The wheel rolls without slipping and its position is control

The wheel rolls without slipping, and its position is controlled by the motion of the slider B. If the center O of the wheel has a constant velocity of 5.1 in/sec to the left, calculate the acceleration of the slider B for the position ? = 0. The acceleration of B is positive if to the right, negative if to the left.

Solution

The kinectic energy transfered to the reciprocating arm from the roller is given by,

KE = 1/2 x m x v^2

Assuming the rollers of neglegible mass KE is 1/2 x v^2 = 1/2 x 5^2 = 12.5 W.

Now this energy tranfer induces a horizontal motion or work on the reciprocating arm,

F x d = KE where d is the distance upto which the force acts. d = 2pi x r = 2 x 3.14 x 6.1 = 38.308 in

Solve for the F,

F = KE / d = 12.5 / 38.308 = 0.32 lbf in / sec2

Since F = m x a assume m = 1

a = F / m = 0.32 / 1 = 0.32 in / sec2