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/4, (3pi)/4], where x is in meters?

Answer 1

The work is `=288.1J`

Explanation:

We need

`intcscxdx=ln|tan(x/2)|+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=6kg`

`F_r=mu_k*mg`

`=6*(2+cscx)g`

The work done is

`W=6gint_(1/4pi)^(3/4pi)(2+cscx)dx`

`=6g*[2x+ln|tan(x/2)|]_(1/4pi)^(3/4pi)`

`=6g((3/2pi+ln|tan(3/8pi)|+sec(pi/6))-(1/2pi+ln|tan(pi/8)|)`

`=6g(*4.9)`

`=288.1J`