You are planning to build a log cabin and will need to pull the logs up a hill t

Question

You are planning to build a log cabin and will need to pull the logs up a hill to the building site by means of a rope attached to a winch. In order to buy the rope, you need to know how strong the rope must be and decide to do a quick calculation for this. The logs weigh 1560 Newtons. at most and the hill is at an angle of 39° with respect to the horizontal. You estimate that the coefficient of kinetic friction between a log and the hill is 0.90. When pulling a log up the hill, you will make sure that the rope stays parallel to the surface of the hill and the acceleration of the log is never more than 0.96 m/s2.

(a) What is the algebraic expression for the maximum force the rope must be able to exert in terms of the maximum weight of the log (W), the angle of the hill (), the coefficient of kinetic friction between the log and the hill(K), the maximum acceleration of the log (a), and the gravitational acceleration (g)? [Note: Don't enter an equation like "x=blah". Just enter the "blah" part. Use capital letters for subscripts. Type "mu" to get the greek letter .]

(b) What is the numerical value for the maximum force the rope must be able to exert?  N

Explanation / Answer

m= W/g = 1560/9.8 = 159.2 kg

a)By Newton’s second law along horizontal,

Fn – mgcos = ma= 0

Fn= mgcos ---------(1)

By Newton’s second law along vertical,

T – mgsin + Fn = ma

From (1),

T = ma + mgsin + uk* mgcos

b)T = ma + mgsin - uk* mgcos = 159.2*0.96 + 1560sin39 + 0.90*1560cos39 = 2225.7 N

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