Work, energy and power

TL;DR:

• Work is force times the distance moved along the force: W = Fd, measured in joules.

• Lifting a mass m a height h at steady speed takes mgh of work and stores that much gravitational potential energy.

• Power is how fast work is done: P = W / t, in watts.

You lift a 2 kg book from the floor to a shelf 1.5 m up. Your arms get tired, the book gains height, and a physicist would say you did work on it. This article gives the exact meaning of that word and shows how work, energy and power fit together with one set of numbers.

The idea in plain words

In everyday speech, work means effort. In physics, work is energy transferred by a force that moves something. If a force F acts along a displacement d in the same direction, the work is W = Fd. The unit is the joule: 1 J = 1 N·m.

Energy is the stored ability to do work. Power is the speed of doing it: P = W / t, with 1 W = 1 J/s. A strong person and a weak person can lift the same book and do the same work. The one who does it faster has more power.

Why work needs motion along the force

Force alone does not count. Hold a heavy bag at rest and you do no work on it, because nothing moves. Carry it horizontally at steady speed and, again, the upward force of your hands is perpendicular to the motion, so no work is done on the bag by that force.

The general version includes the angle: W = F d cos θ. Pushing a box with 10 N at 60° to the floor over 4 m does 10 × cos 60° × 4 = 10 × 0.5 × 4 = 20 J. Only the part of the force along the motion does work.

Worked example: lifting a 2 kg book 1.5 m takes about 29.4 J

Take g = 9.8 m/s². Lift the book slowly, at nearly constant speed, so your upward force just balances its weight.

1. Weight: F = mg = 2 × 9.8 = 19.6 N. Your upward force must equal this.

2. Work: W = Fd = 19.6 × 1.5 = 29.4 J.

3. Where it went: the book now has gravitational potential energy mgh = 2 × 9.8 × 1.5 = 29.4 J. The same expression, because lifting at steady speed turns your work into stored energy.

4. Power, if the lift takes 3 s: P = 29.4 / 3 = 9.8 W. If you did it in 2 s, P = 14.7 W.

Now let it fall back. Potential energy turns into kinetic energy, ½mv². At the floor ½ × 2 × v² = 29.4, so v² = 29.4 and v ≈ 5.4 m/s. Notice the mass cancels in the speed: any object dropped 1.5 m reaches that speed, ignoring air.

It is worth asking why the work equals the energy gained, because that link is the heart of the topic. While you lift at constant speed, the net force on the book is zero: your push up equals gravity down. So the book gains no kinetic energy. All the energy you supply goes into raising it in Earth's gravity field, and it can be handed back later as speed when the book falls.

If instead you jerked the book upward and let it coast, your force would briefly exceed its weight, the book would gain some kinetic energy, and you would do extra work that then trades back as the book slows near the top. The total work for a start-and-stop lift from rest to rest is still mgh.

Power connects to everyday numbers. A 60 W bulb transfers 60 J every second. Over one minute that is 60 × 60 = 3600 J, which is 122 times the 29.4 J book lift (3600 ÷ 29.4 ≈ 122).

How to check your answer

• Units. Newtons times meters is joules. Joules per second is watts. A kg·m²/s² is also a joule.

• Two routes. Compute the work as force times distance and again as mgh. They agree at 29.4 J.

• Scale. Two kilograms is about the mass of two 1 L bottles of water, and 1.5 m is roughly chest height. A few tens of joules is the right size.

• Power sign. The same work in less time must give a larger power, as 14.7 W versus 9.8 W.

Common mistakes

• Using mass instead of weight in W = Fd. The force is mg, not m.

• Counting the horizontal carry as work done against gravity.

• Mixing up energy and power. A kilowatt-hour is a unit of energy, not power: 1 kWh = 3.6 × 10⁶ J.

• Forgetting the angle between force and displacement.

• Saying work is done by a force that does not move its point of application.

Note: Your muscles use energy even when holding the book still, because muscle fibers keep contracting. That is a fact about the body, not about the physics definition of work on the book.

Where you will use it next

The next step is energy conservation, for example on a roller coaster, where the same mgh turns into speed. For a very different use of the idea, momentum and collisions tracks what happens to kinetic energy when objects hit.

Frequently asked questions

Why is a joule a good unit?

It ties together force and distance, so one newton moving something one meter is exactly one joule. Different forms of energy, heat or electrical, are all counted in the same unit.

What is the difference between a watt and a watt-hour?

A watt is a rate, one joule per second. A watt-hour is an amount of energy, the energy used by one watt for one hour, which is 3600 J.