# An object with a mass of 1 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 3x^2+x+2 . How much work would it take to move the object over x in [1, 3], where x is in meters?

Feb 18, 2016

$W = 335 , 54 J$

#### Explanation:

${F}_{f} = {u}_{k} \cdot N$
${F}_{f} = \text{ Friction force}$
$\text{N:Normal force to the contacting surfaces}$
$N = m \cdot g \text{ m:mass of object, g:acceleration of gravity}$
${F}_{f} = {u}_{k} \cdot m \cdot g$
$W = {\int}_{1}^{3} {F}_{f} \cdot d x$
$W : \text{work doing by Friction force}$
$W = {\int}_{1}^{3} {u}_{k} \cdot m \cdot g \cdot d x$
$W = m g {\int}_{1}^{3} \left(3 {x}^{2} + x + 2\right) d x$
$W = m g {\left[3 \cdot \frac{1}{3} {x}^{3} + \frac{1}{2} {x}^{2} + 2 x\right]}_{1}^{3} + C$
$W = 1 \cdot 9 , 81 {\left[{x}^{3} + \frac{1}{2} {x}^{2} + 2 x\right]}_{1}^{3} + C$
u_k(0)=2 ; then C=2#
$W = 9 , 81 \left(\left[27 + 4 , 5 + 6\right] - \left[1 + 0 , 5 + 2\right]\right) + 2$
$W = 9 , 81 \left(37 , 5 - 3 , 5\right) + 2$
$W = 9 , 81 \cdot 34 + 2$
$W = 335 , 54 J$