# An object with a mass of 8 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 2x^2-x . How much work would it take to move the object over x in [2, 3], where x is in meters?

Oct 28, 2017

The work is $= 797.1 J$

#### Explanation:

We need

$\int {x}^{n} \mathrm{dx} = {x}^{n + 1} / \left(n + 1\right) + C \left(n \ne - 1\right)$

The work done is

$W = F \cdot d$

The frictional force is

${F}_{r} = {\mu}_{k} \cdot N$

The normal force is $N = m g$

The mass is $m = 8 k g$

${F}_{r} = {\mu}_{k} \cdot m g$

$= 8 \cdot \left(2 {x}^{2} - x\right) g$

The acceleration due to gravity is $g = 9.8 m {s}^{-} 2$

The work done is

$W = 8 g {\int}_{2}^{3} \left(2 {x}^{2} - x\right) \mathrm{dx}$

$= 8 g \cdot {\left[\frac{2}{3} {x}^{3} - \frac{1}{2} {x}^{2}\right]}_{2}^{3}$

=8g(2/3*3^3-9/2))-(2/3*8-2)#

$= 8 g \left(18 - \frac{9}{2} - \frac{16}{3} + 2\right)$

$= 8 g \left(20 - \frac{59}{6}\right)$

$= 797.1 J$