# 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, 5], where x is in meters?

Mar 5, 2016

$= 2250 J$

#### Explanation:

work done against the frictional force
W $= {\int}_{2}^{5} {u}_{k} \left(x\right) m g \cdot \mathrm{dx}$
$= {\int}_{2}^{5} \left({x}^{2} + 2\right) \cdot 5 \cdot 10 \cdot \mathrm{dx}$ (Given mass m = 5kg and taking g $10 m {s}^{-} 2$)
$= 50 {\int}_{2}^{5} \left({x}^{2} + 2\right) \mathrm{dx}$
$= 50 {\left[{x}^{3} / 3 + 2 x\right]}_{2}^{5}$
$= 50 \left[\left({5}^{3} / 3 + 2 \cdot 5\right) - \left({2}^{3} / 3 + 2 \cdot 2\right)\right]$
$= 50 \left(\frac{117}{3} + 6\right)$
$= 50 \left(\frac{135}{3}\right) 50 \cdot 45 = 2250 J$
Here
$\left\{k g \cdot m {s}^{-} 2 \cdot m = N \cdot m = J\right]$