Work against gravity and friction

Lifting a load

You lift a box from the floor onto a table. How much work do you do?

At a steady speed, your upward force equals the weight of the box, $mg$, with $g = 9.8\ \text{m/s}^2$. The distance is the height $h$. So the work done in lifting is

Does the path matter?

You can lift the box straight up, or push it up a smooth ramp to the same height. The ramp is longer, so you push for longer, but with a smaller force. The work against gravity is the same, $mgh$. Only the height matters.

Pulling along the ground

Now you pull a sledge on level ground. It does not go up, so its weight takes no work. What you work against is friction. At a steady speed your pull equals the friction force $f$, and over a distance $s$

You should find that a longer ramp needs a smaller push, but with no friction the work stays $mgh$. With friction, a longer ramp costs more work, because friction acts all the way along it.

Watch out. $W = F \times s$ needs a force in newtons. If you are given a mass in kilograms, find the weight $mg$ first.

A worked example

A porter lifts a $22\ \text{kg}$ suitcase onto a rack $1.9\ \text{m}$ above the floor. Then he drags a $30\ \text{kg}$ trunk $15\ \text{m}$ along the platform, against a friction force of $90\ \text{N}$, at a steady speed.

1. The suitcase's weight. $22\ \text{kg} \times 9.8\ \text{m/s}^2 = 215.6\ \text{N}$.
2. Lifting. $W = 215.6\ \text{N} \times 1.9\ \text{m} = 410\ \text{J}$.
3. Dragging. The trunk stays level, so only friction counts: $W = 90\ \text{N} \times 15\ \text{m} = 1350\ \text{J}$.

Practise with new numbers

Try it in Python

The program finds the work of lifting loads to different heights.

Remember these

• Lifting at a steady speed: $W = mgh$. Only the height matters, not the path.
• Pulling along level ground at a steady speed: $W = f \times s$.
• Change a mass into a weight, $mg$, before you use $W = F \times s$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How much work is done in lifting a $3\ \text{kg}$ book onto a shelf $1.5\ \text{m}$ high?

2. A box is lifted straight up to a platform, and another one just like it is pushed up a smooth ramp to the same platform. Which needs more work against gravity?

3. A friction force of $25\ \text{N}$ acts on a trolley. How much work do you do in pulling it $8\ \text{m}$ at a steady speed?

4. Why does the weight of a sledge not appear in the work done to pull it along level ground?