Data to justify experimental claims examples

WEBVTT
Kind: captions
Language: en

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- [Instructor] What we have
here are a few data analysis

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questions in a biology context

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from the New York Regents Exam,

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but these are useful example problems

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if you're studying high
school biology in general

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because they might show up in some type

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of an exam that your teacher gives you

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or you might see on a standardized test.

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A student proposes that,
if volunteers warm up

00:00:19.370 --> 00:00:22.620
before squeezing a
clothespin for one minute,

00:00:22.620 --> 00:00:24.780
they will increase the number of times

00:00:24.780 --> 00:00:27.570
that they can squeeze it within tiring.

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He states that this is
because their muscles will

00:00:30.550 --> 00:00:32.980
be better prepared for the exercise.

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The data from an experiment are shown

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in the data table below.

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Which trial from the chart
above provides the best data

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to support his claim?

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Support your answer.

00:00:44.510 --> 00:00:47.690
So, pause this video and
see if you could answer it.

00:00:47.690 --> 00:00:49.750
So, let's just make sure we
understand what's going on here.

00:00:49.750 --> 00:00:51.700
So, there's two trials.

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This is the first trial,
trial one, right over here,

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and in trial one, you had
two groups of 10 students.

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One of them did warm up and
the other 10 did not warm up,

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and then in trial two,
you had two more groups

00:01:07.900 --> 00:01:10.170
and this time, they were groups of 25.

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Once again, 25 who did warm
up and 25 who didn't warm up.

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Now, in trial one, the
group that warmed up

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actually had a lower
number of average squeezes,

00:01:22.360 --> 00:01:25.070
so this goes against the
hypothesis that the student had.

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The hypothesis the student has
is if they warm up beforehand

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that you would actually be
able to do more squeezes,

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so trial one actually goes
against the student's claim.

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Trial two seems to slightly go in favor

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of the student's claim.

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The group that warmed up was
able to eke out a few more.

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Now, we haven't done
the statistical analysis

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to know how probable this is,

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if it was due purely to random chance,

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but just to answer that question,

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one, the data goes, trial
one goes a little bit

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against the student's claim.

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Trial two best supports this claim,

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although it doesn't do it so strongly.

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So, I would say trial two

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and that is because the 25 that warmed,

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warmed up, that is group three,

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group three, were able,

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on average, on average,

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to squeeze more in a minute,

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to squeeze more

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in the minute

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than group four

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that didn't warm up.

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Alright, let's do another question here.

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So, here we're told, students
were asked to design a lab

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that investigated the
relationship between exercise

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and heart rate.

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Heart rate was determined by recording

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the pulse rate in beats per minute.

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The students hypothesized
that increased exercise

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results in an increased heart rate.

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The class results for
the experiment are shown

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in the graph below.

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And then they say, which
statement is best supported

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by this graph?

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And we're given four choices.

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So, pause this video and see if you could

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answer this on your own.

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Alright, now let's go through
statement by statement.

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Statement one, before exercising,

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the average pulse rate was 65.

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Four minutes after exercising,

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the average pulse rate was 65.

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So, let's see.

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Before exercising, the
average pulse rate was 65.

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So, before exercising, this
is time period right over here

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and it does look like over most

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of that time period,
let me do this in red,

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it does look like that is reasonably,

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that is roughly true.

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It looks like the average
pulse rate was 65,

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but now let's read the
second part of the statement.

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Four minutes after exercising,

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the average pulse rate was 65 as well.

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Four minutes after exercising,

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what's the average pulse rate?

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Well, it sure isn't 65, it looks closer

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to being, maybe, 135, not 65,
so we could rule this one out.

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After four minutes of exercising,

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the average pulse rate was 120.

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Let's see.

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So, this is when the exercise starts.

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After four minutes, the
average pulse rate was 120.

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No, it looks like it was like
135 or something like this,

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so this is already looking fishy.

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Two minutes after exercising,

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the average pulse rate was 120.

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So, two minutes after exercising,

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the average pulse rate actually
seems closer to about 90,

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so both of those look fishy,
I'll rule that one out.

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While exercising, the average,

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the highest average pulse rate was 150.

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Before exercising, the
average pulse rate was 65.

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So, let's see, while exercising,

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the highest average pulse
rate happens right over there

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and that does indeed
look pretty close to 150,

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so this part's looking good.

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Before exercising, the
average pulse rate was 65.

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Yeah, we already talked about
this period right over here.

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So, this is looking good,
I'm liking this choice,

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but let's see choice four.

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Two minutes before exercising,

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the average pulse rate was 80.

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So, two minutes before exercising,

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this is when we start exercising

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and each of these hash marks is a minute,

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so two minutes before, the
average pulse rate was not 80,

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so that's false.

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After two minutes of exercise,

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the average pulse rate was 140.

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After two minutes, the average
pulse rate was also not 140,

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so both of those
statements are not correct.

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So, I like statement three,
that's the one that seems

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to be backed up by the data here.


Molecular evidence for evolutionary relationships examples

WEBVTT
Kind: captions
Language: en

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- [Instructor] An
investigation was carried out

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on four different plant species

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to determine which of three species

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was most closely related to
an unknown plant species.

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The results of the investigation

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are shown in the data table below.

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Which plant species appears
to be most closely related

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to the unknown species?

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Support your answer.

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So pause this video and see
if you can figure that out.

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Alright, so here in
this data the first row

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is for our unknown species.

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We know that it tests
positive for Enzyme M.

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This is blanked out because this says,

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how many differences in amino
acid sequences are there

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between the species in that
row and the unknown species,

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and of course the unknown species

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would have zero differences
'cause it is itself.

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And then this is the gel
electrophoresis banding pattern.

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So what we wanna do is look for

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as many similarities as possible

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to our unknown plant species.

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So first, if we looked at the enzyme,

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whether we test positive for it,

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species one does not test positive for it,

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species two and three do.

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So species one is not looking good.

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If we look at differences
in amino acid sequences,

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species one, once again,
has the most differences

00:01:08.720 --> 00:01:09.810
in amino acid sequences,

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so that confirms our belief

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that it might be the most different

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from the unknown plant species.

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But then if we look between
plant species two and three,

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plant species two has fewer differences.

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So right now, plant species two

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is looking like the leading candidate.

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Plant species three has more differences.

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And then when we look

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at the gel electrophoresis
banding pattern,

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plant species two has
the exact same pattern

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as our unknown plant,

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and it's the only one with
the exact same pattern.

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And so because of that,

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I'm feeling really good saying

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that two is most closely
related to the unknown species.

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Two, because same banding pattern,

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and we could say, and least
different amino acid sequences,

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and tests positive for Enzyme M.

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Let's do another question like this.

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Scientists attempted to determine

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the evolutionary relationships

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between three different
finch species, A, B, and C.

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In order to do this,

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they examined the physical characteristics

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and DNA of these species.

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DNA was extracted from all three species

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and analyzed using gel electrophoresis.

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The results are shown in the diagram.

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And then they ask us,

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which statement best describes
the method used above

00:02:56.870 --> 00:02:59.700
to determine the
evolutionary relationships

00:02:59.700 --> 00:03:02.500
between three species of finches?

00:03:02.500 --> 00:03:03.790
So the first choice is,

00:03:03.790 --> 00:03:07.530
examine the structure of
the beaks and compare them.

00:03:07.530 --> 00:03:09.170
Statement two, observe behavioral

00:03:09.170 --> 00:03:11.760
and physical characteristics
of all the finches

00:03:11.760 --> 00:03:13.720
and group them by similarities.

00:03:13.720 --> 00:03:16.570
C, obtain molecular evidence
from all three species

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and identify similarities.

00:03:18.540 --> 00:03:22.660
Four, compare common ancestors
of all three of the species

00:03:22.660 --> 00:03:24.980
and see if they are the same.

00:03:24.980 --> 00:03:29.113
So pause this video and
see which choice you like.

00:03:30.360 --> 00:03:33.740
Alright, well clearly they're
looking at molecular evidence.

00:03:33.740 --> 00:03:36.370
They're taking the DNA
of the different species,

00:03:36.370 --> 00:03:37.480
they're amplifying them,

00:03:37.480 --> 00:03:39.020
which means they're
making a bunch of copies

00:03:39.020 --> 00:03:41.780
of certain sequences of that DNA,

00:03:41.780 --> 00:03:43.140
and then they're running a gel,

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which means they allow
those different segments

00:03:45.140 --> 00:03:47.260
to separate on this gel.

00:03:47.260 --> 00:03:49.690
The smaller segments are
going to migrate down further,

00:03:49.690 --> 00:03:52.740
the larger segments are
going to migrate less far.

00:03:52.740 --> 00:03:54.430
And the more similarities you have

00:03:54.430 --> 00:03:57.830
on this gel electrophoresis
right over here,

00:03:57.830 --> 00:03:59.133
this banding pattern,

00:03:59.133 --> 00:04:04.133
the more similar that DNA probably is.

00:04:04.180 --> 00:04:06.780
And so they're definitely
using molecular evidence,

00:04:06.780 --> 00:04:08.940
and three is the only one that talks

00:04:08.940 --> 00:04:10.740
about molecular evidence,

00:04:10.740 --> 00:04:13.320
so I will go with choice three.

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They did say in the beginning

00:04:15.310 --> 00:04:18.260
that they examined
physical characteristics

00:04:18.260 --> 00:04:20.110
and DNA of these species,

00:04:20.110 --> 00:04:23.610
and so you could maybe say

00:04:23.610 --> 00:04:26.460
examine the structure of
the beaks and compare them,

00:04:26.460 --> 00:04:27.470
but that's very specific,

00:04:27.470 --> 00:04:29.850
and it didn't say for sure
that they're doing that,

00:04:29.850 --> 00:04:32.060
that may be one of the
physical characteristics

00:04:32.060 --> 00:04:33.060
that they're looking at,

00:04:33.060 --> 00:04:34.860
but we know for sure

00:04:34.860 --> 00:04:36.720
that they're obtaining molecular evidence

00:04:36.720 --> 00:04:38.240
from all three species,

00:04:38.240 --> 00:04:41.010
and they're identifying the similarities.

00:04:41.010 --> 00:04:43.180
So now, this next part, it says,

00:04:43.180 --> 00:04:47.220
based on the data they collected
using gel electrophoresis,

00:04:47.220 --> 00:04:49.550
label the branching tree diagram below,

00:04:49.550 --> 00:04:51.120
writing the letters A, B, and C

00:04:51.120 --> 00:04:54.040
to represent the possible
evolutionary relationships

00:04:54.040 --> 00:04:56.380
between species A, B, and C.

00:04:56.380 --> 00:05:00.260
So pause this video and see
if you can figure this out,

00:05:00.260 --> 00:05:03.573
and I'll show you the
banding pattern again.

00:05:04.880 --> 00:05:06.010
Alright, so down there,

00:05:06.010 --> 00:05:07.760
and actually I'll redraw it up here,

00:05:07.760 --> 00:05:11.330
they draw a tree that looks like this,

00:05:11.330 --> 00:05:14.910
and they want us to put A's, B's, and C's

00:05:14.910 --> 00:05:16.530
in these blanks here.

00:05:16.530 --> 00:05:18.060
And one way to think about it is,

00:05:18.060 --> 00:05:19.730
these two are more closely related.

00:05:19.730 --> 00:05:22.950
They have a more recent common ancestor

00:05:22.950 --> 00:05:26.250
than from this one over here.

00:05:26.250 --> 00:05:28.210
So if we look at the banding patterns,

00:05:28.210 --> 00:05:32.230
it's pretty clear we have
this similar band on A and B,

00:05:32.230 --> 00:05:34.980
we have this similar band on A and B,

00:05:34.980 --> 00:05:38.390
and they have this band
right over here in common,

00:05:38.390 --> 00:05:41.400
although C here has that one as well.

00:05:41.400 --> 00:05:44.870
The only other band that
A shares in common with C

00:05:44.870 --> 00:05:47.470
is this one right over here,

00:05:47.470 --> 00:05:51.250
and B doesn't share anything
else in common with C.

00:05:51.250 --> 00:05:53.410
So it's pretty clear that A and B

00:05:53.410 --> 00:05:55.880
share the most common banding patterns.

00:05:55.880 --> 00:05:58.100
So I would write that A and B

00:05:58.100 --> 00:06:00.710
have the most recent common ancestor,

00:06:00.710 --> 00:06:03.903
and that C is more distant.


