# An object with a mass of 1 kg is pushed along a linear path with a kinetic friction coefficient of u_k(x)= 2e^x-x+3 . How much work would it take to move the object over x in [1, 4], where x is in meters?

Aug 14, 2017

$W = 1033$ $\text{J}$

#### Explanation:

We're asked to find the work necessary to push an object on a certain position interval with a varying coefficient of kinetic friction.

The equation for work with a varying force is

$W = {\int}_{{x}_{1}}^{{x}_{2}} {F}_{x} \textcolor{w h i t e}{l} \mathrm{dx}$

where

• ${x}_{1}$ and ${x}_{2}$ are the initial and final positions

• ${F}_{x}$ is the force, which will be equal to the friction force acting (but in the opposite direction):

${F}_{x} = {\mu}_{k} n = {\mu}_{k} m g$

So

$W = {\int}_{{x}_{1}}^{{x}_{2}} {\mu}_{k} m g \textcolor{w h i t e}{l} \mathrm{dx}$

We know:

• ${x}_{1} = 1$ $\text{m}$

• ${x}_{2} = 4$ $\text{m}$

• ${\mu}_{k} = 2 {e}^{x} - x + 3$

• $m = 1$ $\text{kg}$

• $g = 9.81$ ${\text{m/s}}^{2}$

Plugging these in:

W = int_(1color(white)(l)"m")^(4color(white)(l)"m")(1color(white)(l)"kg")(9.81color(white)(l)"m/s"^2)(2e^x-x+3)color(white)(l)dx = color(red)(ulbar(|stackrel(" ")(" "1033color(white)(l)"J"" ")|)