A viewer of my previous physics videos asked me to help settle a dispute over the answer to another
physics problem involving a pulley. I gave it some thought and came up with an answer that
turned out to agree with his, but I'm not sure my reasoning is adequate. Let me know what you think.
A block that slides horizontally along a surface without friction is attached
to a weight by a string that loops over a frictionless pulley which is above the block.
Assuming that the block can't lift off the table, either because it has enough mass or because it's
on a rail that keeps it down, then when the weight is allowed to drop, are the accelerations of the
weight and the block equal or different? And if different, which one has the greater acceleration?
To be clear, this animation is not simulating the actual physics of the setup.
I'm not even sure how to do that, so I'm just shifting the block arbitrarily and
computing the corresponding position of the weight.
Normally, you start these problems by looking at the forces involved. The weight experiences
a gravitational force which is reduced by the force from the tension in the string.
For the block, we only need to consider the horizontal component of the tension force
since vertical forces are balanced by definition. So to figure out what the accelerations will
actually be, we would have to calculate the tension. But that seems really difficult.
Fortunately, I don't think we need it in order to answer the simpler question we've been given.
I think that because for a typical elementary pulley problem which has the pulley at the same
height as the block, the solution begins by simply stating as fact that the string forces
the displacements to be equal, therefore the velocities and the accelerations must also be
equal, which is true. But notice that we've just answered the question of relative acceleration
for this special case just based on how the two masses are constrained by the string.
So I think it will be just as valid to do the same when the pulley is at a height greater than zero.
First though, are the accelerations really always equal?
What if we allow the block to continue sliding to the other side of the pulley?
Aha! All of a sudden the weight changes direction and starts heading upwards. It experiences an
instantaneous change of direction while the block does not, and therefore the acceleration,
whatever it was in the instant before, must no longer be the same as that of the block.
In fact it's infinitely less for an infinitesimal amount of time.
Let's look at the derivative of this displacement graph. This shows the ratio of the velocities of
the weight and the block for a given displacement of the block. At first, the velocities are the
same so the ratio is one, and then the block gets to the other side and it becomes negative one.
Now let's look at the derivative of that function. This shows the change in the
ratios of velocity for a given displacement of the block. When the block is right under the pulley,
this graph shows negative infinity, which means that the weight's acceleration is infinitely
less than that of the block. And everywhere else it's zero, which means that there's
zero change in the relative velocities and that means the accelerations are the same.
Right? Call me out on that if you think my interpretation is off the mark because I'm
about to use this graph to justify my intuition that the block accelerates more than the weight.
Let's go back to the problem we were given.
What we can see from the top graph is that the displacement of the weight is no longer equal to
the displacement of the block. It's always less! At first, the difference is very small but as the
block approaches the pulley the difference gets more dramatic until we can see that the movement
of the block results in barely any movement of the weight. And I think that's the key.
If the block is always moving further than the weight in any range of its displacement, then
assuming they both started with zero velocity, it has to be accelerating more than the weight.
Looking at the bottom graph we reach the same conclusion. That red line is always below zero,
indicating (to me) that the acceleration of the weight is always less than that of the block.
It becomes even clearer with a larger pulley height. If this is the starting configuration,
then when the weight is released, the block will move much farther than the
weight in any range of time, so it must be experiencing a greater acceleration.
Now, who wants to compute the actual acceleration as a function of time? Hmmm, do I hear,
"second order non-linear differential equation"? And what about adding in some friction? Oy!