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 [(pi)/12, (3pi)/8], where x is in meters?

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Answer 1 · 2017-07-20T09:42:57

The work is #=128.7J#

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 N#

The work done is

#W=6gint_(1/12pi)^(3/8pi)(1+cotx)dx#

#=6g*[x+ln|sin(x)|]_(1/12pi)^(3/8pi)#

#=6g((3/8pi+ln|sin(3/8pi)|))-(1/12pi+ln|sin(1/12pi)|))#

#=6g(7/24pi+1.27)#

#=6g(2.19)#

#=128.7J#