Can someone do this problem step-by-step and thouroghly explain why you did what

Question

Can someone do this problem step-by-step and thouroghly explain why you did what you did at each step?

1. d) A device for measuring the coefficient of kinetic friction consists of a FRICTIONLESS ramp of length L whose angle with the horizontal can be ad- justed. A mass is released from rest at the top of the ramp, slides down the ramp, and along a horizontal surface of length d that has a coefficient of friction The angle is adjusted until the mass just stops at the end of the horizonta!surfac Find an expression for K in terrns ou. d. and [13 points] m1NT the uork done by gravity while the mass is on the slope (or the decrease in the potential energy) is he horizontal surface.

Explanation / Answer

let theta = x

Since the ramp makes an angle of x with the horizontal, the height of the ramp = L*sin(theta)

=L*sin(x)

Now, since the ramp is frictionless, we apply law of ocnservation of energy to get the result.

so,

decrease in potential energy during the drop = increase in kinetic energy of the body

or m*g*h = 0.5*mv^2

or v = (2*gh)^0.5 where h = L*sin(x)

now in the horizontal surface,

Normal reaction = mg

so frictional force = u*mg

so, work done by frictional force = F.d

=-u*mg*d

so, by conservation of energy again,

kinetic energy of the body = negative work done by friction

ro 0.5mv^2 = u*mg*d

or g*L*sin(x) = u * g*d

or u = (L/d)*sin(theta)

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