Question

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?

Answer 1

`W = 1033` `"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_xcolor(white)(l)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_kn = mu_kmg`

So

`W = int_(x_1)^(x_2) mu_kmgcolor(white)(l)dx`

We know:

• `x_1 = 1` `"m"`

• `x_2 = 4` `"m"`

• `mu_k = 2e^x - x + 3`

• `m = 1` `"kg"`

• `g = 9.81` `"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"" ")|)`