818
So I'll write this as plus T2
over 2, and then times root 3.

00:11:33.088 --> 00:11:35.524
And you might think this is no better.

00:11:35.524 --> 00:11:37.477
I mean this is still a
horrible mess right here.

00:11:37.477 --> 00:11:40.986
But, look at. This is
in terms of T1 and T2.

00:11:40.986 --> 00:11:42.266
That's what I'm gonna do over here.

00:11:42.266 --> 00:11:44.237
I'm gonna put these in terms of T1 and T2,

00:11:44.237 --> 00:11:45.505
and then we can solve.

00:11:45.505 --> 00:11:48.557
So T1X is T1 over cosine 30,

00:11:48.557 --> 00:11:52.798
so I'm gonna write this
as T1 times cosine 30,

00:11:52.798 --> 00:11:55.381
and cosine 30 is root 3 over 2,

00:11:56.568 --> 00:11:59.576
so this is T1 over 2 times root 3.

00:11:59.576 --> 00:12:02.909
And that should equal T2X is right here,

00:12:03.937 --> 00:12:07.354
That's T2 cosine 60, cosine 60 is a half.

00:12:08.224 --> 00:12:10.641
So T2X is gonna be T2 over 2.

00:12:12.348 --> 00:12:13.946
So T2 over 2.

00:12:13.946 --> 00:12:16.139
So what I'm doing is, if
this doesn't make sense,

00:12:16.139 --> 00:12:18.873
I'm just substituting
what these components are

00:12:18.873 --> 00:12:22.750
in terms of the total
magnitude in the angle.

00:12:22.750 --> 00:12:25.269
And I do this, because
look at what I have now,

00:12:25.269 --> 00:12:26.683
I have got one equation with T1 and T2.

00:12:26.683 --> 00:12:28.663
I've got another equation with T1 and T2.

00:12:28.663 --> 00:12:30.927
So what I'm gonna do to solve these,

00:12:30.927 --> 00:12:32.843
when we have two equations
and two unknowns,

00:12:32.843 --> 00:12:34.925
you have to solve for
one of these variables,

00:12:34.925 --> 00:12:37.198
and then substitute it
into the other equation.

00:12:37.198 --> 00:12:40.127
That way you'll get one
equation with one unknown.

00:12:40.127 --> 00:12:42.220
And you try to get the math right,

00:12:42.220 --> 00:12:43.197
and you'll get the problem.

00:12:43.197 --> 00:12:44.920
So I'm gonna solve this one is easier,

00:12:44.920 --> 00:12:47.572
so I'm gonna solve this
one for, let's just say T2.

00:12:47.572 --> 00:12:48.982
So if we solve this for T2,

00:12:48.982 --> 00:12:52.235
I get that T2 equals, well, I can multiply

00:12:52.235 --> 00:12:56.577
both sides by 2, and
I'll get T1 times root 3.

00:12:56.577 --> 00:13:00.947
So T1 times root 3, because
the 2 here cancels with this 2,

00:13:00.947 --> 00:13:04.182
or when I multiply both
sides by 2 it cancels out.

00:13:04.182 --> 00:13:07.936
So we get that T2 equals
T1 root 3. This is great.

00:13:07.936 --> 00:13:12.103
I can substitute T2 as T1
root 3 into here for T2.

00:13:13.998 --> 00:13:15.431
And the reason I do that,

00:13:15.431 --> 00:13:17.655
is I'll get one equation with one unknown.

00:13:17.655 --> 00:13:19.541
I'll only have T1 in that equation now.

00:13:19.541 --> 00:13:23.168
So if I do this, I'll
get zero equals negative,

00:13:23.168 --> 00:13:25.118
you know what, let's
just move the -30 over.

00:13:25.118 --> 00:13:26.690
This is kind of annoying here.

00:13:26.690 --> 00:13:28.316
Just add 30 to both sides,

00:13:28.316 --> 00:13:30.804
then take this calculation here.

00:13:30.804 --> 00:13:35.326
We get plus 30 equals,
and then we're gonna have

00:13:35.326 --> 00:13:38.409
T1 over 2, from this T1, so T1 over 2

00:13:39.487 --> 00:13:42.570
plus, I've got plus, T2 is T1 root 3.

00:13:44.894 --> 00:13:48.375
So when I plug T1 root 3 in for T2,

00:13:48.375 --> 00:13:52.292
what I'm gonna get is,
I'm gonna get T1 root 3,

00:13:55.086 --> 00:13:56.893
and then times another route 3,

00:13:56.893 --> 00:13:59.560
because T2 itself was T1 root 3.

00:14:00.938 --> 00:14:02.452
So I'm taking this expression here,

00:14:02.452 --> 00:14:05.518
plugging it in for T2, but
I still have to multiply

00:14:05.518 --> 00:14:08.577
that T2 by a root 3 and divide by 2.

00:14:08.577 --> 00:14:10.430
And so, what do we get?

00:14:10.430 --> 00:14:13.226
Root 3 times root 3 is just 3.

00:14:13.226 --> 00:14:16.976
So we have T2 times 3
halves, plus T1 over 2.

00:14:18.133 --> 00:14:19.941
So I'll get 30 equals,

00:14:19.941 --> 00:14:24.232
and then I get T1 over 2,
we're almost there, I promise.

00:14:24.232 --> 00:14:29.122
T1 over 2, plus, and this is
gonna be T1 times 3 over 2,

00:14:29.122 --> 00:14:33.768
so it's gonna be 3 T1 over
2, or what does that equal?

00:14:33.768 --> 00:14:37.435
T1 over 2 plus 3 T1
over 2 is just 4 halves.

00:14:39.037 --> 00:14:42.372
So that's just 2 T1. So
this cleaned up beautifully.

00:14:42.372 --> 00:14:45.929
So this is just 2 times T1,
and now we can solve for T1.

00:14:45.929 --> 00:14:49.346
We get that T1 is simply 30 divided by 2.

00:14:51.128 --> 00:14:54.497
If I divide both sides,
this left hand side by 2,

00:14:54.497 --> 00:14:56.512
and this side here, this right side by 2,

00:14:56.512 --> 00:14:59.012
I get T1 is 30 over 2 Newtons,

00:15:00.037 --> 00:15:02.396
which is just, these should be Newtons,

00:15:02.396 --> 00:15:06.563
I should have units on these,
which is just 15 Newtons.

