Question

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(pi/4-x/6)`. How much work would it take to move the object over #x in [0, 8pi], where x is in meters?

Answer 1

The work is `=1539.6J`

Explanation:

We need to calculate the integral `intcos(1/4pi-x/6)dx`

Perform this integration by substitution

Let `u=1/4pi-x/6`, `=>`, `du=-1/6dx`

`intcos(1/4pi-x/6)dx=-6intcosudu`

`=-6sinu=-6sin(1/4pi-x/6)`

The work done is

`W=F*d`

The frictional force is

`F_r=mu_k*N`

The normal force is `N=mg`

The mass is `m=7kg`

The acceleration due to gravity is `g=9.8ms^-2`

`F_r=mu_k*mg`

`=7*(1-cos(pi/4-x/6))g`

The work done is

`W=7gint_(0)^(8pi)(1-cos(pi/4-x/6))dx`

`=7g[x+6sin(pi/4-x/6)]_0^(8pi)`

`=7g(8pi+6sin(pi/4-8/6pi))-(6sin(pi/4))`

`=(7xxgxx22.44)J`

`=1539.6J`