Resistivity experimentally determined

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:02.950
- [Instructor] A group of
students prepare a large batch

00:00:02.950 --> 00:00:04.880
of conductive dough,

00:00:04.880 --> 00:00:07.820
a soft substance that
can conduct electricity,

00:00:07.820 --> 00:00:10.970
and then mold the dough
into several cylinders

00:00:10.970 --> 00:00:13.870
with various cross-sectional areas A

00:00:13.870 --> 00:00:15.740
and lengths l.

00:00:15.740 --> 00:00:18.930
Each student applies a
potential difference, delta V,

00:00:18.930 --> 00:00:21.460
across the ends of a dough cylinder

00:00:21.460 --> 00:00:24.950
and determines the
resistance of the cylinder.

00:00:24.950 --> 00:00:27.230
The results of their experiments are shown

00:00:27.230 --> 00:00:29.090
in the table below.

00:00:29.090 --> 00:00:29.923
So what they're doing,

00:00:29.923 --> 00:00:31.920
they're taking this material,
this conductive dough,

00:00:31.920 --> 00:00:33.080
and they're making cylinders

00:00:33.080 --> 00:00:36.610
of different cross-sectional
areas and different lengths.

00:00:36.610 --> 00:00:40.010
So they're varying the area,
the cross-sectional area,

00:00:40.010 --> 00:00:42.430
and the length, but it's all being made

00:00:42.430 --> 00:00:44.910
out of this same material,

00:00:44.910 --> 00:00:46.640
and then depending on how they vary that,

00:00:46.640 --> 00:00:49.590
you get different resistances.

00:00:49.590 --> 00:00:53.960
The students want to
determine the resistivity

00:00:53.960 --> 00:00:56.150
of the dough cylinders.

00:00:56.150 --> 00:00:57.919
So just as a reminder,

00:00:57.919 --> 00:01:00.720
resistivity is different than resistance.

00:01:00.720 --> 00:01:03.670
A particular shape of a given material

00:01:03.670 --> 00:01:05.690
will have a length and
cross-sectional area,

00:01:05.690 --> 00:01:08.060
all of that, will have a given resistance.

00:01:08.060 --> 00:01:12.150
But a resistivity is a
property of the material.

00:01:12.150 --> 00:01:14.330
And the relationship between resistance

00:01:14.330 --> 00:01:16.980
and resistivity, you could write it as

00:01:16.980 --> 00:01:20.070
resistance is equal to resistivity,

00:01:20.070 --> 00:01:22.730
which is usually represented
with the Greek letter rho,

00:01:22.730 --> 00:01:27.000
times the length of the
thing that you're measuring

00:01:27.000 --> 00:01:31.860
the resistance of divided
by the cross-sectional area.

00:01:31.860 --> 00:01:33.890
And so if you have a higher resistivity,

00:01:33.890 --> 00:01:36.520
all else equal, that would
make a higher resistance.

00:01:36.520 --> 00:01:37.710
But if you have a higher length,

00:01:37.710 --> 00:01:39.680
that also makes a higher resistance.

00:01:39.680 --> 00:01:42.310
But if you have a higher
cross-sectional area,

00:01:42.310 --> 00:01:44.720
than that lowers the resistance.

00:01:44.720 --> 00:01:46.950
And if you were to solve for rho,

00:01:46.950 --> 00:01:50.430
you could write rho is equal to resistance

00:01:50.430 --> 00:01:54.603
times cross-sectional area over l.

00:01:55.500 --> 00:01:58.130
So it then says indicate
below which quantities

00:01:58.130 --> 00:02:00.870
could be graphed to determine a value

00:02:00.870 --> 00:02:04.370
for the resistivity of
the dough cylinders.

00:02:04.370 --> 00:02:07.580
You may use the remaining
columns in the table above,

00:02:07.580 --> 00:02:11.510
as needed, to record any
quantities, including units,

00:02:11.510 --> 00:02:14.400
that are not already in the table.

00:02:14.400 --> 00:02:17.600
So pause this video and think
about how you would do that.

00:02:17.600 --> 00:02:20.280
Well I can think of two ways to do that.

00:02:20.280 --> 00:02:21.870
You could make a plot,

00:02:21.870 --> 00:02:25.740
you could make a scatter plot
where on the horizontal axis,

00:02:25.740 --> 00:02:28.260
you put l over A,

00:02:28.260 --> 00:02:31.150
and on the vertical
axis, you put resistance.

00:02:31.150 --> 00:02:35.050
And then the resistivity
would be the slope of a line

00:02:35.050 --> 00:02:37.610
that you try to fit to those points.

00:02:37.610 --> 00:02:39.090
So that's one way to do it.

00:02:39.090 --> 00:02:42.896
Another way to do it is you
could make your horizontal axis

00:02:42.896 --> 00:02:46.280
your A over l

00:02:46.280 --> 00:02:51.280
and then you could make
your vertical axis your rho,

00:02:51.670 --> 00:02:55.350
and then you could try
to fit a horizontal line

00:02:55.350 --> 00:02:56.510
to those points.

00:02:56.510 --> 00:02:59.970
Because your resistivity
should not be dependent

00:02:59.970 --> 00:03:02.360
on A divided by l.

00:03:02.360 --> 00:03:04.580
I'm actually going to
pick this second option

00:03:04.580 --> 00:03:07.040
because if I'm doing
this primarily by hand,

00:03:07.040 --> 00:03:10.170
it's at least to me easier
to fit a horizontal line

00:03:10.170 --> 00:03:13.130
than to fit a non-horizontal line,

00:03:13.130 --> 00:03:14.760
as I would have to do there.

00:03:14.760 --> 00:03:15.910
So let's do that.

00:03:15.910 --> 00:03:17.500
So my vertical axis,

00:03:17.500 --> 00:03:20.960
I'm gonna do resistivity, resistivity,

00:03:20.960 --> 00:03:24.563
so that's resistivity.

00:03:25.630 --> 00:03:29.160
And it's going to be in ohm meters.

00:03:29.160 --> 00:03:31.270
And then in my horizontal axis,

00:03:31.270 --> 00:03:35.840
I am going to do my
area divided by length,

00:03:35.840 --> 00:03:39.110
and that's going to be in meters.

00:03:39.110 --> 00:03:40.830
And now to be able to do this,

00:03:40.830 --> 00:03:42.230
I need to calculate these things.

00:03:42.230 --> 00:03:45.490
So I wanna calculate my
area divided by length,

00:03:45.490 --> 00:03:49.590
and I also wanna calculate
my resistivity, rho.

00:03:49.590 --> 00:03:51.807
And so area divided by length,

00:03:51.807 --> 00:03:52.640
.00049

00:03:57.130 --> 00:03:58.550
divided by .03

00:04:00.070 --> 00:04:02.140
is equal to

00:04:02.140 --> 00:04:03.433
0.016.

00:04:05.940 --> 00:04:08.810
So we have 0.016,

00:04:08.810 --> 00:04:10.040
two significant digits here,

00:04:10.040 --> 00:04:11.410
two significant digits in the things

00:04:11.410 --> 00:04:12.570
that I'm dividing with.

00:04:12.570 --> 00:04:14.660
And then I can calculate rho

00:04:14.660 --> 00:04:19.660
by multiplying my area divided
by length times resistance.

00:04:20.590 --> 00:04:24.280
Times my resistance that I measured,

00:04:24.280 --> 00:04:26.880
23.6, it's important to
realize that when you do things

00:04:26.880 --> 00:04:28.720
experimentally things
aren't going to be perfect.

00:04:28.720 --> 00:04:31.400
There's a lot of noise in the real world.

00:04:31.400 --> 00:04:33.940
So I'll round that to,

00:04:33.940 --> 00:04:36.280
I'll round that to since I have
two significant digits here,

00:04:36.280 --> 00:04:39.700
two significant digits here, so 0.39.

00:04:39.700 --> 00:04:41.313
So this is 0.39.

00:04:44.090 --> 00:04:46.160
And then I'll keep doing that calculation,

00:04:46.160 --> 00:04:47.390
and for the sake of time,

00:04:47.390 --> 00:04:49.963
I will just speed up the video now.

00:04:54.480 --> 00:04:57.750
All right and now we can
move to this second part now.

00:04:57.750 --> 00:05:00.250
It says on the grid below,

00:05:00.250 --> 00:05:02.720
plot the appropriate
quantities to determine

00:05:02.720 --> 00:05:05.350
the resistivity of the dough cylinders.

00:05:05.350 --> 00:05:07.750
Clearly scale and label all axes,

00:05:07.750 --> 00:05:10.010
including units as appropriate.

00:05:10.010 --> 00:05:11.240
So as we mentioned,

00:05:11.240 --> 00:05:14.160
on our horizontal axis right over here,

00:05:14.160 --> 00:05:17.890
we want to put our

00:05:17.890 --> 00:05:21.540
cross-sectional area divided by the length

00:05:21.540 --> 00:05:24.020
of our resistors, and
so that's going to be,

00:05:24.020 --> 00:05:25.640
we have meters squared divided by meters,

00:05:25.640 --> 00:05:27.090
so that's going to be in meters.

00:05:27.090 --> 00:05:29.410
And then in our vertical axis,

00:05:29.410 --> 00:05:33.090
we want to plot our resistivity,

00:05:33.090 --> 00:05:34.720
our resistivity rho,

00:05:34.720 --> 00:05:38.410
and that's going to be in ohm meters.

00:05:38.410 --> 00:05:41.340
And now we need to think about the scales.

00:05:41.340 --> 00:05:44.240
And so the scale of this vertical axis,

00:05:44.240 --> 00:05:46.680
if we go back over here,

00:05:46.680 --> 00:05:49.580
we see that all of our
calculations for rho,

00:05:49.580 --> 00:05:53.260
well it's between around
.3, .4, like that,

00:05:53.260 --> 00:05:55.760
or you could say between .30 and I guess

00:05:55.760 --> 00:05:58.400
we have up to .41.

00:05:58.400 --> 00:06:00.940
So we could have a reasonable scale here.

00:06:00.940 --> 00:06:02.760
We have one, two, three, four, five,

00:06:02.760 --> 00:06:05.080
so we could say this is .10,

00:06:05.080 --> 00:06:07.020
which is the same thing as 1/10.

00:06:07.020 --> 00:06:10.077
This is .20, this is .30,

00:06:10.077 --> 00:06:12.630
and this is .40,

00:06:12.630 --> 00:06:14.280
for our rho scale.

00:06:14.280 --> 00:06:18.190
And then for our area
divided by length, we have,

00:06:19.620 --> 00:06:21.720
so let's see if we view
all of this in terms

00:06:21.720 --> 00:06:24.810
of .0010,

00:06:24.810 --> 00:06:27.400
we go between .0038

00:06:29.230 --> 00:06:33.543
all the way to .0160.

00:06:36.130 --> 00:06:37.713
So let's make this .0050,

00:06:38.770 --> 00:06:39.603
0.0050.

00:06:41.518 --> 00:06:44.003
This would be .0100,

00:06:44.003 --> 00:06:44.836
0.0100.

00:06:47.346 --> 00:06:49.679
This would be .0150, 0.0150,

00:06:52.110 --> 00:06:54.700
and this would be .0200, 0.0200.

00:06:57.570 --> 00:06:59.833
And this would be .0250.

00:07:00.910 --> 00:07:02.610
All right I think I have my scales right,

00:07:02.610 --> 00:07:04.393
so now we just plot our points.

00:07:05.420 --> 00:07:07.280
So let's see the first one

00:07:07.280 --> 00:07:11.600
has a resistivity of 0.39,

00:07:11.600 --> 00:07:14.120
and our A divided by l is,

00:07:14.120 --> 00:07:16.880
which is this is going to be in meters,

00:07:16.880 --> 00:07:20.700
this is going to be .0160

00:07:20.700 --> 00:07:22.280
of a meter.

00:07:22.280 --> 00:07:27.280
So we would look up 160 right over here.

00:07:27.870 --> 00:07:29.350
So this is 150,

00:07:29.350 --> 00:07:30.840
and then we would wanna go 1/5 of the way

00:07:30.840 --> 00:07:31.870
between this and this,

00:07:31.870 --> 00:07:34.630
and it's neatly divided
into five sections here.

00:07:34.630 --> 00:07:36.220
So this would be 160,

00:07:36.220 --> 00:07:40.500
and then I would want 0.39 ohm meters

00:07:40.500 --> 00:07:41.960
for the resistivity.

