# An object with a mass of 7 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 3+sin(4x) . How much work would it take to move the object over #x in [pi, 4pi], where x is in meters?

May 13, 2017

The work done is $= 1939.62 J$

#### Explanation:

The work done is

$W = F \cdot d$

The frictional force is

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

$N = m g$

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

$= 7 \left(3 + \sin 4 x\right) g$

The work done is

$W = 7 g {\int}_{\pi}^{4 \pi} \left(3 + \sin 4 x\right) \mathrm{dx}$

$= 7 g \cdot {\left[3 x - \frac{1}{4} \cos 4 x\right]}_{\pi}^{4 \pi}$

$= 7 g \left(\left(12 \pi - \frac{1}{4}\right) - \left(3 \pi - \frac{1}{4}\right)\right)$

$= 7 g \left(9 \pi\right)$

$= 7 g \cdot 28.27$

$= 1939.62 J$