Can someone work out this problem for me. I am having trouble with it. Prob 3 A

Question

Can someone work out this problem for me. I am having trouble with it.

Prob 3 A marble of mass m is placed at the top of a gutter as shown below and released from rest. The surfaces are all frictionless except for a damaged part just before the circular part on the right side where the coefficient of friction is k over a length L -2R, where R Is the radius of the circular part. When the marble reaches the top of the circular loop, it has a speed v = VgR and does not lose contact with the gutter surface. Throughout the motion, the marble slides without rolling. Express your answers below in terms of g, R, m and pk (not all may be necessary (a) Find the work done by friction as the marble crossed the rough patch before the gutter (b) Find the height, h, from which the marble should be released so that it reaches the speed v at the top of the circle.

Explanation / Answer

(a)The work done by the frictional force will be:

Wf = Ff d

d = L = R/2 uk

Ff = uk m g

Wf = uk m g x R/2 uk = m g R/2

Hence, Wf = m g R/2

(b)From conservation of energy

PE = KE + Wf

m g h = m g R/2 + 1/2 m (sqrt(gR))^2

m g h = m g R/2 + 1/2 m g R

h = R/2 + R/2 = R

Hence, H = R

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