An object with a mass of #6 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 1+cotx #. How much work would it take to move the object over #x in [(3pi)/4, (7pi)/8], where x is in meters?

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Answer 1 · 2017-06-29T06:26:15

The work is #=12.8J#

We need

#intcotxdx=ln|sin(x)|+C#

The work done is

#W=F*d#

The frictional force is

#F_r=mu_k*N#

The normal force is #N=mg#

The mass is #m=6kg#

#F_r=mu_k*mg#

#=6(1+cotx)g#

The work done is

#W=6gint_(3/4pi)^(7/8pi)(1+cotx)dx#

#=6g*[x+ln|sin(x)]_(3/4pi)^(7/8pi)#

#=6g((7/8pi+ln|sin(7/8pi)|)-(3/4pi+lnsin|(3/4pi)|))#

#=6g(1/8pi-0.96+0.35)#

#=6g(-0.217)#

#=-12.8J#