# An object with a mass of 5 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= xe^x+x . How much work would it take to move the object over x in [1, 2], where x is in meters?

Nov 7, 2017

The work is $= 435.6 J$

#### Explanation:

We need

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

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 $m = 5 k g$

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

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

The work done is

$W = 5 g {\int}_{1}^{2} \left(x {e}^{x} + x\right) \mathrm{dx}$

$= 5 g \cdot {\left[x {e}^{x} - {e}^{x} + {x}^{2} / 2\right]}_{1}^{2}$

=5g(2e^2-e^2+2)-(e-e+1/2))#

$= 5 g \left({e}^{2} + \frac{3}{2}\right)$

$= 435.6 J$