Many important applications of rotational dynamics involve objects, such as pulleys, that are connected via ropes or belts to other objects. FIGURE 12.29 shows a rope passing over a pulley and connected to an object in linear motion. If the rope does not slip as the pulley rotates, then the rope’s speed $v_{\text{rope}}$ must exactly match the speed of the rim of the pulley, which is $v_{\text{rim}}=\left|\omega \right|R$. If the pulley has an angular acceleration, the rope’s acceleration $a_{\text{rope}}$ must match the tangential acceleration of the rim of the pulley, $a_t=\left|\alpha \right|R$.

FIGURE 12.29 The rope’s motion must match the motion of the rim of the pulley.

The object attached to the other end of the rope has the same speed and acceleration as the rope. Consequently, an object connected to a pulley of radius R by a rope that does not slip must obey the constraints

These constraints are very similar to the acceleration constraints introduced in Chapter 7 for two objects connected by a string or rope.