**force**$\times$

**distance**. Here's a very simple example

Example

Determine the work required to lift a 60 pound object 3 feet. $$W = FD$$ $$=60(3)$$ $$=180 \text{ft-lbs}$$

# Work Done by a Variable Force

## Compressing a Spring

## Propulsion

First, we need to calculate how much it weighs on the moon, which is ${1}{6}*12$ tons $= 2$ tons. $$\text{The force exerted by gravity varies inversely with the square of the distance from the center of the body,} F(x) = \frac{C}{x^2}$$ $$2 = \frac{C}{1100^2}, C = 2,420,000$$ If you move the module a VERY small amount, it's basically linear, so you could say $$\Delta{}W = \frac{2,420,000}{x^2}*\Delta{}x$$ You must integrate it with bounds from the center. $$\int_{1100}^{1150}\mathrm{\frac{2,420,000}{x^2}}\, \mathrm{d}x = -\frac{1,210,000}{x}|_{1100}^{1150} = 95.652 \text{ ton-miles}$$

## Lifting a Chain

Let's say you have a 20 foot chain that weighs 5 pounds/foot. How much work is required to raise one end of the chain to a height of 30 feet, so it's 10 feet off the ground.

David Witten