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

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Answer 1 · 2017-07-13T14:44:17

The work is #=494.6J#

The integration by parts is

#intuv'dx=uv-intu'vdx#

#F_r=mu_k*N#

#dW=F_r*dx#

We start by calculating the integral of

#intxcos(pi/4-x/6)#

We perform this by integration by parts

#u=x#, #=>#, #u'=1#

#v'=cos(pi/4-x/6)#, #=>#, #v=-6sin(pi/4-x/6)#

Therefore,

#intxcos(pi/4-x/6)=-6xsin(pi/4-x/6)+int1*6sin(pi/4-x/6)#

#=-6xsin(pi/4-x/6)+36cos(pi/4-x/6)#

The work is

#W=7gint_0^(4pi)(1+xsin(pi/4-x/6))#

#=7g[x-6xsin(pi/4-x/6)+36cos(pi/4-x/6)]_0^(4pi)#

#=7g((4pi-24pisin(-5/12pi)+36cos(-5/12pi))-(0-0+36cos(pi/4)))#

#=7g(69.259)#

#=494.6J#