00:07:41.960 --> 00:07:45.530
So that would put us right over there.

00:07:45.530 --> 00:07:48.090
So that's my first data point.

00:07:48.090 --> 00:07:51.937
And now if I go to my second one,

00:07:51.937 --> 00:07:54.753
.0098, 0.31.

00:07:56.050 --> 00:08:01.050
So .0098 will be right around here,

00:08:01.960 --> 00:08:04.930
and then at 0.31 would
be right around there.

00:08:04.930 --> 00:08:07.450
So that's my second data point.

00:08:07.450 --> 00:08:12.450
And then I get to .0066 and 0.41.

00:08:14.710 --> 00:08:18.540
So .0066 would get us,

00:08:18.540 --> 00:08:21.910
let's see this would be 60, .0066,

00:08:21.910 --> 00:08:26.250
and then we have 0.41.

00:08:26.250 --> 00:08:30.123
So that puts us right over there.

00:08:31.290 --> 00:08:33.280
Some place like that.

00:08:33.280 --> 00:08:34.757
And then last but not least,

00:08:34.757 --> 00:08:37.603
.0038 and 0.4.

00:08:38.960 --> 00:08:41.017
So then we have

00:08:41.017 --> 00:08:44.360
.0038 so this would be
40, this would be 30,

00:08:44.360 --> 00:08:46.260
so 38 is like here,

00:08:46.260 --> 00:08:49.970
and then we go to .40.

00:08:49.970 --> 00:08:52.680
So it is like that.

00:08:52.680 --> 00:08:55.440
So those are our four data points.

00:08:55.440 --> 00:08:56.810
And then they say,

00:08:56.810 --> 00:08:59.680
use the above graph to estimate a value

00:08:59.680 --> 00:09:03.160
for the resistivity of
the dough cylinders.

00:09:03.160 --> 00:09:03.993
Well as we said,

00:09:03.993 --> 00:09:06.620
the resistivity should be constant.

00:09:06.620 --> 00:09:11.310
It shouldn't depend on our
area divided by our length.

00:09:11.310 --> 00:09:15.200
And so if we wanna fit a
horizontal line to this

00:09:15.200 --> 00:09:16.470
that gets as close as possible,

00:09:16.470 --> 00:09:18.150
ideally we would have a computer,

00:09:18.150 --> 00:09:19.670
but we're gonna hand do it.

00:09:19.670 --> 00:09:21.760
We have three of these
points that are pretty close

00:09:21.760 --> 00:09:25.140
to .40, but then we
have this one down here,

00:09:25.140 --> 00:09:28.190
so I'd wanna get a little bit
closer to these three up there

00:09:28.190 --> 00:09:29.530
because there are so
many data points there.

00:09:29.530 --> 00:09:33.343
So I would say a line
that looks something,

00:09:34.210 --> 00:09:36.600
something like that.

00:09:36.600 --> 00:09:38.840
And so this point right over here,

00:09:38.840 --> 00:09:43.840
this is maybe approximately 0.37.

00:09:43.890 --> 00:09:47.830
So this is 0.37 ohm meters.

00:09:47.830 --> 00:09:52.470
I would say 0.37 ohm meters.

00:09:52.470 --> 00:09:53.303
And we're done.

00:09:53.303 --> 00:09:55.760
And frankly another way
to think about this is,

00:09:55.760 --> 00:09:58.160
you could've done it potentially
even without the graph

00:09:58.160 --> 00:09:59.410
by saying well look,

00:09:59.410 --> 00:10:03.250
a good estimate for my
resistivity of my conductive dough

00:10:03.250 --> 00:10:06.680
is to just take the mean of these values.

00:10:06.680 --> 00:10:08.270
But they wanted us to use a graph,

00:10:08.270 --> 00:10:09.730
and so one technique would be all right,

00:10:09.730 --> 00:10:11.820
let's try to fit a horizontal line.

00:10:11.820 --> 00:10:13.740
The other technique with
the graph would've been

00:10:13.740 --> 00:10:17.590
plotting l over A here and
then plotting resistance

00:10:17.590 --> 00:10:19.000
on our vertical axis,

00:10:19.000 --> 00:10:22.940
and then our estimate of the
slope of the fitted line,

00:10:22.940 --> 00:10:24.900
that would be our estimate of rho.

00:10:24.900 --> 00:10:27.250
But that would've been
frankly harder to do

00:10:27.250 --> 00:10:28.803
than even what we did here.


Velocity, acceleration and distance traveled for points on wave

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:02.660
- [Instructor] We are
told a transverse wave

00:00:02.660 --> 00:00:05.030
travels to the right along a string,

00:00:05.030 --> 00:00:06.620
they draw it right over here.

00:00:06.620 --> 00:00:09.330
Two dots have been painted on the string.

00:00:09.330 --> 00:00:12.980
In the diagrams below, those
dots are labeled P and Q,

00:00:12.980 --> 00:00:14.950
so that's these dots here.

00:00:14.950 --> 00:00:19.180
The figure below shows the
string at an instant in time.

00:00:19.180 --> 00:00:23.620
At the instant shown, dot
P has maximum displacement

00:00:23.620 --> 00:00:27.960
and dot Q has zero
displacement from equilibrium.

00:00:27.960 --> 00:00:32.370
At each of the dots P and Q, draw an arrow

00:00:32.370 --> 00:00:35.640
indicating the direction
of the instantaneous

00:00:35.640 --> 00:00:37.740
velocity of that dot.

00:00:37.740 --> 00:00:40.990
If either dot has zero velocity,

00:00:40.990 --> 00:00:45.160
write V equals zero next to the dot.

00:00:45.160 --> 00:00:47.250
So pause this video and
see if you can have a go

00:00:47.250 --> 00:00:49.900
at it on your own before I
work through it with you.

00:00:49.900 --> 00:00:51.460
All right, now let's work
through this together.

00:00:51.460 --> 00:00:54.110
So, let's think about the
instantaneous velocity

00:00:54.110 --> 00:00:55.940
of this dot right over here.

00:00:55.940 --> 00:00:57.500
And to help us think about it,

00:00:57.500 --> 00:00:59.190
let's think about what's going on

00:00:59.190 --> 00:01:00.990
on either side of this dot.

00:01:00.990 --> 00:01:04.200
So, if we think about this
point right over here,

00:01:04.200 --> 00:01:06.450
what's gonna happen a little bit after

00:01:06.450 --> 00:01:08.390
this moment in time we're looking at?

00:01:08.390 --> 00:01:11.020
Well, the wave is
traveling in that direction

00:01:11.020 --> 00:01:13.770
so in a little while,
the wave is gonna look,

00:01:13.770 --> 00:01:15.970
in a moment after that,
the wave is gonna look

00:01:15.970 --> 00:01:17.800
something like this.

00:01:17.800 --> 00:01:20.150
The wave is gonna look
something like this.

00:01:20.150 --> 00:01:24.480
So, this dot is in the
process of moving down.

00:01:24.480 --> 00:01:27.090
And what about this dot right over here?

00:01:27.090 --> 00:01:28.830
Well, this dot a little bit later

00:01:28.830 --> 00:01:30.780
is going to be higher, so it's moving up.

00:01:30.780 --> 00:01:33.770
And it's this point where
we're at our maximum,

00:01:33.770 --> 00:01:37.490
when we're at our maximum
displacement from equilibrium,

00:01:37.490 --> 00:01:41.130
we are going from moving down to going up

00:01:41.130 --> 00:01:42.250
right at that moment.

00:01:42.250 --> 00:01:44.570
So right at this moment
of maximum displacement,

00:01:44.570 --> 00:01:46.550
we have zero velocity.

00:01:46.550 --> 00:01:48.110
So let me do that in
red so you can see it.

00:01:48.110 --> 00:01:52.290
So here, at point P, we would
write V is equal to zero.

00:01:52.290 --> 00:01:54.710
Now what about at point Q?

00:01:54.710 --> 00:01:57.520
Well, we just sketched
what's going to happen

00:01:57.520 --> 00:01:58.410
a moment later.

00:01:58.410 --> 00:02:02.500
Our transverse wave is
gonna look like this

00:02:02.500 --> 00:02:05.090
and so point Q is going to be up here.

00:02:05.090 --> 00:02:08.220
So point Q is moving upwards.

00:02:08.220 --> 00:02:10.810
So if we want to draw its velocity vector,

00:02:10.810 --> 00:02:15.810
it would look something like that.

00:02:15.810 --> 00:02:17.960
Now let's do the next part.

00:02:17.960 --> 00:02:20.650
The figure below shows the
string at the same instant

00:02:20.650 --> 00:02:23.490
as shown in part A-one.

00:02:23.490 --> 00:02:27.540
At each of the dots P and Q, draw an arrow

00:02:27.540 --> 00:02:30.170
indicating the direction
of the instantaneous

00:02:30.170 --> 00:02:33.060
acceleration of that dot.

00:02:33.060 --> 00:02:35.810
If either dot has zero acceleration,

00:02:35.810 --> 00:02:39.700
write A equals zero next to the dot.

00:02:39.700 --> 00:02:41.780
So pause this video again and see

00:02:41.780 --> 00:02:43.720
if you can have a go at it.

00:02:43.720 --> 00:02:46.740
One way to think about it
is all the points around P

00:02:46.740 --> 00:02:50.220
and actually all the points
in this area right over here

00:02:50.220 --> 00:02:52.950
are going to have a net
upward acceleration.

00:02:52.950 --> 00:02:56.840
So even though this point
for example right over here

00:02:56.840 --> 00:03:00.720
has a velocity downwards,
that downward velocity

00:03:00.720 --> 00:03:03.220
is becoming less and less
and less as you get closer

00:03:03.220 --> 00:03:05.840
to P and then it flips around,
becomes an upward velocity,

00:03:05.840 --> 00:03:08.190
and becomes more and more
upward as we get closer

00:03:08.190 --> 00:03:09.660
and closer to this point.

00:03:09.660 --> 00:03:11.580
So everything in this area right over here

00:03:11.580 --> 00:03:14.060
has a net upward acceleration.

00:03:14.060 --> 00:03:18.540
So P would have a net
upward acceleration as well.

00:03:18.540 --> 00:03:20.700
It has zero instantaneous velocity

00:03:20.700 --> 00:03:23.810
but we're, at that
moment, we're going from

00:03:23.810 --> 00:03:27.070
a downward velocity to an upward velocity.

00:03:27.070 --> 00:03:29.713
So we have an upward acceleration.

00:03:30.620 --> 00:03:32.530
What about at point Q?

00:03:32.530 --> 00:03:34.790
Well, one interesting thing to appreciate

00:03:34.790 --> 00:03:38.760
is that everything in this
region right over here

00:03:38.760 --> 00:03:42.584
has an upward acceleration
and everything over here

00:03:42.584 --> 00:03:45.370
has a downward acceleration,

00:03:45.370 --> 00:03:50.140
and then everything over here
has an upward acceleration.

00:03:50.140 --> 00:03:52.700
And so point Q is at that transition point

00:03:52.700 --> 00:03:55.350
where we're going from
a downward acceleration

00:03:55.350 --> 00:03:57.200
to an upward acceleration.

00:03:57.200 --> 00:04:00.800
And so here, right at
that point at this moment,

00:04:00.800 --> 00:04:03.370
you're going to have zero acceleration.

00:04:03.370 --> 00:04:07.163
So here, we write
acceleration is equal to zero.

00:04:10.380 --> 00:04:12.820
And then it says, the
figure below represents

00:04:12.820 --> 00:04:14.980
the string at time t equals zero,

00:04:14.980 --> 00:04:17.600
the same instant as shown in part A

00:04:17.600 --> 00:04:22.230
where dot P is at its maximum
displacement from equilibrium.

00:04:22.230 --> 00:04:24.940
For simplicity, dot Q is not shown.

00:04:24.940 --> 00:04:29.940
On the grid below, draw
the string at a later time,

00:04:29.960 --> 00:04:33.140
time t is equal to
capital T divided by four,

00:04:33.140 --> 00:04:36.750
where capital T is the period of the wave.

00:04:36.750 --> 00:04:39.390
So pause this video again
and see if you could do that,

00:04:39.390 --> 00:04:41.460
you can even sketch it out
if you don't have access

00:04:41.460 --> 00:04:43.630
to this drawing right here.

