Question

An object with a mass of `5 kg` is pushed along a linear path with a kinetic friction coefficient of `u_k(x)= x^2-3x+6`. How much work would it take to move the object over #x in [2, 3], where x is in meters?

Answer 1

Though not mentioned it looks like the object moves along a plane perpendicular to the the gravitational force. So we can equate the Normal force on the object to its weight as they balance each other.

`N=mg; \qquad F_k(x) = \mu_k(x).N`.
Workdone : `W_{if} \equiv \int_{x_i}^{x_f} F(x).dx = \int_{x_i}^{x_f}\mu_k(x).Nd.x`
`W_{23} = mg.\int_2^3(x^2-3x+6).dx = mg [\frac{x^3}{3}-3/2x^2+6x]_2^3 = 29/6.mg=29/6\times(5 kg)\times(9.8 ms^{-2})`

`=236.83` Joules.