Two Segments Ramp A ramp of height h is divided into two sec

Two Segments Ramp A ramp of height h is divided into two sections, each of height 4, as shown in the figure. The angle the first section makes with the horizon is a while the angle the second half makes with horizon is . A block of mass m is pushed up the ramp; the force pushing the block is directed parallel to each section of the ramp. Find the expression for the work done against friction if the coefficient of kinetic friction between the block and ramp is . Your expression should be in terms of the given variables and any other known constants such as acceleration due to gravity, g. (Greek letters should be spelled out, for example, type \"alpha\" without the quotations for a.) The coefficient of kinetic friction, , is typed as \"muk\" without the quotations. D Submit Answer Tries 0/6

Solution

The distance the block has to cover while climbing up the sections is :

d1 = h/2 sin(alpha)

d2 = h/2 sin(theta)

total distance moved up

D = d1 + d2

D = h/2 [(1/sin(alpha) + 1/sin(theta)]

the frictional force acting on the block in section 1

Ff1 = uk m g cos(alpha)

Ff2 = uk m g cos(theta)

Total frictional force

F = F1 + F2

F = uk m g [cos(alpha) + cos(theta)]

We know from the defination of work done

W = F x D

W = uk m g [cos(alpha) + cos(theta)] x h/2 [(1/sin(alpha) + 1/sin(theta)]

W = {uk m g h/2 [cos(alpha) + cos(theta)] [sin(theta) + sin(alpha)]}/sin(theta)sin(alpha)]