# An object with a mass of 7 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 4+secx . How much work would it take to move the object over #x in [(pi)/8, (5pi)/12], where x is in meters?

Feb 15, 2016

$W \cong 5 , 288 J$

#### Explanation:

$W = {\int}_{\frac{\pi}{8}}^{\frac{5 \pi}{12}} {u}_{k} \left(x\right) \cdot d x = {\int}_{\frac{\pi}{8}}^{\frac{5 \pi}{12}} \left(4 + \sec x\right) \cdot d x$
$W = {\left[4 x + l n \left(\sec x + \tan x\right)\right]}_{\frac{\pi}{8}}^{\frac{5 \pi}{12}}$
$W = \left[\frac{20 \pi}{12} + l n \left(\sec \frac{5 \pi}{12} + \tan \frac{5 \pi}{12}\right)\right] - \left[\frac{4 \pi}{8} + l n \left(\sec \frac{\pi}{8} + \tan \frac{\pi}{8}\right)\right]$
$W = \left[\frac{5 \pi}{3} + l n \left(3 , 864 + 3 , 732\right)\right] - \left[\frac{\pi}{2} + l n \left(1 , 082 + 0 , 414\right)\right]$
$W = \left[\frac{5 \pi}{3} + l n \left(7 , 596\right)\right] - \left[\frac{\pi}{2} + l n \left(1.496\right)\right]$
$W = \left[\frac{5 \pi}{3} + 2 , 028\right] - \left[\frac{\pi}{2} + 0 , 403\right]$
$W = \left[7 , 261\right] - \left[1 , 973\right]$
$W \cong 5 , 288 J$