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)/4, (3pi)/8], where x is in meters?

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Answer 1 · 2017-09-03T07:08:37

The work is #=38.8J#

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_(1/4pi)^(3/8pi)(1+cotx)dx#

#=6g*[x+ln|sin(x)|]_(1/4pi)^(3/8pi)#

#=6g(3/8pi+ln|sin(3/8pi)|-1/4pi-ln|sin(1/4pi)|)#

#=6g(0.66)#

#=38.8J#