# 7. Work by a Variable Force using Integration

by M. Bourne

The work (*W*) done by a constant force (*F*) acting on a body by moving it through a distance (*d*) is given by:

W=F×d

### Example of work done by a constant force

An apple weighs about `1\ "N"`. If you lift the apple `1\ "m"` above a table, you have done approximately `1\ "Newton meter (Nm)"` of work.

## Work done by a Variable Force

If the force varies (e.g. compressing a spring) we need to use calculus to find the work done.

If the force is given by *F*(*x*) (a function of *x*) then the work done by the force along the *x*-axis from *a *to *b* is:

`W=int_a^bF(x)dx`

## Hooke's Law for Springs

The force (F) that it takes to stretch (or compress) a spring *x* units from its normal length is proportional to *x*.

`F = kx`

We can find the **spring constant ***k* from observing what force gives what stretch for each spring. This spring constant is also called the **stiffness** of the spring.

## Interactive Appplet

In this activity, you can see the forces involved, the work done and you can explore the meaning of *k*, the spring constant.

### Things to do...

- Extend or compress the spring. See the force required to do this.
- When you let go, you will see the work done in compressing or extending the spring.
- Change the value of
*k*and see how much this changes the force required and the work done. - Why does the force increase as the amount of stretch (
*x*) increases? - Does the relationship `F = kx` hold?
- Why does the spring slow down?

Grab the ball and stretch (or compress) the spring.

Natural length of spring = 100 cm

Spring constant, k = cm

Stretch/compression = cm

Force = N

Work = N cm

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**NOTE:** The above activity assumes the mass of the ball at the end of the spring is negligible, so the force of gravity is insignificant compared to the forces in the spring.

You can see other interesting spring examples at Applications of Trigonometric Curves and in Composite Trigonometric Curves.

### Example 1

(a) Find the work done on a spring when you compress it from its natural length of 1 m to a length of 0.75 m if the spring constant is *k* = 16 N/m.

(b) What is the work done in compressing the spring a further 30 cm?

**Note:** For a spring,

`int_a^bF(x)dx`

requires that *a* and *b* are the distance from the **natural **position of the spring.

### Example 2

A force of 1200 N compresses a spring from its natural length of 18 cm to a length of 16 cm. How much work is done in compressing it from 16 cm to 14 cm?

### Check your understanding

Go back up to the spring interactive above and calculate the work done in compressing or stretching the spring for various amounts of stretch. Do your answers tally with the answer given?

Let's now look at another example of work done by a variable force.

### Example 3

### Need Graph Paper?

A leaky bucket weighing 5N is lifted 20 m into the air at a constant speed. The rope weighs 0.08 Nm^{-1}. The bucket starts with 2 N of water and leaks at a constant rate. It finishes draining just as it reaches the top. How much work was done:

**a) **lifting the water alone

**b) **lifting the water and bucket together

**c) **lifting the water, bucket and rope?

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