Question

An object with a mass of `1 kg` is pushed along a linear path with a kinetic friction coefficient of `u_k(x)= 3x^2-x+2`. How much work would it take to move the object over #x in [1, 3], where x is in meters?

Answer 1

The work is `=254.8J`

Explanation:

We need

`intx^ndx=x^(n+1)/(n+1)+C (x!=-1)`

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=1kg`

`F_r=mu_k*mg`

`=1*(3x^2-x+2)g`

The work done is

`W=gint_(1)^(3)(3x^2-x+2)dx`

`=g*[3x^3/3-x^2/2+2x]_(1)^(3)`

`=g((3^3-9/2+6)-(1-1/2+2))`

`=g(33-3-4)`

`=(26g)J`

`=254.8J`