# Pulley and two masses - Conditions for acceleration to be different

We have two objects connected over a pulley. If the string is an ideal string most of the times, the acceleration of both the objects will be same. Below are some of the cases. I am trying to think will there be any case, when the objects wont have the same acceleration.? The case that is troubling me is #3 below. In that case, as one of the objects falls vertically down, the other object on the incline has to travel more distance (along the incline)

1. two masses - both hanging on either side of pulley
2. One mass hanging and other one on a horizontal plane
3. One mass hanging and other one on a inclined plane
4. Same as #2 above but object on horizontal plane can roll (e.g. a ball)
5. Same as #3 above but object on inclined plane can roll (e.g. a ball)

• @user31058 Imagine a wheel rolling on a flat surface. If the center of the wheel is traveling at speed $v$, then a point at the top of the wheel is moving at speed $2v$ due to the combination of linear and rotational motion. So, if a string is being wrapped onto the wheel by the rolling motion, the string will move at twice the velocity of the wheel, because that is how fast the point where the string connects moves. For more complicated situations with inclined planes, you need to keep track of how fast the object is spinning to see how the length of the string is changes due to wrapping. Commented Aug 20, 2020 at 19:57
Why do you think the mass on the incline will have to travel more distance? From your comments, it appears that you are assuming they will travel equal vertical distances, thus when hanging mass travels $$h$$, mass on incline will travel $$h*cosec \theta$$. This is an incorrect assumption. There is no reason why the vertical components of their displacements should be equal. Assuming that the string is inextensible, the components of the velocity along the string have to be equal.