00:15:08.227 --> 00:15:10.910
Whoo, I did it, 15 Newtons.

00:15:10.910 --> 00:15:12.554
T1 is 15 Newtons.

00:15:12.554 --> 00:15:14.436
We got T1. That's one of them.

00:15:14.436 --> 00:15:15.881
How do we get the other?

00:15:15.881 --> 00:15:19.200
You start back over at the very beginning.

00:15:19.200 --> 00:15:21.559
No, not really, that would be terrible.

00:15:21.559 --> 00:15:23.130
You actually just take this T1,

00:15:23.130 --> 00:15:25.946
and you plug it right into
here, boop, there it goes.

00:15:25.946 --> 00:15:27.977
So T2, we already got it.

00:15:27.977 --> 00:15:29.426
T2 is just T1 root 3.

00:15:29.426 --> 00:15:32.348
So all I have to do is
multiply root 3 by my T1,

00:15:32.348 --> 00:15:33.181
which I know now.

00:15:33.181 --> 00:15:37.521
And I get that T2 is just
15 times root 3 Newton.

00:15:37.521 --> 00:15:39.908
So once you get one of the forces,

00:15:39.908 --> 00:15:41.386
the next one is really easy.

00:15:41.386 --> 00:15:42.884
This is just T2.

00:15:42.884 --> 00:15:46.051
So T2 is 15 root 3, and T1 is just 15.

00:15:47.500 --> 00:15:49.544
So in case you got lost in the details,

00:15:49.544 --> 00:15:52.015
the big picture recap is this.

00:15:52.015 --> 00:15:55.349
We drew a force diagram,
we used Newton's Second Law

00:15:55.349 --> 00:15:57.643
in the vertical direction
we couldn't solve,

00:15:57.643 --> 00:15:59.152
because there were too many unknowns.

00:15:59.152 --> 00:16:02.501
We used Newton's Second Law
in the horizontal direction,

00:16:02.501 --> 00:16:04.158
we couldn't solve because
there were two unknowns.

00:16:04.158 --> 00:16:06.225
We put all four of these unknowns

00:16:06.225 --> 00:16:09.376
in terms of only two unknowns, T1 and T2,

00:16:09.376 --> 00:16:12.560
by writing how those components depended

00:16:12.560 --> 00:16:13.994
on those total vectors.

00:16:13.994 --> 00:16:17.324
We substituted these expressions
in for each component.

00:16:17.324 --> 00:16:19.256
Once we did that, we had two equations,

00:16:19.256 --> 00:16:23.190
with only T1, T2, and T1 and T2 in them.

00:16:23.190 --> 00:16:27.680
We solved one of these
equations for T2 in terms of T1,

00:16:27.680 --> 00:16:29.923
substituted that into the other equation.

00:16:29.923 --> 00:16:32.794
We got a single equation
with only one unknown.

00:16:32.794 --> 00:16:34.929
We were able to solve for that unknown.

00:16:34.929 --> 00:16:37.508
Once we got that, which is our T1,

00:16:37.508 --> 00:16:39.137
once we have that variable,

00:16:39.137 --> 00:16:40.618
we plug it back into that first equation

00:16:40.618 --> 00:16:42.290
that we had solved for T2.

00:16:42.290 --> 00:16:45.560
We plug this 15 in, we get
what the second tension is.

00:16:45.560 --> 00:16:48.060
So even when it seems
like Newton's Second Law

00:16:48.060 --> 00:16:49.896
won't get you there, if you have faith,

00:16:49.896 --> 00:16:52.725
and you persevere, you will make it.

00:16:52.725 --> 00:16:53.558
Good job.


AP Physics 1 Review of Charge and Circuit

WEBVTT
Kind: captions
Language: en

00:00:00.468 --> 00:00:01.688
- [Voiceover] Electric
Charge is a property

00:00:01.688 --> 00:00:05.622
that some, but not all fundamental
particles in nature have.

00:00:05.622 --> 00:00:06.761
The most commonly talked about

00:00:06.761 --> 00:00:09.562
fundamentally charged
particles are the electrons,

00:00:09.562 --> 00:00:11.564
which orbit the outside of the atom.

00:00:11.564 --> 00:00:13.345
These are negatively charged.

00:00:13.345 --> 00:00:15.146
There's also the protons, which reside

00:00:15.146 --> 00:00:18.484
inside the nucleus, and
these are positively charged.

00:00:18.484 --> 00:00:20.088
And the neutrons inside the nucleus

00:00:20.088 --> 00:00:21.855
don't have any net charge.

00:00:21.855 --> 00:00:24.212
It turns out that all
fundamentally charged particles

00:00:24.212 --> 00:00:26.310
in the universe, have charges that come in

00:00:26.310 --> 00:00:29.258
integer units of the elementary charge.

00:00:29.258 --> 00:00:30.804
So if you find a particle in nature,

00:00:30.804 --> 00:00:33.916
it's gonna have a charge
of one times this number,

00:00:33.916 --> 00:00:37.016
two times this number,
three times this number,

00:00:37.016 --> 00:00:38.717
and it could either be
positive or negative.

00:00:38.717 --> 00:00:41.690
For instance, the electron
has a charge of negative 1.6

00:00:41.690 --> 00:00:44.658
times 10 to the negative 19th Coulombs,

00:00:44.658 --> 00:00:47.542
and the charge of the
proton is positive 1.6

00:00:47.542 --> 00:00:49.889
times 10 to the negative 19th Coulombs.

00:00:49.889 --> 00:00:51.928
However, most atoms in the universe

00:00:51.928 --> 00:00:54.076
are electrically neutral overall,

00:00:54.076 --> 00:00:56.828
since they'll have just
as many negative electrons

00:00:56.828 --> 00:00:59.013
as they do positive protons.

00:00:59.013 --> 00:01:01.307
But if an atom had too many electrons,

00:01:01.307 --> 00:01:04.009
overall that atom would
be negatively charged,

00:01:04.009 --> 00:01:06.501
and if an atom had too few electrons,

00:01:06.501 --> 00:01:08.844
that atom would be overall
positively charged.

00:01:08.844 --> 00:01:10.593
And something that's really
important to remember

00:01:10.593 --> 00:01:13.077
is that the electric
charge is always conserved

00:01:13.077 --> 00:01:15.212
for every process, in other words,

00:01:15.212 --> 00:01:17.539
the total charge initially, is gonna equal

00:01:17.539 --> 00:01:20.586
the total charge finally
after any process.

