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 [(3pi)/4, (7pi)/8], where x is in meters?

Answer 1

The work is `=89.3J`

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_(3/4pi)^(7/8pi)(2+cscx)dx`

`=6g*[2x+ln|tan(x/2)|]_(3/4pi)^(7/8pi)`

`=6g((2*7/8pi+ln|tan(7/16pi)|)-(2*3/4pi+ln|tan(3/8pi)|))`

`=6g(7/4pi-6/4pi+1.61)`

`=6g(1/4pi+0.73)`

`=89.3J`