A bowling ball of mass M and radius R (treated as a uniform sphere) is thrown al

Question

A bowling ball of mass M and radius R (treated as a uniform sphere) is thrown along a level floor with kinetic energy Ko. It is thrown so that initially (at t = 0) it slides with a linear speed Vo but does not rotate. As the ball slides, kinetic friction causes it to begin spinning. Eventually the ball begins to roll without slipping along the floor, with linear speed V and angular speed w. The coefficient of kinetic friction between the ball and the floor is Uk.

a) At what time t > 0 does the bowling ball begin rolling smoothly along the floor? What is the linear speed V at that time? Express your answer in terms of Vo, g, and Uk, as needed.

b) What is the change in K in the bowling ball's kinetic energy during this entire process. (i.e. from sliding motion to rolling without slipping)? Express your answer in terms of Ko.

c) After rolling along the horizontal floor for awhile, the bowling ball rolls without slipping up a ramp made of the same material as the floor (i.e. same Uk >0). What maximum height H above its original location does the bowling ball's center of mass attain? Express your answer in terms of Vo, g, and Uk.

d) Suppose instead that the bowling ball rolls up a frictionless ramp. What maximum height h above its original location does the bowling ball's center of mass attain in this case? Express your answer as a multiple of H, the height it attained on the ramp in part c.

Explanation / Answer

a)

When the ball is thrown:Initial Angular momentum=M*V0*R

When it Starts to roll Final Angular momentum=M*V*R+2/5*MR^2*V/R

where V is the final Velocity and V=w*R

using conservation of angular momentum

V=5/7*V0

using

v=u+at:equation of motion

where a=-Uk , v=V and u=V0 we get t=(2/7*V0)/(Uk*g)

b)

KE final=0.5*M*V^2+0.5*2/5*M*R^2*(V/R)^2=7/10*M*V^2=5/7*K0

Change in KE=2/7 K0

d)

5/7*K0=M*g*H

H=5/14*V0^2/g

please verify regarding part c) whether some more data like angle of incline is given or not.I will repost the answer once more

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