00:01:20.586 --> 00:01:22.086
So what's an example problem involving

00:01:22.086 --> 00:01:23.660
electric charge look like?

00:01:23.660 --> 00:01:26.187
Let's say three identically
sized metal spheres

00:01:26.187 --> 00:01:28.344
start off with the charges seen below.

00:01:28.344 --> 00:01:32.136
Positive five Q, positive
three Q, and negative two Q.

00:01:32.136 --> 00:01:35.793
If we touch sphere X to
sphere Y, and separate them,

00:01:35.793 --> 00:01:39.609
and then touch sphere Y to
sphere Z, and separate them,

00:01:39.609 --> 00:01:42.575
what will be the final
charge on each sphere?

00:01:42.575 --> 00:01:44.547
Well first, when we touch X to Y,

00:01:44.547 --> 00:01:46.693
the total charge has to be conserved.

00:01:46.693 --> 00:01:49.224
There's a total charge
of eight Q amongst them,

00:01:49.224 --> 00:01:50.934
and since they're identically sized,

00:01:50.934 --> 00:01:52.779
they'll both share the total charge,

00:01:52.779 --> 00:01:54.038
which means after they touch,

00:01:54.038 --> 00:01:56.008
they'll both have positive four Q.

00:01:56.008 --> 00:01:57.429
If one of the spheres were larger,

00:01:57.429 --> 00:01:59.028
it would gain more of the charge,

00:01:59.028 --> 00:02:01.378
but the total charge
would still be conserved.

00:02:01.378 --> 00:02:04.193
And now when sphere Y
is touched to sphere Z,

00:02:04.193 --> 00:02:06.414
the total charge amongst
them at that moment

00:02:06.414 --> 00:02:09.443
would be positive four
Q plus negative two Q,

00:02:09.443 --> 00:02:10.976
which is positive two Q.

00:02:10.976 --> 00:02:13.493
They would share it equally,
so sphere Y would have

00:02:13.493 --> 00:02:17.029
positive Q, and sphere Z
would also have positive Q.

00:02:17.029 --> 00:02:19.247
So the answer here is C.

00:02:19.247 --> 00:02:22.288
Opposite charges attract
and like charges repel,

00:02:22.288 --> 00:02:24.874
and what Coulomb's Law does
is it gives you a way to find

00:02:24.874 --> 00:02:28.499
the magnitude of the electric
force between two charges.

00:02:28.499 --> 00:02:30.069
The formula for Coulomb's Law says

00:02:30.069 --> 00:02:32.123
that the magnitude of the electric force

00:02:32.123 --> 00:02:36.003
between two charges Q1
and Q2, is gonna equal the

00:02:36.003 --> 00:02:39.203
electric constant K, which is
nine times 10 to the ninth,

00:02:39.203 --> 00:02:42.619
times the product of the two
charges, measured in Coulombs

00:02:42.619 --> 00:02:44.489
divided by the center to center distance

00:02:44.489 --> 00:02:47.260
between those two charges, squared.

00:02:47.260 --> 00:02:49.542
You can't forget to square this distance.

00:02:49.542 --> 00:02:51.612
And it's gotta be in
meters if you want to find

00:02:51.612 --> 00:02:54.151
SI units of Newtons for the force.

00:02:54.151 --> 00:02:56.716
Also, don't rely on the
negative and positive signs

00:02:56.716 --> 00:02:59.626
of the charges to tell you
which way the force points,

00:02:59.626 --> 00:03:02.064
just use the fact that
opposite charges attract

00:03:02.064 --> 00:03:04.986
and like charges repel,
and use Coulomb's Law

00:03:04.986 --> 00:03:07.175
to get the magnitude of the force.

00:03:07.175 --> 00:03:08.500
So what's an example problem involving

00:03:08.500 --> 00:03:09.790
Coulomb's Law look like?

00:03:09.790 --> 00:03:11.653
Let's say two charges
exert an electric force

00:03:11.653 --> 00:03:13.586
of magnitude F on each other.

00:03:13.586 --> 00:03:16.336
What would be the magnitude
of the new electric force

00:03:16.336 --> 00:03:18.749
if the distance between
the charges is tripled

00:03:18.749 --> 00:03:21.687
and the magnitude of one
of the charges is doubled?

00:03:21.687 --> 00:03:23.375
Well we know the formula for Coulomb's Law

00:03:23.375 --> 00:03:25.265
says that the force between two charges

00:03:25.265 --> 00:03:28.026
is the electric constant
times one of the charges,

00:03:28.026 --> 00:03:30.354
times the other charge
divided by the distance

00:03:30.354 --> 00:03:33.321
between them squared, and now
once we triple the distance

00:03:33.321 --> 00:03:36.154
and double a charge,
the new electric force

00:03:36.154 --> 00:03:39.180
is gonna be the electric constant
times one of the charges,

00:03:39.180 --> 00:03:42.719
multiplied by two times
one of the charges,

00:03:42.719 --> 00:03:46.402
divided by three times the
distance, which is squared,

00:03:46.402 --> 00:03:48.148
so I'm gonna get a factor of two on top,

00:03:48.148 --> 00:03:49.926
and this three will get
squared, which gives me

00:03:49.926 --> 00:03:51.761
a factor of nine on the bottom.

00:03:51.761 --> 00:03:54.148
If I pull up those extra
factors I get that the new force

00:03:54.148 --> 00:03:58.588
is gonna be two ninths
multiplied by K, Q1, Q2,

00:03:58.588 --> 00:04:00.993
over D squared, but this entire quantity

00:04:00.993 --> 00:04:03.971
was just the old force F, so the new force

00:04:03.971 --> 00:04:07.731
is going to be two
ninths of the old force.

00:04:07.731 --> 00:04:10.644
The electrical current I tells
you the amount of Coulombs

00:04:10.644 --> 00:04:14.061
of charge that passes a
point in a wire per second.

00:04:14.061 --> 00:04:16.019
So if you watch some point in a wire,

00:04:16.019 --> 00:04:18.113
and you count how many Coulombs of charge

00:04:18.113 --> 00:04:21.328
pass by that point per second,
that would be the current.

00:04:21.328 --> 00:04:23.659
Or in equation form we
can see that the current I

00:04:23.659 --> 00:04:25.628
is gonna be the amount
of charge that flows

00:04:25.628 --> 00:04:27.897
past a point in a wire per time.

