A block of mass M slides down a friction less incline from a height h. Assume Fi

Question

A block of mass M slides down a friction less incline from a height h. Assume Figure 5: Diagram for problem B2. at the bottom of this incline the block is moving to the right as in Fig. 5. It slides over a rough surface of length l with coefficient of kinetic friction Iik such that mukl = 0.16h. If the block passes through the rough patch, it is able to slide up a second facing frictionless incline. Give answers in terms of M, h, muk, l and g, the acceleration due to gravity, as nccessary. Determine how fast the block is moving. v, at the bottom of the incline and before it reaches the rough patch. Determine the change in kinetic energy A' after passing over the rough patch. Find how high up the second incline the block will travel. How many times will the block cross the rough patch before it stops and how far from the left edge of the rough patch will the block finally stop?

Explanation / Answer

a)
for conservation of energy, we have:
(1/2)Mv2 = mgh

=>v = (2gh)

b)

the friction force is Fr = -mgk, the work done by friction force is: Wf = Fr*l = -mgk*l

the kinetic energy after the rough surface is:(kl = 0.16h)

K1 = (1/2)Mv2 - Mgkl = Mgh - Mgkl = Mgh - 0.16Mgh = 0.84Mgh

the change in kinetic energy is:

K = (1/2)Mv2 - K1 = Mgh - 0.84Mgh = 0.16Mgh

c)

let h1 be the high up that the block will travel in second incline, since the kinetic energy after the rough surface is 0.84Mgh, for energy conservation,

mgh1 = 0.84Mgh

=>h1 = 0.84h

d)

everytime the block across the rough surface, it lost 0.16Mgh of its energy

so we have, after 6 times it passed the rough surface, it only have kinetic energy

K2 = 0.04Mgh,

this energy will not enough to make another pass,

so after 6 pass and it will stop in the attempts for 7 pass (it will stop in the middle of this last pass).

let the distance from the left edge of the rough patch it will stop be l1, we have:

0.04Mgh = mgk *l1

=>l1 = 0.04h/k

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