An object with a mass of #7 kg# is pushed along a linear path with a kinetic friction coefficient of #u_k(x)= 1-cos(pi/4-x/6) #. How much work would it take to move the object over #x in [0, 8pi], where x is in meters?

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Answer 1 · 2017-11-26T07:08:15

The work is #=1539.6J#

We need to calculate the integral #intcos(1/4pi-x/6)dx#

Perform this integration by substitution

Let #u=1/4pi-x/6#, #=>#, #du=-1/6dx#

#intcos(1/4pi-x/6)dx=-6intcosudu#

#=-6sinu=-6sin(1/4pi-x/6)#

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=7kg#

The acceleration due to gravity is #g=9.8ms^-2#

#F_r=mu_k*mg#

#=7*(1-cos(pi/4-x/6))g#

The work done is

#W=7gint_(0)^(8pi)(1-cos(pi/4-x/6))dx#

#=7g[x+6sin(pi/4-x/6)]_0^(8pi)#

#=7g(8pi+6sin(pi/4-8/6pi))-(6sin(pi/4))#

#=(7xxgxx22.44)J#

#=1539.6J#