- Good morning, today we are going
to solve free response question number one
from the 2015 AP Physics I Exam.
♫ Flipping Physics ♫
- We have block one and block two
connected by a string of negligible mass.
That string goes over two pulleys.
Those pulleys have negligible mass
and negligible friction, and the mass
of block two is greater than the mass of block one.
Part A asks us to draw the free-body diagrams
for the forces acting on each block.
Billy, could you start please?
- Well each block has a force of gravity
acting downwards on it, and the mass
of block two is greater than the mass
of block one, so the length of the arrow
for the force of gravity acting on block two
should be longer than the length of the arrow
for the force of gravity acting on block one.
- Bo, please continue.
- There is a tension force in the string
and it acts up on both blocks.
- And because the string and pulleys
have no mass, the magnitudes of those two tension forces
are the same, oh, and because the pulleys have no friction.
- But how long should we draw the tension forces?
- Well, because block two has a larger force
of gravity acting on it than block one,
block one will accelerate upward
and block two will accelerate downward.
- Therefore the tension force has to be larger
in magnitude than the force of gravity one
to cause block one to accelerate upward
and the tension force also has to be smaller
in magnitude than the force of gravity two
to cause block two to accelerate downward.
- Great, that is the answer to part A.
Now, I do want to take a moment to pause
to reiterate the importance of free-body diagrams
or force diagrams.
On the AP test, your free-body diagrams
are a part of your solution, therefore,
you need to draw them carefully and legibly.
You need to make sure that the lengths of your arrows
matches the relative lengths of those force vectors.
You need to make sure you name your vectors,
you label your force vectors so they can identify
which forces you're talking about.
And if you need to break forces into components,
first draw a component-free free-body diagram,
then break forces into components
and then re-draw a free-body diagram with those components.
- Did anybody else notice the soapbox Mr. P is standing on?
- Crikey Bo, quit that codswallop, he's only trying to help.
- All right, let's move on.
Part B asks for us to derive the magnitude
of the acceleration of block two.
- Mr. P? - [Voiceover] Yes Bobby?
- Don't both blocks have the same acceleration
because they're attached by the string?
- Bobby that is correct, that is actually why
we're going to find the magnitude of the acceleration
of both blocks simultaneously by summing
the forces on the whole system.
Oh but first, we need to identify
the positive direction which is going to be up
for block one and down for block two,
the direction we know both blocks are going to accelerate.
Bobby, please?
- The net force on both blocks in the positive direction
you just showed is,
positive force of gravity two minus force of tension
plus force of tension minus force of gravity one
and the net force also equals mass times acceleration
which would be the total mass times the acceleration
of both blocks, the two tension forces cancel one another
out and we can substitute in mass two times acceleration
due to gravity for force of gravity two,
and mass one times acceleration due to gravity
for force of gravity one and mass one plus mass two
for the total mass.
Divide both sides by mass one plus mass two,
and we have our answer.
Acceleration equals mass two times acceleration due
to gravity minus mass one times acceleration due
to gravity, all divided by the quantity
mass one plus mass two.
- Very nice, you'll notice I actually went one step further.
I pulled the acceleration due to gravity out
which isn't absolutely necessary
but it looks a little bit cleaner.
Okay, part C adds a third mass.
There is no friction between block three and the table.
Bo, how does adding block three change
the free-body diagrams?
- Well, there is a force of gravity three
down on block three, and a force normal up
on block three caused by the table.
And adding block three changes the tension forces.
Now we need to label the tension force up
on block one, tension force one,
which has the same magnitude as the tension force one
which acts to the left on block three,
and we need to label the tension force up
on block two, tension force two,
which has the same magnitude as the tension force two,
which acts to the right on block three.
- Billy, could you please solve
for the new acceleration of the system?
- The net force on all three blocks
in the same positive direction as before
is force of gravity two minus tension force two
plus tension force two minus tension force one
plus tension force one minus force of gravity one
which all equals the total mass times acceleration.
All the tension forces cancel out,
and we can substitute in the same equations
as before, only now the total mass also includes
the mass of block three, so the acceleration
expression is almost the same, it just adds the mass three
in the denominator, therefore, adding block three
increases the value of the denominator,
which will decrease the acceleration of the block.
- Correct, now I will point out, because the question
only asks you to explain how you arrived at your answer
for what happens to the magnitude of the acceleration
of block two when you add block three.
You do not have to derive the acceleration
like we did right here.
You could just explain it using Newton's Second Law.
You could explain that, adding block three
does not change the net force acting on the system,
therefore because it increases the mass of the system,
it must therefore decrease the acceleration of the system
and therefore decrease the acceleration of block two.
Again, that is an explanation which is what they asked for.
I did a derivation because I feel like you understand
it a little bit better when we walk through the whole thing.
For example, you understand why the net force,
the addition of block three does not change
the net force acting on the system.
One other thing.
Notice that free response question number one
had absolutely no numbers in it.
So as I've said before, please, let go
of your numbers dependency.
Thank you very much for learning with me today.
I enjoyed learning with you.