Why are the accelerations of blocks A and B different in this pulley problem?

[PineApple2]

It is not actually homework, but a physics problem I have encountered. For ease of discussion, I am attaching the question [University Physics, Young and Freedman, 12th edition] and it's solution as appeared in the solutions manual.

What I don't understand is: In section (c), how can the accelerations of blocks A and B be different? They are connected by the same string. Doesn't it imply (by geometrical considerations, the string length remains the same) that the accelerations are also the same?

 

[Alucinor]

But they're on a pulley--the string isn't fixed to anything but the two masses.

Look at part B as well--you're applying enough force to lift only one of the masses [EVEN THOUGH the total force (294N) is the same as the combined two gravitational forces 9.8(20 + 10) = 294 N]. But this force is divded equally between the two, so each feels 147N. The smaller mass will surely lift then. However you're not applying enough for the other mass. This mass will stay on the ground. This implies different accelerations. It is the same in part C, but you're now applying just enough force to lift the bigger block, but that is still more than enough to lift the smaller one.

It is applying an equal force to each of the two masses (different masses + equal forces = different accelerations)If both masses were simply tied to a string your analysis would be in the right direction because they're basically one mass. You'd be able to model it as if they were one mass of combined mass m_a + m_b. The pulley, however, makes them independent.

 

[PineApple2]

But they're on a pulley--the string isn't fixed to anything but the two masses.

Look at part B as well--you're applying enough force to lift only one of the masses [EVEN THOUGH the total force (294N) is the same as the combined two gravitational forces 9.8(20 + 10) = 294 N]. But this force is divded equally between the two, so each feels 147N. The smaller mass will surely lift then. However you're not applying enough for the other mass. This mass will stay on the ground. This implies different accelerations. It is the same in part C, but you're now applying just enough force to lift the bigger block, but that is still more than enough to lift the smaller one.

It is applying an equal force to each of the two masses (different masses + equal forces = different accelerations)


If both masses were simply tied to a string your analysis would be in the right direction because they're basically one mass. You'd be able to model it as if they were one mass of combined mass m_a + m_b. The pulley, however, makes them independent.

but it IS the same rope connecting the 2 masses over the pulley. Do the different accelerations mean the rope is not taut?

 

[Alucinor]

It is the same rope but it can slide along the pulley--meaning you can be pulling up the lighter mass, making that part of the string shorter, while increasing the amount of string on the side of the heavier. The rope is taut the whole time, it just isn't able to lift up the heavier mass while it is able to lift up the lighter mass.

Do you see the difference? In a situation where the pulley was replaced by a knot, you couldn't have the rope moving like that, but since the pulley is there, you can, and since the rope can move, the accelerations on the masses can be different.

This is the kind of problem that you should try to model in the real world, if you're in a physics class go ask to borrow a pulley and do an analog to this experiment and you'll be able to see that one mass can move while the other stays stationary (due to the different accelerations)

You're thinking about the system like this:

                |
                | <-- rope
                |
               & <-- knot
              /   \
             /     \
          M1     M2

But it is this:

                |
                | <-- rope
                |
               O <-- pulley
              /   \
             /     \
          M1     M2

With parts b and c of that problem making it look like this eventually:

                |
                | <-- rope
                |
               O <-- pulley
              /   \
             /    M2
            /
          M1

 

[PineApple2]

You're right. My problem was that I imagined the rope not slipping. But since it can slip, the accelerations can be different.
Thanks!

 

[Alucinor]

No problem, sorry my wording was confusing in the first post

 

[thebrainstorm]

Funny...