00:04:43.630 --> 00:04:44.920
To think about a full period,

00:04:44.920 --> 00:04:47.690
we can look at a full
wavelength of this wave

00:04:47.690 --> 00:04:50.570
and so one way to do it
is you start at a point

00:04:50.570 --> 00:04:52.410
and then you see, well
how far do you have to go

00:04:52.410 --> 00:04:53.870
to get to this same point again?

00:04:53.870 --> 00:04:56.020
So we went up, we go down,

00:04:56.020 --> 00:04:57.286
you could say we're at the same point

00:04:57.286 --> 00:04:59.700
but here, we're going up,
and here we're going down.

00:04:59.700 --> 00:05:02.300
So we wanna go back to
when we're going back up.

00:05:02.300 --> 00:05:04.950
So that is a full wavelength.

00:05:04.950 --> 00:05:08.910
A full wavelength here is 24 centimeters.

00:05:08.910 --> 00:05:11.110
So after one-fourth of a period,

00:05:11.110 --> 00:05:14.890
this wave would've shifted
one-fourth of this length.

00:05:14.890 --> 00:05:17.670
And so, one-fourth would
be one of these sections

00:05:17.670 --> 00:05:18.760
right over here.

00:05:18.760 --> 00:05:20.980
So all I will do is draw the same wave

00:05:20.980 --> 00:05:23.180
but instead of starting
here, I'm gonna start here.

00:05:23.180 --> 00:05:25.570
One-fourth of a wave length to the right.

00:05:25.570 --> 00:05:27.530
But then I'll draw the exact same wave.

00:05:27.530 --> 00:05:29.060
It just shifted.

00:05:29.060 --> 00:05:32.980
So notice, it has shifted one-fourth

00:05:32.980 --> 00:05:35.323
of a wavelength to the right.

00:05:37.505 --> 00:05:40.520
And so it will look something

00:05:40.520 --> 00:05:42.150
like that.

00:05:42.150 --> 00:05:45.210
And then if we look at the
next parts of the question,

00:05:45.210 --> 00:05:49.090
on your drawing above, draw a
dot to indicate the position

00:05:49.090 --> 00:05:52.170
of point P on the string
at time t is equal

00:05:52.170 --> 00:05:55.110
to the period divided by
four and clearly label

00:05:55.110 --> 00:05:57.180
the dot with the letter P.

00:05:57.180 --> 00:06:00.890
So let's just remind ourselves
of where P was before.

00:06:00.890 --> 00:06:04.270
P is at this point that's halfway

00:06:04.270 --> 00:06:07.280
between 12 and 24 centimeters.

00:06:07.280 --> 00:06:08.960
So that's gonna be at 18 centimeters.

00:06:08.960 --> 00:06:09.940
So it's right over here.

00:06:09.940 --> 00:06:14.930
So at this point, P is
going to be right over here.

00:06:14.930 --> 00:06:17.810
So from T is equal to zero to T is equal

00:06:17.810 --> 00:06:19.600
to the period divided by four,

00:06:19.600 --> 00:06:22.713
P went from down here up to here.

00:06:23.790 --> 00:06:26.210
And then last but not least, it says

00:06:26.210 --> 00:06:30.710
now consider the wave at time
T is equal to a full period.

00:06:30.710 --> 00:06:34.030
Determine the distance
traveled, not the displacement,

00:06:34.030 --> 00:06:37.510
by dot P between the times t equals zero

00:06:37.510 --> 00:06:39.850
and t is equal to capital T.

00:06:39.850 --> 00:06:43.170
So pause this video and see
if you can figure it out.

00:06:43.170 --> 00:06:46.860
So we already saw at
time t is equal to zero,

00:06:46.860 --> 00:06:48.780
P is right over here.

00:06:48.780 --> 00:06:51.180
And so what's gonna
happen over a full period?

00:06:51.180 --> 00:06:53.440
Well, over that time
period, P is gonna move

00:06:53.440 --> 00:06:56.770
all the way up to positive
eight centimeters on our Y axis

00:06:56.770 --> 00:07:01.360
and then all the way back
down over a full period.

00:07:01.360 --> 00:07:03.790
And so if you this displacement,
you'd say it's zero,

00:07:03.790 --> 00:07:05.940
you're back to where you
started but the distance,

00:07:05.940 --> 00:07:09.180
well it would've traveled
eight, 16 centimeters up

00:07:09.180 --> 00:07:11.900
and then 16 centimeters back down.

00:07:11.900 --> 00:07:16.223
So the distance traveled
would be 32 centimeters.

00:07:17.500 --> 00:07:18.373
And we're done.


Angular velocity graphs due to multiple torques

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.650
- [Instructor] A disk is
initially rotating clockwise

00:00:03.650 --> 00:00:08.650
around a fixed axis with
angular speed omega nought.

00:00:09.440 --> 00:00:12.030
At time t equals zero,

00:00:12.030 --> 00:00:15.610
the two forces of F sub
one is equal to 20 newtons

00:00:15.610 --> 00:00:19.340
and F sub two is equal to 10
newtons are exerted on the disk

00:00:19.340 --> 00:00:20.600
as shown in the figure below.

00:00:20.600 --> 00:00:21.640
So these are the magnitudes,

00:00:21.640 --> 00:00:23.730
but here they show the
magnitude and the direction.

00:00:23.730 --> 00:00:26.560
And they say, which graph
could show the angular velocity

00:00:26.560 --> 00:00:28.050
of the disk over time?

00:00:28.050 --> 00:00:29.960
So before I even show you the options,

00:00:29.960 --> 00:00:31.940
let's see if we can just
sketch it ourselves.

00:00:31.940 --> 00:00:33.100
So pause this video

00:00:33.100 --> 00:00:35.660
and see if you can sketch
the angular velocity

00:00:35.660 --> 00:00:37.283
of the disk over time.

00:00:38.330 --> 00:00:42.090
All right, so in our vertical axis here,

00:00:42.090 --> 00:00:44.930
let's put our angular velocity.

00:00:44.930 --> 00:00:47.010
And then in our horizontal axis here,

00:00:47.010 --> 00:00:50.490
we wanna see how that changes over time.

00:00:50.490 --> 00:00:53.320
So what is our initial angular velocity?

00:00:53.320 --> 00:00:57.290
Well, they tell us that our
angular speed is omega nought

00:00:57.290 --> 00:01:00.500
and the direction is in
the clockwise direction.

00:01:00.500 --> 00:01:02.080
They tell it right over there.

00:01:02.080 --> 00:01:05.930
And the typical convention
is that clockwise,

00:01:05.930 --> 00:01:08.500
clockwise is negative

00:01:08.500 --> 00:01:11.240
and counterclockwise would be positive.

00:01:11.240 --> 00:01:15.280
So we're starting at a
negative angular velocity.

00:01:15.280 --> 00:01:17.680
So we're starting at a
negative angular velocity.

00:01:17.680 --> 00:01:19.530
Let's see right over there.

00:01:19.530 --> 00:01:23.220
And it actually would be
negative omega nought.

00:01:23.220 --> 00:01:24.710
I'll just write negative omega nought.

00:01:24.710 --> 00:01:28.570
Remember, omega nought is
the speed, the angular speed,

00:01:28.570 --> 00:01:31.170
so it's the magnitude
of the angular velocity.

00:01:31.170 --> 00:01:33.400
But since we're going
clockwise, it would be negative,

00:01:33.400 --> 00:01:35.860
so we're gonna start right there.

00:01:35.860 --> 00:01:39.220
And now let's see what the net torque is.

00:01:39.220 --> 00:01:42.000
F one, we're assuming, goes straight down

00:01:42.000 --> 00:01:45.530
and it's perpendicular to the radius.

00:01:45.530 --> 00:01:47.420
Let's call that r, that length r.

00:01:47.420 --> 00:01:49.520
And F two is straight down.

00:01:49.520 --> 00:01:51.100
It's perpendicular to the radius.

00:01:51.100 --> 00:01:52.890
But F one is providing a torque

00:01:52.890 --> 00:01:54.970
in the counterclockwise direction,

00:01:54.970 --> 00:01:57.730
while F two is providing a torque

00:01:57.730 --> 00:01:59.670
in the clockwise direction.

00:01:59.670 --> 00:02:02.190
And the way it's
depicted, F one is larger,

00:02:02.190 --> 00:02:04.900
and they're both r away
from the center of rotation.

00:02:04.900 --> 00:02:09.900
So your net torque is going
to be counterclockwise.

00:02:10.110 --> 00:02:12.700
It's going to be against the direction

00:02:12.700 --> 00:02:15.450
or the starting direction of rotation.

00:02:15.450 --> 00:02:18.080
If your net torque is counterclockwise.

00:02:18.080 --> 00:02:21.610
So we could say our net torque
is going to be positive.

00:02:21.610 --> 00:02:23.150
Positive.

00:02:23.150 --> 00:02:25.540
And if our net torque is positive,

00:02:25.540 --> 00:02:28.020
that means that our angular acceleration

00:02:28.020 --> 00:02:30.630
is going to be positive.

00:02:30.630 --> 00:02:34.090
And our angular acceleration
is just the rate of change

00:02:34.090 --> 00:02:35.630
of our angular velocity.

00:02:35.630 --> 00:02:37.240
And this is going to be constant,

00:02:37.240 --> 00:02:39.920
assuming that these forces are constant.

00:02:39.920 --> 00:02:41.310
You're going to see this.

00:02:41.310 --> 00:02:44.370
We're going to be rotating clockwise

00:02:44.370 --> 00:02:46.470
at a slower and slower rate

00:02:46.470 --> 00:02:49.970
until all of a sudden the
wheel switches direction

00:02:51.214 --> 00:02:52.360
or the disk switches directions

00:02:52.360 --> 00:02:56.590
and then all of a sudden we
are rotating counterclockwise.

00:02:56.590 --> 00:02:58.490
So it would look something like that.

00:02:58.490 --> 00:03:00.533
So let's see which of these choices fit.

00:03:02.500 --> 00:03:03.680
So if we look at all of 'em,

00:03:03.680 --> 00:03:06.250
this looks almost exactly like we drew,

00:03:06.250 --> 00:03:08.920
but let's just review what
these other ones are saying.

00:03:08.920 --> 00:03:11.370
This one is implying here

00:03:11.370 --> 00:03:14.660
that you're starting off going clockwise,

00:03:14.660 --> 00:03:17.340
but then you're accelerating
in the clockwise direction.

00:03:17.340 --> 00:03:18.580
And that wasn't the direction

00:03:18.580 --> 00:03:20.170
that the net torque was being applied in,

00:03:20.170 --> 00:03:21.770
so we rule that out.

00:03:21.770 --> 00:03:23.800
This assumes that you're starting

00:03:23.800 --> 00:03:26.500
in the counterclockwise direction,

00:03:26.500 --> 00:03:28.710
which we know is not the case.

00:03:28.710 --> 00:03:29.870
This assumes that you're starting

00:03:29.870 --> 00:03:32.163
in the counterclockwise direction.

00:03:34.256 --> 00:03:38.100
And then you're accelerating
in the clockwise direction.

00:03:38.100 --> 00:03:39.470
And you would've picked this

00:03:39.470 --> 00:03:41.020
if you got your convention mixed up

00:03:41.020 --> 00:03:43.310
between counterclockwise and clockwise

00:03:43.310 --> 00:03:44.800
and which one is positive or negative,

00:03:44.800 --> 00:03:47.300
but we can rule that one
out, and the choice is A.


Momentum collision graphs

WEBVTT
Kind: captions
Language: en

00:00:01.120 --> 00:00:04.170
- [Instructor] A cart of
mass, m, moving rightward

00:00:04.170 --> 00:00:05.850
at speed two v,

00:00:05.850 --> 00:00:08.170
hits a slower moving cart

00:00:08.170 --> 00:00:09.600
of mass, m,

00:00:09.600 --> 00:00:12.600
moving rightward at speed v.

00:00:12.600 --> 00:00:14.490
When the carts collide,

00:00:14.490 --> 00:00:16.210
they hook together.

00:00:16.210 --> 00:00:19.130
There is friction between
the track and carts

00:00:19.130 --> 00:00:22.180
and between the moving parts of the carts.

00:00:22.180 --> 00:00:25.470
Which of the following best
represents the momentum

00:00:25.470 --> 00:00:29.290
of the left-side cart
as a function of time

00:00:29.290 --> 00:00:32.310
before and after the collision?

00:00:32.310 --> 00:00:34.480
So before we even look at the choices,

00:00:34.480 --> 00:00:36.090
let's see if we can
sketch it out ourselves.

