Consider a roofing contractor is pulling on a rope to raise a load of roof shing

Question

Consider a roofing contractor is pulling on a rope to raise a load of roof shingles as shown. The contractor is standing on a roof with a pitch of 15 degrees. Each bundle of shingles weighs 30 lbs and the platform that carries the shingles weighs 20 lbs. Draw a Free Body Diagram and Vector Diagram for the Contractor and the platform carrying the shingles. How many packs of shingles can the contractor lift before he begins to slide down the roof assuming the following Contractor weights and values for the coefficient of static friction exists between the contractor and the roof?

Explanation / Answer

sum forces parallel to slope
friction - T - m g sin theta = 0
sum forces perp to slope

N - mg cos theta = 0

N = m g cos theta

so

u m g cos theta - T - mg sin theta = 0

For shingles

T - c Mg - M platform g = 0 where M is the mass of shingle and c is number of shingles

so

u m g cos theta - c M g - M platform g- m g sin theta = 0

c= (u mg cos theta - m g sin theta - M platform g)/(Mg)

1) c = (0.2*160*cos(15) - 160*sin(15)-20)/30=-1 so no shingles can be lifted

2) (0.2*280*cos(15) - 280*sin(15)-20)/30 <0 so no shingles

3) (0.55*160*cos(15) - 160*sin(15)-20)/30=0.786

4) (0.55*280*cos(15) - 280*sin(15)-20)/30=1.876

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