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

Answer 1

The work is `=377.5J`

Explanation:

`"Reminder : "`

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

The work done is

`W=F*d`

The frictional force is

`F_r=mu_k*N`

The coefficient of kinetic friction is `mu_k=(2+csc(x))`

The normal force is `N=mg`

The mass of the object is `m=6kg`

`F_r=mu_k*mg`

`=6*(2+csc(x))g`

The work done is

`W=6gint_(1/8pi)^(3/4pi)(2+csc(x))dx`

`=6g*[2x+ln|(tan(x/2))|]_(1/8pi)^(3/4pi)`

`=6g(3/2pi+ln(tan(3/8pi)))-(1/4pi+3ln(tan(1/16pi))`

`=6g(6.42)`

`=377.5J`