00:04:27.897 --> 00:04:30.588
This gives the units of
I as Coulombs per second,

00:04:30.588 --> 00:04:32.848
which we abbreviate as an Ampere.

00:04:32.848 --> 00:04:34.681
And since charge and time aren't vectors,

00:04:34.681 --> 00:04:36.499
current is not a vector either.

00:04:36.499 --> 00:04:37.736
Something that's kind of strange

00:04:37.736 --> 00:04:40.908
is that the so-called
conventional direction of current

00:04:40.908 --> 00:04:43.475
would be the direction that
positive charges flow within a

00:04:43.475 --> 00:04:47.171
wire, however positive charges
don't flow within a wire.

00:04:47.171 --> 00:04:49.458
The only charges that
actually flow in a wire

00:04:49.458 --> 00:04:51.658
are negative charges, but it turns out

00:04:51.658 --> 00:04:55.019
that negative charges flowing
to the left is physically

00:04:55.019 --> 00:04:57.932
the same as positive charges
flowing to the right.

00:04:57.932 --> 00:05:00.498
So in physics problems
we pretend as if it were

00:05:00.498 --> 00:05:02.819
the positive charges moving, however

00:05:02.819 --> 00:05:04.889
it's really the electrons,
which are negative,

00:05:04.889 --> 00:05:06.464
that are moving within the wire.

00:05:06.464 --> 00:05:07.410
So what's an example problem

00:05:07.410 --> 00:05:09.295
involving electrical current look like?

00:05:09.295 --> 00:05:11.387
Let's say three amps
flows within a circuit.

00:05:11.387 --> 00:05:14.014
How much charge would pass
by a point in that wire

00:05:14.014 --> 00:05:16.505
during a time interval of five minutes?

00:05:16.505 --> 00:05:17.904
Well we know the definition of current

00:05:17.904 --> 00:05:20.341
is the charge per time,
that means the charge

00:05:20.341 --> 00:05:23.285
is gonna be the amount of
current multiplied by the time,

00:05:23.285 --> 00:05:25.303
so we take our current of three amps,

00:05:25.303 --> 00:05:27.305
and we multiply by the time, but we can't

00:05:27.305 --> 00:05:30.295
multiply by five because
that's in units of minutes,

00:05:30.295 --> 00:05:32.486
since amps is Coulombs per second,

00:05:32.486 --> 00:05:34.535
we've got to convert five
minutes into seconds,

00:05:34.535 --> 00:05:37.934
which would be five minutes,
multiplied by 60 seconds

00:05:37.934 --> 00:05:39.833
per minute, which would
give us a total amount

00:05:39.833 --> 00:05:42.000
of charge of 900 Coulombs.

00:05:42.890 --> 00:05:45.794
The resistance of a circuit
element measures how much

00:05:45.794 --> 00:05:48.703
that element will restrict
the flow of current.

00:05:48.703 --> 00:05:51.132
The larger the resistance,
the less current

00:05:51.132 --> 00:05:52.763
there will be allowed to flow.

00:05:52.763 --> 00:05:55.978
And this definition of
resistance is given by Ohm's Law.

00:05:55.978 --> 00:05:57.794
Ohm's Law states that
the amount of current

00:05:57.794 --> 00:05:59.759
that you'll get through
a portion of a circuit,

00:05:59.759 --> 00:06:01.476
it's gonna be proportional to the voltage

00:06:01.476 --> 00:06:04.371
across that portion,
divided by the resistance

00:06:04.371 --> 00:06:06.006
of that portion of the circuit.

00:06:06.006 --> 00:06:08.331
So between these two points,
the amount of current

00:06:08.331 --> 00:06:10.198
that will flow, is gonna be equal to

00:06:10.198 --> 00:06:12.536
the voltage between those two points,

00:06:12.536 --> 00:06:15.604
divided by the resistance
between those two points.

00:06:15.604 --> 00:06:17.889
So the larger the
resistance, the less current

00:06:17.889 --> 00:06:20.958
will flow, but the greater
the voltage supplied,

00:06:20.958 --> 00:06:22.418
the greater the current will be.

00:06:22.418 --> 00:06:24.107
And this is what Ohm's Law says.

00:06:24.107 --> 00:06:26.316
Even though Ohm's Law
gives you a way to define

00:06:26.316 --> 00:06:29.082
the resistance, you can
determine the resistance

00:06:29.082 --> 00:06:31.043
of a circuit element by knowing the size

00:06:31.043 --> 00:06:32.949
and shape of that circuit element.

00:06:32.949 --> 00:06:35.568
In other words, the resistance
of a cylindrical resistor,

00:06:35.568 --> 00:06:37.872
is gonna be equal to the resistivity,

00:06:37.872 --> 00:06:40.214
which is a measure of an
object's natural resistance

00:06:40.214 --> 00:06:43.126
to current, multiplied by
the length of that resistor,

00:06:43.126 --> 00:06:45.768
the longer the resistor,
the greater the resistance

00:06:45.768 --> 00:06:48.072
and the more it will
resist the flow of current,

00:06:48.072 --> 00:06:50.199
and then divide it by
the cross sectional area

00:06:50.199 --> 00:06:52.860
of that resistor, which would
be this area right here,

00:06:52.860 --> 00:06:55.583
the current is either
flowing into or out of.

00:06:55.583 --> 00:06:58.450
If the resistor is cylindrical,
the area of this circle

00:06:58.450 --> 00:07:01.313
would be Pi times r
squared, where little r

00:07:01.313 --> 00:07:04.356
would be the radius of
this cross sectional area.

00:07:04.356 --> 00:07:07.521
The units of resistance is
Ohms, and it is not a vector.

00:07:07.521 --> 00:07:09.746
It is always positive or zero.

00:07:09.746 --> 00:07:11.888
So what's an example
problem involving Ohm's Law,

00:07:11.888 --> 00:07:14.699
or the resistance of a
cylindrical resistor look like?

00:07:14.699 --> 00:07:16.776
Let's say a battery of
voltage V is hooked up

00:07:16.776 --> 00:07:19.744
to a single cylindrical
resistor of length L

00:07:19.744 --> 00:07:21.813
and radius little r, and when that's done,

00:07:21.813 --> 00:07:24.482
a current I is flowing
through the battery.

00:07:24.482 --> 00:07:27.913
What is the resistivity
Rho, of that resistor?

