7. Work by a Variable Force
by M. Bourne
interactive spring activity
later in this section...
The work (W) done by a constant force (F) acting on a body by moving it through a distance (d) is given by:
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:
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.
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?
- This activity assumes you are looking at the spring from above and that the "ball" has zero mass. (The ball is there so you have got something to grab.)
- You can see other interesting spring examples at Applications of Trigonometric Curves and in Composite Trigonometric Curves.
(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,
requires that a and b are the distance from the natural position of the spring.
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.
Need Graph Paper?
A leaky 5N bucket 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?
Didn't find what you are looking for on this page? Try search:
Online Algebra Solver
This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)
Go to: Online algebra solver
Ready for a break?
Play a math game.
(Well, not really a math game, but each game was made using math...)
The IntMath Newsletter
Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!
Short URL for this Page
Save typing! You can use this URL to reach this page:
Calculus Lessons on DVD
Easy to understand calculus lessons on DVD. See samples before you commit.
More info: Calculus videos