Video Transcript
The figure shows two blocks, block
A and block B. The blocks have equal masses. Block A is on a rough surface, and
block B is on block A. A force 𝐹 acts on block B, parallel to the surface. Initially, block B does not move,
and 𝐹 is steadily increased. When 𝐹 has increased enough to
make block B start to move, both blocks move in the direction of 𝐹 at the same
speed. Which of the following is a correct
statement about the relation of 𝜇 sub one, the coefficient of static friction
between block A and the surface, to 𝜇 sub two, the coefficient of static friction
between the blocks? (A) 𝜇 sub one is equal to 𝜇 sub
two. (B) 𝜇 sub one is less than 𝜇 sub
two. (C) 𝜇 sub one is greater than 𝜇
sub two.
Alright, in this example, we have
these two blocks: block B rests on block A and then block A is resting on this rough
surface. While the two blocks are
stationary, an applied force 𝐹 is exerted on block B, parallel to the interface
between the blocks. At first, nothing happens. But then, as 𝐹 is increased,
eventually block B does start to move. At that exact instant though, block
A is also set in motion. And both blocks move to the left at
the same speed we’re told. So, this is interesting. Even though the force 𝐹 is being
applied to block B, what ends up happening is both blocks move as a unit to the
left.
Knowing this, we want to make a
comparison between the coefficient of static friction that exists between the two
blocks, that’s called 𝜇 sub two, and also the coefficient of static friction that
exists between block A and the rough surface; that is called 𝜇 sub one. Basically, we want to make a
comparison between these two coefficients of static friction. As we think about what’s happening
physically in this situation, we see that even though we’re applying a force just to
block B, apparently, the frictional force between block B and block A is strong
enough that the whole entire mass of the two blocks together begins to move before
block B starts to slide across the top of block A.
We could say then that this static
frictional force here that acts between block A and the surface is weaker than the
static frictional force that acts between the two blocks. We can write out equations for
these two forces of static friction. We can recall that the magnitude of
the static frictional force acting on some object that’s at rest on a flat surface
is equal to that object’s mass times the acceleration due to gravity times the
coefficient of static friction between the object and the surface it rests on. If we call the frictional force in
our scenario that exists between block A and the surface 𝐹 sub one, then we know
that’s equal to the mass of block A plus the mass of block B, because block B rests
on top of block A, multiplied by 𝑔 times 𝜇 sub one, the coefficient of static
friction between block A and the surface. Compared to this, if we call the
static frictional force between the two blocks 𝐹 sub two, then that’s equal simply
to the mass of block B times 𝑔 times 𝜇 sub two.
In our problem statement, we were
told that as these blocks move, they move as a unit. We saw that this implies that the
frictional force between the blocks must be greater than that between block A and
the surface. In other words, 𝐹 sub two must be
greater than 𝐹 sub one. We can then replace 𝐹 sub two in
this inequality with this expression and 𝐹 sub one with this one. And once we’ve done that, let’s
recall an important bit of information given to us earlier on. We weren’t told what the masses of
blocks B and A are, but we were told that they’re equal. Off to the side then, we can write
that 𝑚 sub B is equal to 𝑚 sub A, and we can call them both just 𝑚. If we make that substitution in our
inequality, then we have this expression where 𝑚 and 𝑔 are common to both
sides.
If we then divide both sides by 𝑚
times 𝑔, then this cancels out those terms on either side of the equality while
keeping its direction the same. And we find that 𝜇 sub two is
greater than two times 𝜇 sub one. This means that even if we double
𝜇 sub one, it’s still less than 𝜇 sub two. Looking at our three answer
options, we see that this corresponds to option (B) 𝜇 sub one is less than 𝜇 sub
two.