00:00:36.090 --> 00:00:37.140
So pause this video

00:00:37.140 --> 00:00:38.900
and see if you can make a sketch

00:00:38.900 --> 00:00:41.120
of what the momentum of the left-side cart

00:00:41.120 --> 00:00:43.600
would look like as a function of time.

00:00:43.600 --> 00:00:45.340
And actually try to do it
for the right-side cart

00:00:45.340 --> 00:00:46.980
while you're at it.

00:00:46.980 --> 00:00:48.450
All right now let's do this together.

00:00:48.450 --> 00:00:50.790
So just to make sure we
understand what's going on,

00:00:50.790 --> 00:00:52.060
this cart is moving,

00:00:52.060 --> 00:00:53.510
they have the same mass.

00:00:53.510 --> 00:00:55.500
This left cart is moving to the right

00:00:55.500 --> 00:00:57.090
with a velocity of two v,

00:00:57.090 --> 00:00:59.340
the cart on the right has the same mass

00:00:59.340 --> 00:01:02.470
and it's moving to the right
with half the velocity.

00:01:02.470 --> 00:01:05.040
And since momentum is mass times velocity,

00:01:05.040 --> 00:01:07.780
this will also have half the momentum.

00:01:07.780 --> 00:01:09.490
So one way to think about it is,

00:01:09.490 --> 00:01:11.810
if on our vertical axis,

00:01:11.810 --> 00:01:13.620
let me draw a straighter line.

00:01:13.620 --> 00:01:15.580
If on our vertical axis,

00:01:15.580 --> 00:01:17.990
we have momentum.

00:01:17.990 --> 00:01:20.030
Momentum.

00:01:20.030 --> 00:01:22.400
And on our horizontal axis,

00:01:22.400 --> 00:01:24.033
we have time.

00:01:24.920 --> 00:01:26.720
So this is

00:01:26.720 --> 00:01:27.710
time,

00:01:27.710 --> 00:01:29.400
right over there.

00:01:29.400 --> 00:01:33.080
So this left cart will
start with some momentum

00:01:33.080 --> 00:01:34.730
but since there's friction in the system,

00:01:34.730 --> 00:01:37.840
especially in the mechanics of this cart,

00:01:37.840 --> 00:01:39.310
you're going to see it slow down.

00:01:39.310 --> 00:01:40.143
The force of friction

00:01:40.143 --> 00:01:41.770
is going against the direction of motion

00:01:41.770 --> 00:01:43.270
so it's going to decelerate it.

00:01:43.270 --> 00:01:46.530
And then its going to hit
the other slower moving cart

00:01:46.530 --> 00:01:48.210
and that other slower,

00:01:48.210 --> 00:01:49.480
and one way to think about it is

00:01:49.480 --> 00:01:51.490
some of the momentum will get transferred

00:01:51.490 --> 00:01:53.110
to the slower moving cart

00:01:53.110 --> 00:01:55.300
because then they're going
to be moving together

00:01:55.300 --> 00:01:56.780
at the same velocity.

00:01:56.780 --> 00:01:58.350
And so, you could imagine,

00:01:58.350 --> 00:02:00.120
and then at that slower velocity

00:02:00.120 --> 00:02:01.300
will continue to decelerate

00:02:01.300 --> 00:02:02.800
because of the friction.

00:02:02.800 --> 00:02:06.870
And so the graph of momentum
over time of the left cart

00:02:06.870 --> 00:02:09.400
might look something like that.

00:02:09.400 --> 00:02:10.290
Now, just for kicks,

00:02:10.290 --> 00:02:11.123
if you wanna think about,

00:02:11.123 --> 00:02:12.900
what about the right cart?

00:02:12.900 --> 00:02:15.170
Well, the right cart has the same mass

00:02:15.170 --> 00:02:16.890
but half the velocity.

00:02:16.890 --> 00:02:19.450
So it's going to have
half the initial momentum.

00:02:19.450 --> 00:02:22.480
And so it would also degrade,

00:02:22.480 --> 00:02:24.620
it would look something like this.

00:02:24.620 --> 00:02:26.710
But then right at the collision,

00:02:26.710 --> 00:02:28.800
the right cart would gain momentum.

00:02:28.800 --> 00:02:31.270
And then they should
go together like that.

00:02:31.270 --> 00:02:33.480
So these are the lines
for each individual one.

00:02:33.480 --> 00:02:35.870
And if you wanted the combined system,

00:02:35.870 --> 00:02:39.000
well, then you would add these momentums

00:02:39.000 --> 00:02:41.050
and you would have something
that just would be a line

00:02:41.050 --> 00:02:42.020
that degrades.

00:02:42.020 --> 00:02:44.270
But anyway, they asked
us about the left cart.

00:02:44.270 --> 00:02:46.170
And so we said hey, something

00:02:46.170 --> 00:02:49.500
something that looks like
this should work out.

00:02:49.500 --> 00:02:51.493
So let's look at the choices.

00:02:54.040 --> 00:02:55.750
So,

00:02:55.750 --> 00:03:00.120
this first choice has
two problems with it.

00:03:00.120 --> 00:03:02.900
First, it shows a world in
which you have no friction.

00:03:02.900 --> 00:03:05.080
It's just, the momentum stays constant,

00:03:05.080 --> 00:03:07.130
and so that means its
velocity to the right

00:03:07.130 --> 00:03:08.020
is staying constant.

00:03:08.020 --> 00:03:10.570
And then when its trying
to show the collision,

00:03:10.570 --> 00:03:12.600
it makes the momentum negative.

00:03:12.600 --> 00:03:15.090
And this would imply that all of a sudden,

00:03:15.090 --> 00:03:17.350
that cart would be moving to the left.

00:03:17.350 --> 00:03:19.500
Where the convention that we typically use

00:03:19.500 --> 00:03:21.720
is right is positive and left is negative.

00:03:21.720 --> 00:03:23.320
And so we can rule this one out.

00:03:24.200 --> 00:03:26.590
This second one looks a little bit better.

00:03:26.590 --> 00:03:29.510
But once again, it's assuming
a world without friction.

00:03:29.510 --> 00:03:31.540
Where the velocity stays constant

00:03:31.540 --> 00:03:32.940
and so the momentum stays constant.

00:03:32.940 --> 00:03:34.010
And still you have the collision,

00:03:34.010 --> 00:03:36.280
and then you still have
it after the collision

00:03:36.280 --> 00:03:37.490
the momentum stays constant.

00:03:37.490 --> 00:03:38.480
So we can rule that out.

00:03:38.480 --> 00:03:39.890
If they said it was frictionless,

00:03:39.890 --> 00:03:41.900
this is what we would have picked.

00:03:41.900 --> 00:03:45.510
Choice D, it's looking
good in the beginning

00:03:45.510 --> 00:03:47.790
but then it shows that
the momentum goes to zero.

00:03:47.790 --> 00:03:51.070
Implying that the left cart
just comes to a standstill.

00:03:51.070 --> 00:03:53.100
And once again, we would not expect that.

00:03:53.100 --> 00:03:54.710
And so we would rule that out.

00:03:54.710 --> 00:03:58.373
And so this is very close
to what we drew, choice B.


Kinematics and force example

WEBVTT
Kind: captions
Language: en

00:00:00.237 --> 00:00:02.980
- [Instructor] A 1,900 kilogram truck

00:00:02.980 --> 00:00:06.550
has an initial speed of
12 meters per second.

00:00:06.550 --> 00:00:08.440
The driver applies the brakes

00:00:08.440 --> 00:00:11.670
and the truck stops in 3.1 seconds.

00:00:11.670 --> 00:00:13.600
What is the best estimate

00:00:13.600 --> 00:00:18.600
of the magnitude of the average
braking force on the truck?

00:00:18.660 --> 00:00:21.010
Pause this video, see if
you can work this out.

00:00:22.270 --> 00:00:24.900
All right, probably the
simplest way to approach this

00:00:24.900 --> 00:00:26.230
is to say, well we could figure out

00:00:26.230 --> 00:00:28.040
the magnitude of the acceleration,

00:00:28.040 --> 00:00:30.280
the average acceleration on that truck

00:00:30.280 --> 00:00:32.640
as it comes to a stop.

00:00:32.640 --> 00:00:34.570
And then we could use Newton's second law,

00:00:34.570 --> 00:00:35.740
F equals ma

00:00:35.740 --> 00:00:36.740
'cause we know its mass.

00:00:36.740 --> 00:00:37.580
And if we know the mass

00:00:37.580 --> 00:00:39.050
and the magnitude of the acceleration,

00:00:39.050 --> 00:00:40.120
we could use Newton's second law

00:00:40.120 --> 00:00:43.520
to come up with the magnitude
of the average braking force.

00:00:43.520 --> 00:00:45.200
And they just want us to estimate it.

00:00:45.200 --> 00:00:47.770
And we can see that these
choices are pretty far apart.

00:00:47.770 --> 00:00:49.780
So an estimate will serve us well.

00:00:49.780 --> 00:00:51.780
So first let's think
about the acceleration.

00:00:51.780 --> 00:00:52.840
The acceleration

00:00:52.840 --> 00:00:54.410
is going to be our change in velocity

00:00:54.410 --> 00:00:55.800
over change in time.

00:00:55.800 --> 00:00:58.410
And so that's just going
to be our final velocity

00:00:58.410 --> 00:01:00.270
minus our initial velocity, the numerator.

00:01:00.270 --> 00:01:03.460
So our final velocity is
zero meters per second.

00:01:03.460 --> 00:01:05.030
We come to a stop.

00:01:05.030 --> 00:01:06.660
Minus our initial velocity,

00:01:06.660 --> 00:01:09.210
that's 12 meters per second.

00:01:09.210 --> 00:01:10.910
And the convention that we'll assume,

00:01:10.910 --> 00:01:11.743
and is typical,

00:01:11.743 --> 00:01:13.340
is that if we're moving to
the right, it's positive,

00:01:13.340 --> 00:01:15.570
if we're moving to the
left, it's negative.

00:01:15.570 --> 00:01:18.070
And so it's moving to the
right at 12 meters per second

00:01:18.070 --> 00:01:20.920
so we're subtracting that
positive velocity out.

00:01:20.920 --> 00:01:22.680
So this is our change in velocity

00:01:22.680 --> 00:01:24.560
over our change in time.

00:01:24.560 --> 00:01:26.750
Well, 3.1 seconds elapses.

00:01:26.750 --> 00:01:28.980
So over 3.1 seconds.

00:01:28.980 --> 00:01:31.510
And so what is this going to be

00:01:31.510 --> 00:01:33.760
approximately equal to?

00:01:33.760 --> 00:01:34.593
Well, let's see.

00:01:34.593 --> 00:01:36.090
Negative 12 over 3.1

00:01:37.330 --> 00:01:40.340
is going to be approximately four.

00:01:40.340 --> 00:01:42.130
And once again, I am estimating.

00:01:42.130 --> 00:01:44.530
It might be a little bit closer to,

00:01:44.530 --> 00:01:45.363
oh, and it's going to be

00:01:45.363 --> 00:01:47.210
approximately equal to negative four.

00:01:47.210 --> 00:01:50.710
It might be a little bit
closer to negative 3.9,

00:01:50.710 --> 00:01:51.543
around there.

00:01:51.543 --> 00:01:52.560
But I'll go with negative four.

00:01:52.560 --> 00:01:56.370
And then the units are
meters per second squared.

00:01:56.370 --> 00:01:59.140
And now we use Newton's
second law to take this

00:01:59.140 --> 00:02:02.400
to think about the magnitude
of the braking force.

00:02:02.400 --> 00:02:03.233
Which is going to be,

00:02:03.233 --> 00:02:05.460
and this makes sense that
the acceleration is negative

00:02:05.460 --> 00:02:06.950
that our velocity's in this direction

00:02:06.950 --> 00:02:09.380
but our acceleration is
in the other direction,

00:02:09.380 --> 00:02:10.710
it is slowing down.

00:02:10.710 --> 00:02:12.870
We're getting lower and
lower and lower velocities.

00:02:12.870 --> 00:02:15.130
And so our force is going
to be the same direction,

00:02:15.130 --> 00:02:17.530
our net force is in the same
direction as that acceleration.

00:02:17.530 --> 00:02:18.950
It's going to be to the left.

00:02:18.950 --> 00:02:20.670
So if we had a sign on it,

00:02:20.670 --> 00:02:21.940
it would be negative.