00:07:27.913 --> 00:07:30.278
Well we know Ohm's Law states
that the current that flows

00:07:30.278 --> 00:07:32.446
through a portion of a
circuit will be equal to

00:07:32.446 --> 00:07:33.972
the voltage across that portion,

00:07:33.972 --> 00:07:36.278
divided by the resistance of that portion.

00:07:36.278 --> 00:07:38.603
And this means the resistance
of this resistor is gonna be

00:07:38.603 --> 00:07:40.883
the voltage of the battery
divided by the current.

00:07:40.883 --> 00:07:43.268
To factor resistivity
into this, we have to use

00:07:43.268 --> 00:07:46.153
the formula for the resistance
of a cylindrical resistor,

00:07:46.153 --> 00:07:48.222
which is Rho times L over A.

00:07:48.222 --> 00:07:50.637
This gives us the
resistance of the resistor,

00:07:50.637 --> 00:07:52.735
which is gotta equal V over I,

00:07:52.735 --> 00:07:55.023
and now we can solve
for the resistivity Rho.

00:07:55.023 --> 00:07:58.381
We get V times A over I
L, but since we're given

00:07:58.381 --> 00:08:01.117
the radius little r,
we gotta write the area

00:08:01.117 --> 00:08:03.204
in terms of that radius,
so this is gonna be

00:08:03.204 --> 00:08:06.858
V times Pi, r squared,
divided by I times L,

00:08:06.858 --> 00:08:09.210
which gives us an answer of C.

00:08:09.210 --> 00:08:12.285
When dealing with complicated
circuits with many resistors,

00:08:12.285 --> 00:08:14.214
you often have to reduce those resistors

00:08:14.214 --> 00:08:17.098
into smaller, equivalent
amounts of resistors.

00:08:17.098 --> 00:08:18.862
And the two ways you
do this are by finding

00:08:18.862 --> 00:08:21.680
two resistors that are
in series or in parallel.

00:08:21.680 --> 00:08:24.164
Resistors will be in
series if the same current

00:08:24.164 --> 00:08:25.891
that flows through the same resistor,

00:08:25.891 --> 00:08:27.688
flows through the next resistor.

00:08:27.688 --> 00:08:30.157
If the current branched
off in between them,

00:08:30.157 --> 00:08:32.791
these resistors would
no longer be in series,

00:08:32.791 --> 00:08:34.153
but if they're in series you can find

00:08:34.153 --> 00:08:37.044
the equivalent resistance
of this section of wire

00:08:37.044 --> 00:08:40.036
by just adding up the two
individual resistances.

00:08:40.036 --> 00:08:41.777
So the current for resistors in series

00:08:41.777 --> 00:08:45.112
must be the same, but the
voltage might be different,

00:08:45.112 --> 00:08:47.244
since they could have
different resistances.

00:08:47.244 --> 00:08:49.104
Two resistors will be in parallel,

00:08:49.104 --> 00:08:52.000
if a current comes in,
splits into two parts,

00:08:52.000 --> 00:08:54.803
goes through one resistor
each, and then rejoins

00:08:54.803 --> 00:08:56.818
before hitting anything
else in the circuit,

00:08:56.818 --> 00:08:58.768
and if this is the case,
you can find the equivalent

00:08:58.768 --> 00:09:00.882
resistance of this portion of the circuit,

00:09:00.882 --> 00:09:04.093
i.e. between these two
points, by saying that one

00:09:04.093 --> 00:09:06.519
over the equivalent
resistance is gonna equal

00:09:06.519 --> 00:09:08.893
one over the resistance
of the first resistor,

00:09:08.893 --> 00:09:11.556
plus one over the resistance
of the second resistor.

00:09:11.556 --> 00:09:14.275
But be careful, one
over R1 plus one over R2

00:09:14.275 --> 00:09:16.544
just gives you one over R equivalent.

00:09:16.544 --> 00:09:18.663
If you want R equivalent,
you're gonna have to take

00:09:18.663 --> 00:09:22.627
one over this entire side,
in order to get R equivalent.

00:09:22.627 --> 00:09:24.604
So what's an example
problem involving resistors

00:09:24.604 --> 00:09:26.386
in series and parallel look like?

00:09:26.386 --> 00:09:28.357
Let's say we have this
circuit shown below,

00:09:28.357 --> 00:09:30.164
and we want to know what current flows

00:09:30.164 --> 00:09:31.726
through the eight Ohm resistor.

00:09:31.726 --> 00:09:33.280
Now you might be tempted to say this,

00:09:33.280 --> 00:09:36.861
since Ohm's Law says that the
current is delta V over R,

00:09:36.861 --> 00:09:38.729
we can just plug in the
voltage of the battery,

00:09:38.729 --> 00:09:41.330
which is 24 volts,
divided by the resistance

00:09:41.330 --> 00:09:43.469
of the resistor, which is eight Ohms,

00:09:43.469 --> 00:09:45.220
and that would give us three Amps.

00:09:45.220 --> 00:09:46.487
But that's not right at all.

00:09:46.487 --> 00:09:48.549
When using Ohm's Law,
the current that flows

00:09:48.549 --> 00:09:51.511
through a resistor R, is gonna be equal to

00:09:51.511 --> 00:09:54.382
the voltage across that
resistor divided by

00:09:54.382 --> 00:09:56.406
the resistance of that resistor.

00:09:56.406 --> 00:09:59.382
So if we plug eight Ohms into
the denominator, we've gotta

00:09:59.382 --> 00:10:02.567
plug in the voltage across
that eight Ohm resistor.

00:10:02.567 --> 00:10:04.456
But the voltage across
the eight Ohm resistor

00:10:04.456 --> 00:10:07.556
is not gonna be the full
24 volts of the battery,

00:10:07.556 --> 00:10:09.595
it's gonna be less than 24 volts.

00:10:09.595 --> 00:10:12.056
In other words, the battery
provides a voltage between

00:10:12.056 --> 00:10:15.144
this point and this point of 24 volts,

00:10:15.144 --> 00:10:16.611
but there's gonna be voltage drops

00:10:16.611 --> 00:10:18.855
across the six and 12 Ohm resistors,

00:10:18.855 --> 00:10:20.138
which make it so that the voltage

00:10:20.138 --> 00:10:21.776
across the eight Ohm resistor is not

00:10:21.776 --> 00:10:23.701
gonna be the full 24 volts.

00:10:23.701 --> 00:10:26.396
So we have to reduce these
resistors to a single resistance.

