Please show all work and explain all the components. A bowling ball of mass M an

Question

Please show all work and explain all the components. A bowling ball of mass M and radius Ii is released onto the surface of a bowling lane with a forward velocity center of mass translational speed v_i and a backspin rotation, angular speed omega_i. (A backspin moans that the rotation is against the direction of motion.) There is a coefficient of kinetic friction of mu_s between the ball and the surface. What is the angular speed of the ball when its motion becomes a pure rolling motion? How far down the bowling lane does the ball travel before this happens?

Explanation / Answer

friction force, f = uk N = uk mg

a = f/m = - uk g

suppose after time, its start pure rolling,

vf = vi + at = vi - (uk g t)

and torque = I x alpha

r f = (2m r^2 / 5) alpha

alpha = (5 uk g / 2 r )

wf = wi + alpha t

wf = wi + (2.5 uk g t / r )

and for pure rolling,

vf = wf R
vi - (uk g t) = R ( wi + (2.5 uk g t / R ) )

vi - R wi = 3.5 uk g t

t = (vi - R wi ) / (3.5 uk g )

wf = wi + [ (2.5 uk g (vi - Rwi) / 3.5 uk g R]

wf = wi - 0.714wi + 0.714vi/R   = (0.714 vi / R) + (0.286 wi )

b) wf^2 - wi^2 = 2(alpha)(theta)

0.082 wi^2 + 0.51vi^2 / R^2 + 0.408Viwi/R - wi^2 = 2((5 uk g / 2 R ))d

theta = (0.51vi^2 + 0.408 vi wi R - 0.918(wi R)^2 ) / (R (5ukg))

d = theta R = (0.51vi^2 + 0.408 vi wi R - 0.918(wi R)^2 ) / (5ukg)

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