00:02:21.940 --> 00:02:23.620
But we just care about the magnitude.

00:02:23.620 --> 00:02:25.210
So when we think about
Newton's second law,

00:02:25.210 --> 00:02:28.070
we'll also just look at the
magnitude of the acceleration.

00:02:28.070 --> 00:02:29.130
So Newton's second law,

00:02:29.130 --> 00:02:31.120
the magnitude of the force,

00:02:31.120 --> 00:02:32.650
needs to be equal to

00:02:32.650 --> 00:02:35.900
the mass times the magnitude
of the acceleration.

00:02:35.900 --> 00:02:38.490
So this is going to be equal to,

00:02:38.490 --> 00:02:39.323
this is,

00:02:39.323 --> 00:02:40.860
let me write an approximation here.

00:02:40.860 --> 00:02:42.720
Because I approximated this.

00:02:42.720 --> 00:02:46.280
It's going to be
approximately 1,900 kilograms

00:02:46.280 --> 00:02:50.530
times four meters per second squared.

00:02:50.530 --> 00:02:52.490
I didn't feel the need to
write the negative there

00:02:52.490 --> 00:02:54.450
'cause I just wanna get the magnitude.

00:02:54.450 --> 00:02:56.920
I care about absolute value right now.

00:02:56.920 --> 00:02:58.470
So if I do this,

00:02:58.470 --> 00:03:01.000
this is going to be approximately what?

00:03:01.000 --> 00:03:04.550
7,600 kilograms.

00:03:04.550 --> 00:03:05.710
Now if I look at the choices,

00:03:05.710 --> 00:03:07.510
I don't see 7,600,

00:03:07.510 --> 00:03:10.210
but the closest one over here is 7,400.

00:03:10.210 --> 00:03:12.010
And I feel good about that

00:03:12.010 --> 00:03:15.330
because the real value
here might be closer

00:03:15.330 --> 00:03:18.320
to negative 3.9 meters per second squared

00:03:18.320 --> 00:03:21.300
and if this was 3.9 right over here,

00:03:21.300 --> 00:03:23.640
3.9 times 1,900,

00:03:23.640 --> 00:03:26.370
well that gets us a lot
closer to this right here.

00:03:26.370 --> 00:03:27.740
So I like this choice.

00:03:27.740 --> 00:03:30.723
And all the other ones are
way off from our estimate.


Conservation of momentum and energy example

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:02.520
- [Instructor] Blocks A
and B are pressed together

00:00:02.520 --> 00:00:04.550
with a spring between them.

00:00:04.550 --> 00:00:06.670
When the blocks are released from rest,

00:00:06.670 --> 00:00:09.000
the spring pushes the blocks apart

00:00:09.000 --> 00:00:12.280
so that the 75/100 of a kilogram block A

00:00:12.280 --> 00:00:15.010
moves up the 30 degree ramp to the left

00:00:15.010 --> 00:00:18.840
and the 0.25 kilogram block B

00:00:18.840 --> 00:00:22.920
moves to the right at a velocity
of three meters per second.

00:00:22.920 --> 00:00:25.060
Assume friction is negligible.

00:00:25.060 --> 00:00:27.170
Which of the following
is the best estimate

00:00:27.170 --> 00:00:31.360
of the maximum distance d that
block A slides up the ramp?

00:00:31.360 --> 00:00:34.550
So pause this video and see
if you can figure that out.

00:00:34.550 --> 00:00:35.620
All right now let's just first think about

00:00:35.620 --> 00:00:37.620
what would happen intuitively.

00:00:37.620 --> 00:00:40.860
So we have A and B, could
imagine you're squeezing

00:00:40.860 --> 00:00:42.450
them together so you
have that energy stored

00:00:42.450 --> 00:00:43.283
in that spring.

00:00:43.283 --> 00:00:44.780
And then when you let go,

00:00:44.780 --> 00:00:46.590
we know that B moves to the right

00:00:46.590 --> 00:00:48.700
and you would intuitively
think that A would move

00:00:48.700 --> 00:00:49.910
to the left.

00:00:49.910 --> 00:00:54.030
And it seems like the law
of conservation of momentum

00:00:54.030 --> 00:00:55.910
is going to be at play here.

00:00:55.910 --> 00:01:00.590
Because if the system of A
and B is initially at rest,

00:01:00.590 --> 00:01:03.220
each of these blocks' velocities are zero,

00:01:03.220 --> 00:01:04.980
doesn't matter what their masses are.

00:01:04.980 --> 00:01:07.580
That means that the total
momentum for the system

00:01:07.580 --> 00:01:10.370
of A and B is going to be zero.

00:01:10.370 --> 00:01:13.890
So even after we let go
and B moves to the right

00:01:13.890 --> 00:01:15.580
and A moves to the left,

00:01:15.580 --> 00:01:17.810
the total momentum of this system

00:01:17.810 --> 00:01:19.600
should still be zero.

00:01:19.600 --> 00:01:21.810
And so maybe we can use that to figure out

00:01:21.810 --> 00:01:24.930
what A's velocity is as it's sliding

00:01:24.930 --> 00:01:26.750
to the left horizontally.

00:01:26.750 --> 00:01:29.280
And then using that velocity,

00:01:29.280 --> 00:01:31.610
well we could think about
what's its kinetic energy.

00:01:31.610 --> 00:01:33.570
And if we know its kinetic energy,

00:01:33.570 --> 00:01:35.180
we could think about well,

00:01:35.180 --> 00:01:36.570
as it goes up the ramp,

00:01:36.570 --> 00:01:37.940
that kinetic energy's going to be turned

00:01:37.940 --> 00:01:39.100
into potential energy.

00:01:39.100 --> 00:01:41.760
And so how high up the ramp would it go?

00:01:41.760 --> 00:01:43.440
And then from that and that angle,

00:01:43.440 --> 00:01:45.860
we could figure out our distance.

00:01:45.860 --> 00:01:47.240
So let's do that.

00:01:47.240 --> 00:01:51.450
So let's start off with the law
of conservation of momentum.

00:01:51.450 --> 00:01:53.340
We know that after releasing,

00:01:53.340 --> 00:01:55.750
A's momentum plus B's momentum

00:01:55.750 --> 00:01:57.690
should add up to zero

00:01:57.690 --> 00:01:59.720
because the sum of
their momentums was zero

00:01:59.720 --> 00:02:01.330
when they were at rest together,

00:02:01.330 --> 00:02:02.680
when they were compressed.

00:02:02.680 --> 00:02:05.460
So what we could say is the momentum for A

00:02:05.460 --> 00:02:06.980
is going to be A's mass,

00:02:06.980 --> 00:02:09.140
so let me write it right over here,

00:02:09.140 --> 00:02:13.817
so 0.75 kilograms is A's mass

00:02:14.810 --> 00:02:16.220
times its velocity.

00:02:16.220 --> 00:02:17.240
Well we don't know what this is.

00:02:17.240 --> 00:02:19.360
Let's call this velocity sub

00:02:19.360 --> 00:02:22.660
or final velocity for A,

00:02:22.660 --> 00:02:24.870
and that's going to be
in meters per second,

00:02:24.870 --> 00:02:29.870
plus B's mass, which is 0.25 kilograms,

00:02:30.440 --> 00:02:32.800
times B's velocity.

00:02:32.800 --> 00:02:35.090
Well that's going to be
three meters per second,

00:02:35.090 --> 00:02:38.010
times three meters per second.

00:02:38.010 --> 00:02:41.090
This should be equal to zero.

00:02:41.090 --> 00:02:43.930
And so now we just have to solve for that.

00:02:43.930 --> 00:02:45.490
And so what do we get?

00:02:45.490 --> 00:02:48.760
Let's see, if we subtract this part

00:02:48.760 --> 00:02:50.800
from both sides,

00:02:50.800 --> 00:02:51.960
we will get,

00:02:51.960 --> 00:02:55.700
and this is just 0.75
kilogram meters per second,

00:02:55.700 --> 00:02:59.910
so we'll get 0.75 kilograms

00:02:59.910 --> 00:03:04.096
times the A's velocity
after it gets released

00:03:04.096 --> 00:03:08.530
is equal to negative 0.75

00:03:10.930 --> 00:03:14.550
kilogram meters per second.

00:03:14.550 --> 00:03:19.270
And so if you divide both
sides by 0.75 kilograms,

00:03:19.270 --> 00:03:21.500
on the left, you're just
going to be left with

00:03:21.500 --> 00:03:22.580
the velocity of A,

00:03:22.580 --> 00:03:23.600
and on the right,

00:03:23.600 --> 00:03:26.150
you would be left with negative one,

00:03:26.150 --> 00:03:28.150
then the kilograms would go away,

00:03:28.150 --> 00:03:30.430
negative one meters per second.

00:03:30.430 --> 00:03:32.920
So A's velocity as it
lets go from the spring

00:03:32.920 --> 00:03:35.768
or as it loses contact with that spring

00:03:35.768 --> 00:03:39.240
is going to be negative
one meters per second.

00:03:39.240 --> 00:03:40.575
Now what does the negative mean?

00:03:40.575 --> 00:03:44.200
Well if to the right is positive
three meters per second,

00:03:44.200 --> 00:03:46.170
then to the left is going to be negative.

00:03:46.170 --> 00:03:47.480
So it's going to be moving to the left

00:03:47.480 --> 00:03:49.620
at one meter per second.

00:03:49.620 --> 00:03:51.760
So let's use that
information to think about

00:03:51.760 --> 00:03:54.000
how far up the ramp it's going to go,

00:03:54.000 --> 00:03:56.062
and we're assuming that
friction is negligible

00:03:56.062 --> 00:03:58.840
and nothing really
interesting happens here,

00:03:58.840 --> 00:04:00.450
although we know in the
real world it might bump

00:04:00.450 --> 00:04:01.710
a little bit, but we're
just going to assume

00:04:01.710 --> 00:04:03.370
that that's not going
to be a major factor.

00:04:03.370 --> 00:04:05.310
And so all of that kinetic energy

00:04:05.310 --> 00:04:07.560
when it's moving to the
left at one meter per second

00:04:07.560 --> 00:04:11.210
is going to be converted to
potential energy at some height.

00:04:11.210 --> 00:04:13.440
So let's say that once
all of the kinetic energy

00:04:13.440 --> 00:04:15.150
turns into potential energy,

00:04:15.150 --> 00:04:19.200
A is at an altitude of h.

00:04:19.200 --> 00:04:21.330
So what's its potential energy going to be

00:04:21.330 --> 00:04:22.760
at that point in time?

00:04:22.760 --> 00:04:24.800
Well it's going to be the mass of A

00:04:25.670 --> 00:04:28.530
times the gravitational field,

00:04:28.530 --> 00:04:30.690
roughly 9.8 meters per second squared,

00:04:30.690 --> 00:04:31.690
if we wanna be even rougher

00:04:31.690 --> 00:04:34.140
say 10 meters per second squared,

00:04:34.140 --> 00:04:35.760
times that height.

00:04:35.760 --> 00:04:37.950
So that's going to be once
all of our kinetic energy

00:04:37.950 --> 00:04:40.010
has been turned into potential energy.

00:04:40.010 --> 00:04:42.630
Well this should be equal to the amount

00:04:42.630 --> 00:04:45.270
of kinetic energy we had at
this point right over here.

00:04:45.270 --> 00:04:46.680
Well what is that going to be?

00:04:46.680 --> 00:04:48.720
Well that's going to be 1/2 m v squared.

00:04:48.720 --> 00:04:52.440
So it's 1/2 times the mass of A

00:04:52.440 --> 00:04:54.690
times the velocity of A squared.

00:04:54.690 --> 00:04:58.800
So the velocity is negative
one meters per second,

00:04:58.800 --> 00:05:00.940
and then we're going to square it.

00:05:00.940 --> 00:05:04.250
Well we could divide both
sides by the mass of A.

00:05:04.250 --> 00:05:07.430
Negative one squared is
just going to be a one.

00:05:07.430 --> 00:05:10.880
So this is just going
to be a one eventually.

00:05:10.880 --> 00:05:12.900
And so you're left with
and then we can divide

00:05:12.900 --> 00:05:14.280
both sides by g,

00:05:14.280 --> 00:05:19.180
and we see that our height is equal to

00:05:19.180 --> 00:05:23.190
1/2 and then we're going to have,

00:05:23.190 --> 00:05:24.693
divided by g,

00:05:25.630 --> 00:05:28.850
divided by g, and then
this is going to be meters.