00:10:26.396 --> 00:10:28.335
The six and the 12 are in parallel,

00:10:28.335 --> 00:10:31.351
so we can say that one
over six, plus one over 12,

00:10:31.351 --> 00:10:32.907
would equal one over the resistance

00:10:32.907 --> 00:10:34.400
of that portion of the circuit.

00:10:34.400 --> 00:10:37.540
This is gonna equal three
twelves, which is one fourth,

00:10:37.540 --> 00:10:39.280
so that means that parallel
portion of the circuit

00:10:39.280 --> 00:10:41.857
has an equivalent resistance of four Ohms.

00:10:41.857 --> 00:10:44.159
So between this point and this point,

00:10:44.159 --> 00:10:46.184
there are four Ohms of resistance,

00:10:46.184 --> 00:10:48.469
and that equivalent
resistance is in series

00:10:48.469 --> 00:10:50.141
with this eight Ohm resistor.

00:10:50.141 --> 00:10:51.652
So we can add four and eight,

00:10:51.652 --> 00:10:54.217
and get 12 Ohms of total resistance.

00:10:54.217 --> 00:10:57.720
And now I can say that the
full 24 volts of the battery

00:10:57.720 --> 00:11:00.135
is applied across this
entire equivalent resistance

00:11:00.135 --> 00:11:03.685
of 12 Ohms, so if I come up
here and change this eight Ohms

00:11:03.685 --> 00:11:06.980
to 12 Ohms of equivalent
resistance for the total circuit,

00:11:06.980 --> 00:11:08.440
I'll get the correct current that flows

00:11:08.440 --> 00:11:10.322
through the battery of two Amps.

00:11:10.322 --> 00:11:12.143
And since that's the current
that's flowing through

00:11:12.143 --> 00:11:13.879
the battery, that had to be the current

00:11:13.879 --> 00:11:16.343
that's flowing through the
eight Ohm resistor as well.

00:11:16.343 --> 00:11:20.059
Since this eight Ohm resistor
and the batter are in series.

00:11:20.059 --> 00:11:23.152
Elements in a circuit
often use Electrical Power.

00:11:23.152 --> 00:11:25.892
That is to say, when current
runs through a resistor,

00:11:25.892 --> 00:11:27.726
the electrons moving through that resistor

00:11:27.726 --> 00:11:30.428
turn some of their
electrical potential energy

00:11:30.428 --> 00:11:33.766
into energies like heat, light, or sound.

00:11:33.766 --> 00:11:35.408
And the rate at which these electrons

00:11:35.408 --> 00:11:37.978
are turning their energy
into other forms of energy,

00:11:37.978 --> 00:11:39.854
is called the electrical power.

00:11:39.854 --> 00:11:41.815
So the rate at which a resistor is turning

00:11:41.815 --> 00:11:43.974
electrical potential energy into heat,

00:11:43.974 --> 00:11:46.481
is the electrical power
used by that resistor.

00:11:46.481 --> 00:11:47.940
In other words, the amount of energy

00:11:47.940 --> 00:11:50.084
converted into heat, divided by the time

00:11:50.084 --> 00:11:52.380
it took to convert that
energy, is the definition

00:11:52.380 --> 00:11:54.358
of the power, and there's
a way to determine

00:11:54.358 --> 00:11:57.404
this number of Joules per
second, in terms of quantities

00:11:57.404 --> 00:12:00.199
like the current, the
voltage, and the resistance.

00:12:00.199 --> 00:12:02.425
The power used by a
resistor can be written as

00:12:02.425 --> 00:12:04.410
the current through that resistor

00:12:04.410 --> 00:12:07.436
multiplied by the voltage
across that resistor,

00:12:07.436 --> 00:12:10.661
or if you substituted Ohm's
Law into this formula,

00:12:10.661 --> 00:12:12.181
you see that this is equivalent to

00:12:12.181 --> 00:12:14.319
the current through that resistor squared,

00:12:14.319 --> 00:12:16.759
multiplied by the
resistance of the resistor,

00:12:16.759 --> 00:12:18.132
or we could rearrange these formulas

00:12:18.132 --> 00:12:20.025
to get that the power used by a resistor

00:12:20.025 --> 00:12:23.124
would also be the voltage
across that resistor squared,

00:12:23.124 --> 00:12:25.521
divided by the resistance
of that resistor.

00:12:25.521 --> 00:12:27.540
All three of these, if used correctly,

00:12:27.540 --> 00:12:30.130
will give you the same
number for the power used

00:12:30.130 --> 00:12:32.074
by a resistor, and if
you wanted to determine

00:12:32.074 --> 00:12:34.605
the number of Joules of
heat energy converted,

00:12:34.605 --> 00:12:37.054
you could set any one of
these equal to the amount

00:12:37.054 --> 00:12:40.122
of energy per time, and
solve for that energy.

00:12:40.122 --> 00:12:42.561
The units of Electrical Power
are the same as the regular

00:12:42.561 --> 00:12:46.262
units of power, which is
Watts, i.e. Joules per second,

00:12:46.262 --> 00:12:48.333
and Electrical Power is not a vector.

00:12:48.333 --> 00:12:49.836
So what's an example problem involving

00:12:49.836 --> 00:12:51.452
Electrical Power look like?

00:12:51.452 --> 00:12:53.296
Let's say a light bulb of resistance R

00:12:53.296 --> 00:12:55.416
is hooked up to a source of voltage V,

00:12:55.416 --> 00:12:57.694
and a second light bulb of resistance 2R,

00:12:57.694 --> 00:13:00.246
is hooked up to a source of voltage 2V.

00:13:00.246 --> 00:13:02.786
How does the power used
by the second light bulb

00:13:02.786 --> 00:13:05.285
compare to the power used
by the first light bulb?

00:13:05.285 --> 00:13:07.553
Since we have the
information about R and V,

00:13:07.553 --> 00:13:09.382
I'll use the version of the power formula

00:13:09.382 --> 00:13:11.645
that says that the
power used by a resistor

00:13:11.645 --> 00:13:13.980
is gonna be delta V squared over R.

00:13:13.980 --> 00:13:15.937
So in terms of quantities
given the power used

00:13:15.937 --> 00:13:18.797
by the first light bulb is
gonna be V squared over R.

