# Work: Integral or average

## Homework Statement

A uniform board of length L and mass M lies near a boundary that separates two regions. In region 1, the coefficient of kinetic friction between the board and the surface is μ1, and in region 2, the coefficient is μ2. The positive direction is shown in the figure.

Find the net work W done by friction in pulling the board directly from region 1 to region 2. Assume that the board moves at constant velocity.

http://imgur.com/tFIDobO

W = Fs

## The Attempt at a Solution

Force of friction in region 1: μ1Mg
Force of friction in region 2: μ2Mg
Average force = μ1Mg + μ2Mg/2 = (Mg/2) x (μ1 + μ2)
W = Average force x distance traveled = (-gML/2) x (μ1 + μ2)

So I was able to get the correct answer using average force, but I was wondering how I would approach this if I wanted to use an integral instead. I know the integral of force finds the work done, but I'm not sure what exactly force would be. Is a way to integrate this if an equation is not already given for force?

Related Introductory Physics Homework Help News on Phys.org
BvU
Homework Helper
You want to integrate ##\displaystyle \int \vec F \cdot \vec {ds}## where now ##\vec F =\vec F(s)##. So I would say you can't do this without an expression for F.

If the contact between board and surface is uniform, you could write ##F = -\Bigl ( \mu_2 mgs/L + \mu_1 mg(L-s)/L \Bigr ) ## where s runs from 0 to L

haruspex