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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)
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2 Answers 2

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If the motion of each object is parallel to the string it is attached to, then the acceleration of the objects will always be equal. However, if the objects have motion perpendicular to the string, this will induce no motion in the string at the pulley, and the other object will have no acceleration. For an extreme example, see this video. One mass travels in a circle and the other mass does not move at all.

As for the situations with rolling, it depends on whether the string wraps around the rolling mass as it rolls.

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  • $\begingroup$ in scenarion #3, both the motions are parallel to string but the object on the incline will have to travel more distance. For example if the the hanging block on pulley falls by distance h, block on the incline has to move h/Sintheta in the same time. SO both objects cover different distance in equal time. how can their acceleration be same. I am confused. $\endgroup$
    – user31058
    Commented Aug 19, 2020 at 20:19
  • $\begingroup$ @user31058 If both motions are parallel to the string, then the travel distances have be be the same, otherwise the string will change length. If the travel distances are the same, then the accelerations must be the same. $\endgroup$
    – Mark H
    Commented Aug 19, 2020 at 21:39
  • $\begingroup$ thx. I get this scenario. For the rolling case assuming the string wraps around how will that work. ? I am trying to solve a problem where a cylinder rolls up a 30 degree incline (connected to a pulley which has falling mass). acceleration of cyclinder here seems to be half of falling weight which I am not able to understand. $\endgroup$
    – user31058
    Commented Aug 20, 2020 at 15:47
  • $\begingroup$ @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. $\endgroup$
    – Mark H
    Commented Aug 20, 2020 at 19:57
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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.

Thus, to answer your original question, the magnitudes of the accelerations of the two masses can be unequal if their velocity is not along the direction of the string. For example, imagine a hole in a horizontal table through which the string passes. If the mass on the table performs circular motion (with the appropriate velocity) it will have some acceleration towards the centre, however the hanging mass will be at rest, thus having zero acceleration.

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