# An object with a mass of 5 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= x^2+2 . How much work would it take to move the object over #x in [2, 3], where x is in meters?

May 23, 2017

The work is $= 408.3 J$

#### Explanation:

The work done is

$W = F \cdot d$

The frictional force is

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

$N = m g$

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

$= 5 \left({x}^{2} + 2\right) g$

The work done is

$W = 5 g {\int}_{2}^{3} \left({x}^{2} + 2\right) \mathrm{dx}$

$= 5 g \cdot {\left[{x}^{3} / 3 + 2 x\right]}_{2}^{3}$

$= 5 g \left(\left(9 + 6\right) - \left(\frac{8}{3} + 4\right)\right)$

$= 5 g \left(15 - \frac{20}{3}\right)$

$= 5 g \cdot \frac{25}{3}$

$= 408.3 J$