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PROFESSOR: Now we're going to do a second example.
This one is of a similar block, but it's now on an accelerating plank.
So the problem itself looks something like this.
We have a block.
It sits on a plank, and this is at a construction site, imagine, and so
there will be a couple of ropes that pull up on the plank, maybe just two
people standing on the roof.
And then this whole thing is going to accelerate with some
acceleration upwards.
Now, if we want to draw the force diagram for this, we would draw
separate force diagrams on each of these two bodies.
So we would have the block over here.
And as before, it would have a normal force on it and a gravity force.
And more traditionally, I'm going to label the gravity force mg instead of
the weight, and I'm going to label the upward force the normal force on the
block due to the plank.
Now, the other body is, of course, the plank, and the plank is
going to sit over here.
And now is the critical point or a very critical point.
Now, in order to satisfy Newton's third law, we have this normal force
acting between the block and the plank, and it acts equally on the two
but in opposite directions.
So if that's the normal force, then we have a normal force down here on the
plank, and I'll label this one normal force on the plank due to the block.
And these two forces, this one and this one, are Newton's third law pair
forces; that one right there and that one right there.
Whatever happens in this problem, these two forces are due to one
interaction, and they must be equal in magnitude and opposite in direction.
Again, we're ignoring friction here, but if we did have friction, it would
also act in the opposite direction.
Imagine that the block, someone tried to slide it to the right, then there
would be a friction force to the left on the block, but that would mean that
there would be a friction force to the right on the plank.
OK.
Now, the force diagram for the plank isn't done because we didn't put in
the force due to the cables.
And so the cables will give an upward force here and an upward force
on the other end.
And I emphasize that this whole thing is accelerating upwards with
acceleration a.
So now let's just go over our check.
Are we satisfying Newton's second law?
Well, let's look at the block first.
Clearly, the normal force that we've drawn here is bigger than the gravity
force, and so the combination of those two forces is in the
net direction upwards.
And so, indeed, the block could be accelerating upwards.
The actual details of applying Newton's law would be to take these
two systems now and separate them.
So you would have something like for the plank, which we're discussing--
I'd better label these forces the force due to the cable on the right
side and the force due to the cable on the left side.
And so we have the sum of all the forces in the y direction equals the
mass, and this is for the plank now, so that's the mass of the plank times
the acceleration.
Because acceleration is upward direction, we're taking positive to be
in the upward direction as one normally does, and now we simply have
to write out what the sum means.
The sum means add together every single force that acts on the body or
the object, which in this case is the plank.
And so we're going to have for the sum of the forces upwards.
We have the force due to the cable on the left side plus the force due to
the cable on the right side minus the mass of the plank times gravity, minus
the normal force on the plank due to the box or the block.
And this has to equal--
I don't have room-- equals m of the plank times a.
And in order to satisfy Newton's second law, well, this is an
expression of Newton's second law, which is mathematically correct and
accurate, and this is the relation that you have to work with.
And so when I said before that the forces of the two cables added
together had to be greater than the mass of the plank times gravity plus
the normal force on the plank due to the box, we can see that that was just
looking ahead to this equation and observing that for this to be
positive, so that the acceleration would be positive, that relation had
to occur, namely, that the cable forces were greater.
So now, finally, let's just do this similar procedure
for the block itself.
The sum of all the y forces on the block equals, well, there are only
two, the normal force on the block due to the plank, minus the mass, which is
the mass of the block, times gravity.
And this must equal the mass of the block times the acceleration.
So now, we have two equations, and there are two connections between
these equations.
We write down these equations separately because we're applying
Newton's second law separately with the block into the plank.
But there are mathematical relations that must hold between these two.
The first one is that the two normal forces are equal, because of Newton's
third law, where they're actually equal and opposite.
But the oppositeness has been accounted for by putting in this minus
sign right here.
And then there is the same acceleration.
Both of these things in the real world over here, they're constrained to move
together, and therefore, they have the same acceleration.
So therefore, this acceleration and that acceleration are equal.
And so those are mathematical, so it's wise for us, it was OK for us to have
used the same acceleration for the two different objects, because we know
from the constraint to the motion that those two objects will have to move
with the same acceleration.