00:05:28.850 --> 00:05:31.210
And so what is this going to roughly be?

00:05:31.210 --> 00:05:33.360
Well h is going to be approximately,

00:05:33.360 --> 00:05:36.330
let's say that g is approximately
10 meters per second,

00:05:36.330 --> 00:05:39.500
because they talk about estimating here.

00:05:39.500 --> 00:05:44.500
So this would be one over two times 10,

00:05:46.370 --> 00:05:48.090
and then we're gonna get meters.

00:05:48.090 --> 00:05:49.990
So this is gonna be 1/20 of a meter.

00:05:49.990 --> 00:05:54.990
So h is approximately 0.05 meters.

00:05:55.370 --> 00:05:56.620
So you might be tempted to say oh look

00:05:56.620 --> 00:05:57.453
there's an answer there,

00:05:57.453 --> 00:05:59.100
but remember they're not asking us this h.

00:05:59.100 --> 00:06:00.800
They're asking us this d.

00:06:00.800 --> 00:06:02.280
And so here we just you have to use

00:06:02.280 --> 00:06:05.410
a little bit of trigonometry.

00:06:05.410 --> 00:06:08.250
Where we say okay what trig ratio deals

00:06:08.250 --> 00:06:11.300
with the opposite of an
angle and the hypotenuse?

00:06:11.300 --> 00:06:13.360
This is a right triangle right over here.

00:06:13.360 --> 00:06:14.830
Well that would be sine.

00:06:14.830 --> 00:06:16.563
So sine of 30 degrees,

00:06:17.400 --> 00:06:19.670
sine of 30 degrees,

00:06:19.670 --> 00:06:24.400
would be equal to the opposite
side which is h over d.

00:06:24.400 --> 00:06:27.530
We know that h is 0.05.

00:06:27.530 --> 00:06:28.860
Sine of 30 degrees,

00:06:28.860 --> 00:06:31.300
if you don't have a calculator
this is a good one to know,

00:06:31.300 --> 00:06:33.958
is 1/2, so you have

00:06:33.958 --> 00:06:37.770
1/2 is equal to 0.05 divided by d.

00:06:37.770 --> 00:06:40.320
We can multiply both sides by d,

00:06:40.320 --> 00:06:43.330
and then you'll have d over two

00:06:43.330 --> 00:06:45.260
is equal to 0.05

00:06:45.260 --> 00:06:48.040
or multiply both sides
by two and you get d

00:06:48.040 --> 00:06:50.400
is equal to 0.10

00:06:50.400 --> 00:06:52.460
or I could say roughly equal to

00:06:52.460 --> 00:06:54.010
since we are estimating.

00:06:54.010 --> 00:06:55.780
We had an estimate for g there.

00:06:55.780 --> 00:06:59.623
And then this is choice
D right over there.


Power dissipation in resistors in series versus in parallel

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:02.440
- [Instructor] A student
builds a circuit with a battery

00:00:02.440 --> 00:00:05.410
and two light bulbs in series.

00:00:05.410 --> 00:00:08.840
Then the student builds
a second circuit with two

00:00:08.840 --> 00:00:11.440
light bulbs in parallel.

00:00:11.440 --> 00:00:15.000
Which battery runs out of power first?

00:00:15.000 --> 00:00:18.590
Assume all bulbs have equal resistance.

00:00:18.590 --> 00:00:23.190
Assume both batteries have
equal electrical potential.

00:00:23.190 --> 00:00:24.640
All right, so let's just think about this

00:00:24.640 --> 00:00:26.100
before we even look at the choices.

00:00:26.100 --> 00:00:29.750
So the first scenario,
you have your battery,

00:00:29.750 --> 00:00:31.840
positive terminal, negative terminal.

00:00:31.840 --> 00:00:35.320
And you have two light bulbs in series.

00:00:35.320 --> 00:00:36.800
So let's just draw it this way.

00:00:36.800 --> 00:00:40.210
And traditional light bulbs
really are just resistors

00:00:40.210 --> 00:00:42.300
that when a current goes through them

00:00:42.300 --> 00:00:45.450
they get hot enough, that's
because the energy dissipated,

00:00:45.450 --> 00:00:47.060
that they emit visible light.

00:00:47.060 --> 00:00:50.200
So that's the first resistor,
or the first light bulb,

00:00:50.200 --> 00:00:51.880
second one, and it goes back.

00:00:51.880 --> 00:00:53.420
So that's the first scenario.

00:00:53.420 --> 00:00:55.840
The second scenario looks like this.

00:00:55.840 --> 00:00:59.370
Plus minus, where they

00:00:59.370 --> 00:01:01.820
are in parallel.

00:01:01.820 --> 00:01:05.060
They are in parallel.

00:01:05.060 --> 00:01:06.140
Just like that.

00:01:06.140 --> 00:01:08.870
And they say assume all
bulbs have equal resistance.

00:01:08.870 --> 00:01:13.350
So could just call their
resistances R, just like that.

00:01:13.350 --> 00:01:15.680
And we assume that both of these batteries

00:01:15.680 --> 00:01:18.630
are, for the sake of argument, identical.

00:01:18.630 --> 00:01:20.280
So one way to think about it is,

00:01:20.280 --> 00:01:22.390
what is going to be the
equivalent resistance

00:01:22.390 --> 00:01:23.700
for these circuits?

00:01:23.700 --> 00:01:26.430
This one when you have
two resistors in series,

00:01:26.430 --> 00:01:29.020
you could simplify, or
you could do this circuit

00:01:29.020 --> 00:01:34.020
as you have your voltage
source, which is our battery,

00:01:34.090 --> 00:01:37.220
and you could model it as a circuit

00:01:37.220 --> 00:01:41.900
with one resistor where
it's resistance is two R.

00:01:41.900 --> 00:01:45.330
If you wanna think of an
equivalent for this parallel

00:01:45.330 --> 00:01:48.283
circuit, you have your
same voltage source,

00:01:50.190 --> 00:01:52.830
and you could model it as
having a resistor there.

00:01:52.830 --> 00:01:54.250
And what would this resistance be?

00:01:54.250 --> 00:01:57.610
What is equivalent to these
two resistances in parallel?

00:01:57.610 --> 00:02:02.610
So if we call this resistance
sub two, we know that

00:02:02.710 --> 00:02:05.760
one over resistance sub two is equal to

00:02:05.760 --> 00:02:10.760
one over R plus one over
R, and this is the formula

00:02:10.830 --> 00:02:12.340
for equivalent resistance when you have

00:02:12.340 --> 00:02:13.630
resistors in parallel.

00:02:13.630 --> 00:02:17.830
And so this is going to
be equal to two over R.

00:02:17.830 --> 00:02:21.150
And so if you just take the
reciprocal of this and this,

00:02:21.150 --> 00:02:26.130
you get R two is equal to R over two.

00:02:26.130 --> 00:02:29.720
So this is equal to R over two.

00:02:29.720 --> 00:02:33.920
Now, what is the power
dissipated across a resistor?

00:02:33.920 --> 00:02:36.610
We know, let me just do
this in a different color,

00:02:36.610 --> 00:02:40.330
power dissipated across
a resistor is going to be

00:02:40.330 --> 00:02:43.150
the voltage across the
resistor times the current

00:02:43.150 --> 00:02:45.930
going through that resistor.

00:02:45.930 --> 00:02:49.290
Well, we know that the
voltage across both of these

00:02:49.290 --> 00:02:50.890
resistances is going to be the same.

00:02:50.890 --> 00:02:54.620
So V is going to be the
same in either situation.

00:02:54.620 --> 00:02:56.033
But what about the current?

00:02:57.280 --> 00:03:00.065
Well we know that V is equal to IR.

00:03:00.065 --> 00:03:02.160
Or another way to think about it is

00:03:02.160 --> 00:03:05.960
current is equal to
voltage over resistance.

00:03:05.960 --> 00:03:08.130
So our current in this first scenario

00:03:08.130 --> 00:03:11.330
is going to be V over
two R, while our current

00:03:11.330 --> 00:03:16.330
in this second scenario is
going to be V over R over two,

00:03:16.470 --> 00:03:20.990
which is the same thing as two V over R.

00:03:20.990 --> 00:03:24.280
So our current in this second scenario

00:03:24.280 --> 00:03:26.950
is going to be four times as large.

00:03:26.950 --> 00:03:29.950
So if you quadruple the
current and you have the same

00:03:29.950 --> 00:03:33.050
voltage, well you're
gonna quadruple the power.

00:03:33.050 --> 00:03:35.730
And so if you're quadrupling the power,

00:03:35.730 --> 00:03:38.140
you're quadrupling the amount of energy

00:03:38.140 --> 00:03:40.310
that's dissipated per unit of time.

00:03:40.310 --> 00:03:43.295
And so in this second
scenario where we quadrupled

00:03:43.295 --> 00:03:45.700
the power, we're quadrupling the energy

00:03:45.700 --> 00:03:47.130
dissipated over time.

00:03:47.130 --> 00:03:49.560
That battery is going
to run out a lot faster.

00:03:49.560 --> 00:03:52.620
So they ask which battery
runs out of power first,

00:03:52.620 --> 00:03:54.270
well that's gonna be the
batter in the circuit

00:03:54.270 --> 00:03:57.610
with two light bulbs in parallel.

00:03:57.610 --> 00:03:58.593
And we're done.


Energy dissipation across two resistors in series example

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:02.910
- [Instructor] A student
builds a circuit with a battery

00:00:02.910 --> 00:00:05.990
and two resistors in series.

00:00:05.990 --> 00:00:10.683
The resistance of R2 is
double the resistance of R1.

00:00:11.720 --> 00:00:16.720
Below is the graph of the
energy lost at R1 over time.

00:00:16.840 --> 00:00:18.600
So that's this graph.

00:00:18.600 --> 00:00:23.390
Which of the following shows
the energy lost at R2 over time

00:00:23.390 --> 00:00:24.960
in terms of E1.

00:00:24.960 --> 00:00:26.950
We have these four choices here.

00:00:26.950 --> 00:00:28.790
So pause this video and see if you can

00:00:28.790 --> 00:00:31.340
figure it out on your own.

00:00:31.340 --> 00:00:32.920
All right, now let's do this together.

00:00:32.920 --> 00:00:34.820
So first let's visualize the circuit.

00:00:34.820 --> 00:00:36.825
So you have your voltage source,

00:00:36.825 --> 00:00:39.090
this is our battery here, so
this is the positive terminal,

00:00:39.090 --> 00:00:41.810
negative terminal, and
then this could be our

00:00:41.810 --> 00:00:44.320
first resistor, and then
that's our second resistor.

00:00:44.320 --> 00:00:47.070
They tell us these two
resistors are in series

00:00:47.070 --> 00:00:49.310
and we complete the circuit here.

00:00:49.310 --> 00:00:54.310
This is going to be R1, this is R2,

00:00:54.450 --> 00:00:57.620
and then if you just have
one circuit like this,

00:00:57.620 --> 00:01:00.680
where you don't have any parts
of it that are in parallel,

00:01:00.680 --> 00:01:02.990
your current throughout this circuit

00:01:02.990 --> 00:01:04.100
is going to be constant.

00:01:04.100 --> 00:01:05.860
So we could say at any point here,

00:01:05.860 --> 00:01:09.660
our current, let's just call that I.

00:01:09.660 --> 00:01:12.120
Now they tell us how do these resistances

00:01:12.120 --> 00:01:13.340
relate to each other.

00:01:13.340 --> 00:01:17.480
If R1 has a resistance
of, let's call it X ohms,

00:01:17.480 --> 00:01:19.830
they tell us that the
resistance of R2 is double

00:01:19.830 --> 00:01:21.560
the resistance of R1.

00:01:21.560 --> 00:01:25.410
So this is X ohms, this
is going to be two X ohms.

00:01:25.410 --> 00:01:27.410
And then to think about, what would be,

00:01:27.410 --> 00:01:30.760
if this is the energy
dissipated across R1 over time,

00:01:30.760 --> 00:01:33.810
to think about the energy
dissipated across R2 over time,

00:01:33.810 --> 00:01:35.850
let's think about power and energy

00:01:35.850 --> 00:01:37.450
and what we know about them.

00:01:37.450 --> 00:01:39.510
So we know that power is equal

00:01:39.510 --> 00:01:41.500
to energy dissipated over time.

