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

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Answer 1 · 2017-05-01T08:24:28

The work is #=146.7J#

The work done is

#W=F*d#

The frictional force is

#F_r=mu_k*N#

#N=mg#

#F_r=mu_k*mg#

#=6(1+3cotx)g#

The work done is

#W=int_(pi/6)^(3/8pi)6(1+3cotx)gdx#

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

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

#=6g(3/8pi+3lnsin(3/8pi)-(pi/6+3lnsin(pi/6))#

#=6g(5/24pi+1.84)#

#=6g*2.49#

#=146.7J#