In Fig. 10-28, wheel A of radius rA is coupled by belt B to wheel C of radius rC

Question

In Fig. 10-28, wheel A of radius rA is coupled by belt B to wheel C of radius rC. The angular speed of wheel A is increased from rest at a constant rate . Find the time needed for wheel C to reach angular speed assuming the belt does not slip. (Hint: If the belt does not slip, the linear speeds at the two rims must be equal.) State your answer in terms of the given variables.

Hint to solve the problem: The instantaneous angular acceleration is the time derivative of the angular speed. The angular speed is 2 divided by the period. Here the period is a function of time. How then is the angular acceleration related to the period?

Explanation / Answer

Let the time be t for wheel C to reach angular speed, The linear speed of the rim of C,                    vc = .rc The linear spee of the rim of wheel A is the same.                         vA= vc = .rc But         vA= A.rA Hence, A.rA =.rc              A      = . .rc/rA Since, i = 0            A  = 0 + .t Time, t =  A / =.rc/rA              = (/).rc/rA

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