00:13:18.797 --> 00:13:21.225
And the power used by the
second light bulb is gonna be

00:13:21.225 --> 00:13:23.779
equal to the voltage across
the second light bulb,

00:13:23.779 --> 00:13:26.852
which is two times the voltage
across the first light bulb,

00:13:26.852 --> 00:13:28.953
and we square that,
divided by the resistance

00:13:28.953 --> 00:13:30.678
of the second light
bulb, which is gonna be

00:13:30.678 --> 00:13:33.015
two times the resistance
of the first light bulb.

00:13:33.015 --> 00:13:35.547
The two squared on top is
gonna give me a factor of four,

00:13:35.547 --> 00:13:37.708
and I'll have another
factor of two on the bottom.

00:13:37.708 --> 00:13:40.032
So if I factor out this
four divided by two,

00:13:40.032 --> 00:13:42.269
I get that the power used
by the second light bulb

00:13:42.269 --> 00:13:44.666
is gonna be two times V squared over R,

00:13:44.666 --> 00:13:46.663
but V squared over R was the power used

00:13:46.663 --> 00:13:48.658
by the first light bulb, so the power used

00:13:48.658 --> 00:13:50.653
by the second light bulb is gonna be two

00:13:50.653 --> 00:13:52.838
times the power used by
the first light bulb,

00:13:52.838 --> 00:13:54.681
and so if the light
bulb of resistance two R

00:13:54.681 --> 00:13:57.662
has twice the power, and
that means it'll be brighter.

00:13:57.662 --> 00:13:59.591
The quantity that
determines the brightness

00:13:59.591 --> 00:14:03.124
of a light bulb, is the electrical
power of that light bulb.

00:14:03.124 --> 00:14:06.318
It's not necessarily the
resistance or the voltage,

00:14:06.318 --> 00:14:07.821
it's the combination of the two

00:14:07.821 --> 00:14:10.895
in this formula that will
tell you the electrical power,

00:14:10.895 --> 00:14:13.649
and therefore the brightness
of the light bulb.

00:14:13.649 --> 00:14:15.519
Two of the most useful ideas in circuits

00:14:15.519 --> 00:14:17.767
are referred to as Kirchhoff's Rules.

00:14:17.767 --> 00:14:19.854
The first rule is called
the Junction rule,

00:14:19.854 --> 00:14:22.040
and it states that all the
current entering a junction

00:14:22.040 --> 00:14:25.140
must equal all the current
exiting that junction.

00:14:25.140 --> 00:14:26.961
In other words, if you add
all the current that flows

00:14:26.961 --> 00:14:28.997
into a junction, that has to equal

00:14:28.997 --> 00:14:31.260
all the current that flows
out of that junction,

00:14:31.260 --> 00:14:33.263
because current is just flowing charge,

00:14:33.263 --> 00:14:35.897
and charge is conserved,
so charge can't be

00:14:35.897 --> 00:14:38.964
created or destroyed at
any point in the circuit.

00:14:38.964 --> 00:14:41.506
No more than water can
get created or destroyed

00:14:41.506 --> 00:14:43.213
within a series of pipes.

00:14:43.213 --> 00:14:45.080
And the second rule is
called the Loop rule,

00:14:45.080 --> 00:14:47.515
which states that if you
add up all the changes

00:14:47.515 --> 00:14:51.235
in electric potential, i.e.
voltages around any closed

00:14:51.235 --> 00:14:54.238
loop in a circuit, it'll
always add up to zero.

00:14:54.238 --> 00:14:56.464
So if you add up all
the voltages encountered

00:14:56.464 --> 00:14:58.191
through a closed loop through a circuit,

00:14:58.191 --> 00:14:59.768
it always adds up to zero.

00:14:59.768 --> 00:15:02.273
And this is just a result
of conservation of energy.

00:15:02.273 --> 00:15:04.408
The electrons will gain
energy when they flow

00:15:04.408 --> 00:15:06.410
through the battery,
and they'll lose energy

00:15:06.410 --> 00:15:08.396
every time they flow through a resistor,

00:15:08.396 --> 00:15:09.919
but the total amount of energy they gain

00:15:09.919 --> 00:15:12.443
from the battery, has to be
equal to the total amount

00:15:12.443 --> 00:15:14.956
of energy they lose due to the resistors.

00:15:14.956 --> 00:15:17.345
In other words, if we
consider a complicated circuit

00:15:17.345 --> 00:15:19.759
that has a batter and three resistors,

00:15:19.759 --> 00:15:23.125
the total current flowing
into a junction I1,

00:15:23.125 --> 00:15:25.065
will have to be equal to the total current

00:15:25.065 --> 00:15:28.293
coming out of that junction, I2 and I3.

00:15:28.293 --> 00:15:30.951
Since no charge gets created or destroyed.

00:15:30.951 --> 00:15:32.986
And that means when these
two currents combine again,

00:15:32.986 --> 00:15:34.482
the total current flowing out

00:15:34.482 --> 00:15:37.194
of that section is gonna again be I1.

00:15:37.194 --> 00:15:39.990
And if we follow a closed
loop through this circuit,

00:15:39.990 --> 00:15:42.372
the sum of all the
voltages around that loop

00:15:42.372 --> 00:15:45.919
have to add up to zero, i.e.
the voltage of the battery

00:15:45.919 --> 00:15:48.939
minus the voltage drop
across the first resistor,

00:15:48.939 --> 00:15:51.784
minus the voltage drop
across the second resistor,

00:15:51.784 --> 00:15:53.480
would have to equal zero.

00:15:53.480 --> 00:15:54.894
So what's an example problem involving

00:15:54.894 --> 00:15:56.496
Kirchhoff's Rules look like?

00:15:56.496 --> 00:15:58.513
Let's say we have the circuit
below and we wanted to

00:15:58.513 --> 00:16:01.702
determine the voltage
across the six Ohm resistor.

00:16:01.702 --> 00:16:03.409
To do this, we could use the Loop rule,

00:16:03.409 --> 00:16:05.633
I'll start behind the
battery, and I'll go through

00:16:05.633 --> 00:16:08.392
the resistor, I want to
determine the voltage across.

00:16:08.392 --> 00:16:10.932
I'll add up all the
voltages across that loop,

00:16:10.932 --> 00:16:12.364
and set it equal to zero.

00:16:12.364 --> 00:16:13.556
So the voltage across the battery

00:16:13.556 --> 00:16:15.703
is gonna be positive 24 volts,

00:16:15.703 --> 00:16:18.276
minus the voltage across
the six Ohm resistor,

00:16:18.276 --> 00:16:19.745
and then minus the voltage across the

00:16:19.745 --> 00:16:22.220
eight Ohm resistor has to equal zero.

