Question

An object with a mass of `6 kg` is pushed along a linear path with a kinetic friction coefficient of `u_k(x)= 2+cscx`. How much work would it take to move the object over #x in [pi/8, (7pi)/8], where x is in meters?

Answer 1

The work is `=467J`

Explanation:

We need

`intcscxdx=ln|tan(x/2)|`

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(2+cscx)g`

The work done is

`W=6gint_(1/8pi)^(7/8pi)(2+cscx)dx`

`=6g*[2x+ln|tan(x/2)|]_(1/8pi)^(7/8pi)`

`=6g((2*7/8pi+ln|tan(7/16pi)|)-(2*1/8pi+ln|tan(1/16pi)|))`

`=6g(7/4pi-1/4pi+3.22)`

`=6g(3/2pi+3.22)`

`=467J`