00:01:41.500 --> 00:01:44.670
So we could write, for
example, power is equal

00:01:44.670 --> 00:01:48.970
to energy dissipated over time,

00:01:48.970 --> 00:01:52.020
which can be expressed
as your change in voltage

00:01:52.020 --> 00:01:55.430
across a resistor times the current

00:01:55.430 --> 00:01:57.750
going through that resistor.

00:01:57.750 --> 00:02:00.620
And so if you want an
expression for energy dissipated

00:02:00.620 --> 00:02:03.870
over time, you can look at
this part, right over there.

00:02:03.870 --> 00:02:06.750
We can just multiply
both sides by delta T.

00:02:06.750 --> 00:02:10.270
And so we could write that
energy dissipated over time

00:02:10.270 --> 00:02:12.710
is going to be equal to
our change in voltage

00:02:12.710 --> 00:02:15.240
across the resistor times the current

00:02:15.240 --> 00:02:20.190
going through the resistor,
times our change in time.

00:02:20.190 --> 00:02:23.610
Now, if we are starting
at time equals zero,

00:02:23.610 --> 00:02:25.900
our change in time is
just going to be whatever

00:02:25.900 --> 00:02:28.060
time we are at minus
zero, so it's going to be

00:02:28.060 --> 00:02:30.240
whatever time we're at,
so I'm just gonna write

00:02:30.240 --> 00:02:32.960
times T right over here.

00:02:32.960 --> 00:02:36.460
Now your change in
voltage across a resistor

00:02:36.460 --> 00:02:38.440
we have learned before,
your change in voltage

00:02:38.440 --> 00:02:39.940
across a resistor is just going to be

00:02:39.940 --> 00:02:41.770
the current going through that resistor

00:02:41.770 --> 00:02:43.560
times the resistance.

00:02:43.560 --> 00:02:46.150
So we could take this and
substitute it back for

00:02:46.150 --> 00:02:48.300
delta V, and what do we get?

00:02:48.300 --> 00:02:52.550
We get energy dissipated
is going to be equal

00:02:52.550 --> 00:02:55.620
to our current going through a resistor

00:02:55.620 --> 00:02:58.560
times the resistance times
the current going through

00:02:58.560 --> 00:03:01.560
the resistor times time, or we could write

00:03:01.560 --> 00:03:04.710
that energy dissipated as a function of,

00:03:04.710 --> 00:03:06.960
well as a function of all
three of these things,

00:03:06.960 --> 00:03:09.560
is going to be the current squared

00:03:09.560 --> 00:03:13.340
times the resistance times time.

00:03:13.340 --> 00:03:17.320
Now, we already know
from this that at time T1

00:03:17.320 --> 00:03:19.790
we have an energy dissipated of E1,

00:03:19.790 --> 00:03:22.100
and that's what the resistance of X ohms.

00:03:22.100 --> 00:03:26.700
So we can say we have
energy dissipated of E1

00:03:26.700 --> 00:03:30.610
is equal to I squared times X ohms,

00:03:30.610 --> 00:03:32.943
that's the resistance, times T1.

00:03:34.300 --> 00:03:36.770
But now let's think about
what the energy dissipated

00:03:36.770 --> 00:03:41.770
at time T sub one for our
second resistor would be.

00:03:42.660 --> 00:03:44.960
At the energy dissipated,
if we just go back

00:03:44.960 --> 00:03:47.840
to this equation right over here,

00:03:47.840 --> 00:03:52.840
is going to be equal to T
sub one times our current,

00:03:53.430 --> 00:03:55.130
remember we have the same
current going through

00:03:55.130 --> 00:03:57.380
both resistors, they're in series,

00:03:57.380 --> 00:04:00.500
so we have the same
current there, I squared,

00:04:00.500 --> 00:04:02.590
and then what is going
to be the resistance?

00:04:02.590 --> 00:04:05.230
Well instead of an X,
we have a two X here.

00:04:05.230 --> 00:04:08.500
And so we can rewrite
this, we could rewrite this

00:04:08.500 --> 00:04:13.500
as two time I squared times X times T1.

00:04:15.640 --> 00:04:18.960
Well, this right over
here is the same thing

00:04:18.960 --> 00:04:23.140
as the amount of energy
dissipated at time T sub one

00:04:23.140 --> 00:04:24.190
in our first resistor.

00:04:24.190 --> 00:04:26.940
So this is E sub one.

00:04:26.940 --> 00:04:29.780
So at time T sub one,
the energy dissipated

00:04:29.780 --> 00:04:32.160
and our second resistor
is going to be twice

00:04:32.160 --> 00:04:33.600
as much energy.

00:04:33.600 --> 00:04:36.460
And so what we wanna look
at is at time T sub one

00:04:36.460 --> 00:04:37.980
we wanna have twice as much energy.

00:04:37.980 --> 00:04:40.330
This dissipated, this
only has a fourth as much,

00:04:40.330 --> 00:04:42.060
so we rule that out, this has half as much

00:04:42.060 --> 00:04:43.930
energy dissipated, rule that out.

00:04:43.930 --> 00:04:46.400
This has the same amount
of energy dissipated,

00:04:46.400 --> 00:04:48.160
we rule that out, but then here,

00:04:48.160 --> 00:04:52.020
at time T sub one, we have
dissipated twice as much energy.

00:04:52.020 --> 00:04:55.200
So we like that choice right over there.

00:04:55.200 --> 00:04:57.170
Another way to think about it is,

00:04:57.170 --> 00:04:59.600
if you have the same current
going through two resistors,

00:04:59.600 --> 00:05:01.800
and the second resistor
has twice the resistance,

00:05:01.800 --> 00:05:04.950
and any given amount of time
for the second resistor,

00:05:04.950 --> 00:05:07.260
you're going to dissipate
twice as much energy.

00:05:07.260 --> 00:05:08.760
Another way to think about
it, you'll have twice

00:05:08.760 --> 00:05:10.820
the slope of this line.

00:05:10.820 --> 00:05:13.130
So this one right over here
clearly has twice the slope

00:05:13.130 --> 00:05:16.770
of this, or you could say
for any given amount of time,

00:05:16.770 --> 00:05:20.130
so from time equals zero to T sub one,

00:05:20.130 --> 00:05:22.030
you would dissipate twice as much energy.

00:05:22.030 --> 00:05:23.810
So instead of E sub one,

00:05:23.810 --> 00:05:26.233
you would dissipate two times E sub one.


Newton's third law conceptual worked example

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:02.130
- [Instructor] Block A, with mass M,

00:00:02.130 --> 00:00:06.800
sits on top of block B,
with mass 2M in an elevator.

00:00:06.800 --> 00:00:10.060
The elevator is moving
downward and slowing down.

00:00:10.060 --> 00:00:11.910
All right, when we have
this diagram over here,

00:00:11.910 --> 00:00:15.130
it's moving downward and slowing down.

00:00:15.130 --> 00:00:18.380
So that means it's accelerating upwards.

00:00:18.380 --> 00:00:21.800
The magnitude of the
Earth's force of gravity

00:00:21.800 --> 00:00:26.800
on block A, is F sub G sub A.

00:00:27.150 --> 00:00:30.730
The force magnitude exerted
on block A by block B

00:00:30.730 --> 00:00:32.630
is F sub A.

00:00:32.630 --> 00:00:36.100
And the force magnitude
exerted on block B by block A

00:00:36.100 --> 00:00:37.500
is F sub B.

00:00:37.500 --> 00:00:40.370
How do the force magnitudes compare?

00:00:40.370 --> 00:00:42.010
This is fascinating.

00:00:42.010 --> 00:00:44.090
So pause this video and see
if you can figure it out

00:00:44.090 --> 00:00:44.923
on your own.

00:00:46.500 --> 00:00:48.880
All right, well let's just
draw a free-body diagram

00:00:48.880 --> 00:00:51.070
for block A.

00:00:51.070 --> 00:00:52.250
So,

00:00:52.250 --> 00:00:55.420
I have block A right over here.

00:00:55.420 --> 00:00:57.400
Now there's a couple of things we know.

00:00:57.400 --> 00:01:01.710
We know that this whole
system is moving downward

00:01:01.710 --> 00:01:04.010
and slowing down.

00:01:04.010 --> 00:01:04.990
And we know that 'cause it says

00:01:04.990 --> 00:01:07.420
the elevator is moving
downward and slowing down

00:01:07.420 --> 00:01:09.190
and block A is sitting on block B

00:01:09.190 --> 00:01:11.460
is sitting on the elevator.

00:01:11.460 --> 00:01:13.970
And so if you're moving
downward and slowing down,

00:01:13.970 --> 00:01:17.740
that means that your net
acceleration is upwards.

00:01:17.740 --> 00:01:22.270
Which tells you that
your net force on block A

00:01:22.270 --> 00:01:23.670
needs to be upwards.

00:01:23.670 --> 00:01:25.740
Which tells you if we have some forces

00:01:25.740 --> 00:01:27.140
that are acting upwards,

00:01:27.140 --> 00:01:29.460
the magnitude of those are gonna be larger

00:01:29.460 --> 00:01:32.750
than the forces acting downwards.

00:01:32.750 --> 00:01:34.470
So for block A,

00:01:34.470 --> 00:01:37.900
what are the forces that
are acting downwards on it?

00:01:37.900 --> 00:01:41.040
Well, the force of gravity
is acting on block A.

00:01:41.040 --> 00:01:42.620
And that's that right over there.

00:01:42.620 --> 00:01:47.080
That is, its magnitude
is capital F sub G sub A,

00:01:48.050 --> 00:01:52.630
and what's the upward
force acting on block A?

00:01:52.630 --> 00:01:54.530
Well, they say the force magnitude exerted

00:01:54.530 --> 00:01:57.730
on block A by block B is F sub A.

00:01:57.730 --> 00:02:01.840
So the magnitude here is F sub A.

00:02:01.840 --> 00:02:02.720
So we immediately know,

00:02:02.720 --> 00:02:05.080
because we have a net
acceleration upwards,

00:02:05.080 --> 00:02:07.400
remember, we're moving down
but we're slowing down.

00:02:07.400 --> 00:02:09.710
So it's a net acceleration upwards.

00:02:09.710 --> 00:02:11.350
That immediately tells us

00:02:11.350 --> 00:02:14.310
that the magnitude of F sub A

00:02:14.310 --> 00:02:16.810
is going to be greater than

00:02:16.810 --> 00:02:20.610
the magnitude of the
force of gravity on A.

00:02:20.610 --> 00:02:22.080
And so we can immediately,

00:02:22.080 --> 00:02:24.540
we can get rid of some of these choices.

00:02:24.540 --> 00:02:27.420
But they also are comparing F sub B here.

00:02:27.420 --> 00:02:31.120
So how can we involve F
sub B in this inequality?

00:02:31.120 --> 00:02:34.370
Well, the key here is
in Newton's Third Law.

00:02:34.370 --> 00:02:37.540
Whatever force B is exerting on A,

00:02:37.540 --> 00:02:40.790
and we know we call that F sub A,

00:02:40.790 --> 00:02:42.570
or it's magnitude is F sub A.

00:02:42.570 --> 00:02:44.530
A is going to exert
from Newton's Third Law

00:02:44.530 --> 00:02:47.090
and equal and opposite force on B.

00:02:47.090 --> 00:02:49.400
So this is going to be magnitude,

00:02:49.400 --> 00:02:50.620
the arrows are,

00:02:50.620 --> 00:02:52.520
this arrow points downward,

00:02:52.520 --> 00:02:54.550
but it's gonna have the same magnitude.

00:02:54.550 --> 00:02:55.950
F sub B.

00:02:55.950 --> 00:02:59.730
So the magnitude of these
two forces are the same.

00:02:59.730 --> 00:03:01.410
So another way to think about it is

00:03:01.410 --> 00:03:04.890
F sub A is equal to F sub B

00:03:04.890 --> 00:03:06.390
by Newton's Third Law.

00:03:06.390 --> 00:03:10.050
And since we know that the whole system

00:03:10.050 --> 00:03:11.820
is accelerating upwards,

00:03:11.820 --> 00:03:14.300
it's moving downwards
but it's slowing down,

00:03:14.300 --> 00:03:16.480
we know that this thing is going to be

00:03:16.480 --> 00:03:19.630
greater than the force of gravity on A.

00:03:19.630 --> 00:03:22.943
And we see that that is
choice B, and we're done.