# An object with a mass of 7 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 1-cos(x/6) . How much work would it take to move the object over #x in [0, 8pi], where x is in meters?

Jan 4, 2018

The work is $= 2078.6 J$

#### Explanation:

$\text{Reminder : }$

$\int \cos a x \mathrm{dx} = \frac{1}{a} \sin a x + C$

The work done is

$W = F \cdot d$

The frictional force is

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

The coefficient of kinetic friction is ${\mu}_{k} = \left(1 - \cos \left(\frac{x}{6}\right)\right)$

The normal force is $N = m g$

The mass of the object is $m = 7 k g$

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

$= 7 \cdot \left(1 - \cos \left(\frac{x}{6}\right)\right) g$

The work done is

$W = 7 g {\int}_{0}^{8 \pi} \left(1 - \cos \left(\frac{x}{6}\right)\right) \mathrm{dx}$

$= 7 g \cdot {\left[x - 6 \sin \left(\frac{1}{6} x\right)\right]}_{0}^{8 \pi}$

$= 7 g \left(8 \pi - 6 \sin \left(\frac{4}{3} \pi\right) - \left(0\right)\right)$

$= 7 g \left(30.3\right)$

$= 2078.6 J$