Question

An object with a mass of `7 kg` is pushed along a linear path with a kinetic friction coefficient of `u_k(x)= 1+x+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?

Answer 1

The work is `=10166.5J`

Explanation:

We need

`intxcos(pi/4-x/6)dx=18sqrt2(x/6sin(x/6)-x/6cos(x/6)+cos(x/6)+sin(x/6))+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=7kg`

`F_r=mu_k*mg`

`=7*(1+x+xcos(pi/4-x/6))g`

The work done is

`W=7gint_(0)^(4pi)(1+x+xcos(pi/4-x/6))dx`

`=7g*[x+x^2/2+18sqrt2(x/6sin(x/6)-x/6cos(x/6)+cos(x/6)+sin(x/6))]_(0)^(4pi)`

`=7g(148.2)`

`=10166.5J`