00:16:22.220 --> 00:16:24.676
But we're given this current,
so we know that two Amps

00:16:24.676 --> 00:16:26.706
flows through the eight Ohm resistor,

00:16:26.706 --> 00:16:27.847
and you can always determine the voltage

00:16:27.847 --> 00:16:29.939
across the resistor using Ohm's Law,

00:16:29.939 --> 00:16:31.870
so the voltage across
the eight Ohm resistor

00:16:31.870 --> 00:16:33.870
is gonna be two Amps, which is flowing

00:16:33.870 --> 00:16:35.436
through the eight Ohm resistor,

00:16:35.436 --> 00:16:39.129
multiplied by eight Ohms,
and we get 16 volts.

00:16:39.129 --> 00:16:41.837
Which I can plug into
here, and this gives me

00:16:41.837 --> 00:16:45.686
24 volts minus the voltage
across the six Ohm resistor,

00:16:45.686 --> 00:16:48.389
minus 16, has to equal zero.

00:16:48.389 --> 00:16:49.583
And if I solve this for the voltage

00:16:49.583 --> 00:16:52.253
across the six Ohm
resistor, I get 24 volts

00:16:52.253 --> 00:16:54.871
minus 16 volts, which is eight volts.

00:16:54.871 --> 00:16:56.721
So the voltage across the six Ohm resistor

00:16:56.721 --> 00:16:58.308
would be eight volts.

00:16:58.308 --> 00:17:01.392
Note, because the 12 Ohm
resistor and the six Ohm resistor

00:17:01.392 --> 00:17:04.095
are in parallel, the voltage
across the 12 Ohm resistor

00:17:04.095 --> 00:17:07.078
would also be eight volts,
because the voltage across

00:17:07.078 --> 00:17:11.633
any two elements in parallel,
have to be the same.

00:17:11.633 --> 00:17:13.549
Voltmeters are the device that you use

00:17:13.549 --> 00:17:16.398
to measure the voltage between
two points in a circuit.

00:17:16.398 --> 00:17:18.113
When hooking up a voltmeter you've gotta

00:17:18.113 --> 00:17:20.580
hook it up in parallel
between the two points

00:17:20.580 --> 00:17:22.600
you wanna find the voltage across.

00:17:22.600 --> 00:17:24.015
In other words, to determine the voltage

00:17:24.015 --> 00:17:26.388
between this point and
this point, which would be

00:17:26.388 --> 00:17:29.235
the voltage across R3, you
would hook up the voltmeter

00:17:29.235 --> 00:17:31.465
in parallel with R3.

00:17:31.465 --> 00:17:33.351
Ammeters are the devices we use to measure

00:17:33.351 --> 00:17:34.908
the electrical current that pass

00:17:34.908 --> 00:17:36.780
through a point in a circuit, and ammeters

00:17:36.780 --> 00:17:39.595
have to be hooked up in series
with the circuit element

00:17:39.595 --> 00:17:41.373
you want to determine the current through.

00:17:41.373 --> 00:17:43.261
In other words, if we wanted
to determine the current

00:17:43.261 --> 00:17:47.267
through R1, we would hook up
the ammeter in series with R1.

00:17:47.267 --> 00:17:49.877
Note that for these electrical
devices to work well,

00:17:49.877 --> 00:17:53.185
the ammeter should have almost
zero internal resistance,

00:17:53.185 --> 00:17:55.407
thereby not affecting
the current that flows

00:17:55.407 --> 00:17:57.479
through the circuit, and
voltmeters should have

00:17:57.479 --> 00:18:00.440
near infinite resistance,
so that it doesn't draw

00:18:00.440 --> 00:18:02.674
any of the current from the resistor.

00:18:02.674 --> 00:18:04.969
In reality, ammeters have a very small,

00:18:04.969 --> 00:18:07.148
but non-zero internal resistance,

00:18:07.148 --> 00:18:08.463
and voltmeters have a very high,

00:18:08.463 --> 00:18:10.653
but not infinite internal resistance.

00:18:10.653 --> 00:18:12.107
So what would an example problem involving

00:18:12.107 --> 00:18:14.108
voltmeters and ammeters look like?

00:18:14.108 --> 00:18:15.865
Let's say we have the circuit shown below,

00:18:15.865 --> 00:18:18.558
and these numbered circles
represent possible places

00:18:18.558 --> 00:18:21.173
we could stick a voltmeter
to measure the voltage

00:18:21.173 --> 00:18:23.165
across the eight Ohm resistor.

00:18:23.165 --> 00:18:25.380
Which two of these
voltmeters would correctly

00:18:25.380 --> 00:18:28.079
give the voltage across
the eight Ohm resistor?

00:18:28.079 --> 00:18:30.124
And you have to be
careful, some AP problems

00:18:30.124 --> 00:18:31.844
are gonna require you to select

00:18:31.844 --> 00:18:33.838
two correct answers for
the multiple choice,

00:18:33.838 --> 00:18:36.509
so be sure to read the
instructions carefully.

00:18:36.509 --> 00:18:38.229
Voltmeter number four
is a terrible choice,

00:18:38.229 --> 00:18:40.368
you never hook up your
voltmeter in series,

00:18:40.368 --> 00:18:42.331
but the circuit element
you're trying to find

00:18:42.331 --> 00:18:44.382
the voltage across, and
voltmeter number one

00:18:44.382 --> 00:18:46.524
is doing nothing really,
because its' measuring

00:18:46.524 --> 00:18:48.989
the voltage between two points in a wire

00:18:48.989 --> 00:18:51.206
with nothing in between that wire.

00:18:51.206 --> 00:18:52.952
So the voltage measured by voltmeter one

00:18:52.952 --> 00:18:56.034
should just be zero, since
the voltage across a wire

00:18:56.034 --> 00:18:59.412
of zero resistance should
just give you zero volts.

00:18:59.412 --> 00:19:01.627
So the correct choices would
be voltmeter number two,

00:19:01.627 --> 00:19:04.736
which gives you the voltage
across the eight Ohm resistor,

00:19:04.736 --> 00:19:06.429
and voltmeter number
three, which also gives you

00:19:06.429 --> 00:19:08.304
an equivalent measurement of the voltage

00:19:08.304 --> 00:19:10.870
across the resistor eight Ohms.