Work, power and energy

Energy is the most universal idea in physics: it changes from one form to another but is never lost and never appears from nothing. Work measures how much energy passes from one body to another or from one form to another. The energy method often lets you solve a problem without finding forces and accelerations, just by comparing the initial and final states. That makes this topic the most powerful “tool” of mechanics, and it is used again and again in heat, electricity and nuclear physics.

§1. Mechanical work

A force that moves a body does work:

• α is the angle between the force and the displacement.
• α < 90°: the work is positive (the force “helps” the motion).
• α = 90°: the work is zero. In circular motion the centripetal force does no work; in horizontal motion neither weight nor the normal reaction does any work.
• α > 90°: the work is negative. Sliding friction always does negative work.

If the force varies, the work equals the area under the F(s) graph.

• Work done by gravity does not depend on the path, only on the height difference: A = mg(h₁ − h₂).
• Work done by a spring deformed from x₁ to x₂: A = kx₁²/2 − kx₂²/2. It comes from the triangular area under the F = kx graph.
• Work done by friction: A = −F_fr·s; it depends on the length of the path.

Forces whose work around a closed path is zero (gravity, elastic force) are called conservative; friction is not conservative.

§2. Power and efficiency

Power is the rate of doing work:

The last formula explains a car’s gearbox: engine power is limited, so when a large pulling force is needed uphill, the speed is reduced. Electrical energy is billed in kilowatt-hours: 1 kW·h = 3.6·10⁶ J. An old unit is the horsepower: 1 hp ≈ 736 W (metric horsepower).

In every machine part of the input work is lost to friction and heating.

Efficiency is always below 100 %; this follows from conservation of energy.

Worked example

A pump lifts 600 kg of water through 10 m in 2 minutes with 80 % efficiency. What power does it consume?

1. Useful work: A_u = mgh = 600·10·10 = 6·10⁴ J.
2. Input work: A_in = A_u/η = 6·10⁴/0.8 = 7.5·10⁴ J.
3. Power: N = A_in/t = 7.5·10⁴/120 = 625 W.

§3. Kinetic energy and the work–energy theorem

A moving body can do work, so it has energy:

Kinetic energy is proportional to the square of speed: doubling the speed quadruples the energy. That is why a car’s braking distance grows very quickly with speed.

The total work of all forces on a body equals the change in its kinetic energy. The theorem follows from Newton’s second law and the formula v² − v₀² = 2as. With it, braking-distance problems take one line: F_fr·s = mv²/2.

Worked example

A 1500 kg car moves at 20 m/s; when braking, the friction force is 3000 N. Find the braking distance.

1. Initial kinetic energy: E_k = 1500·20²/2 = 300 000 J.
2. The work of friction removes all of it: F_fr·s = E_k.
3. s = 300 000/3000 = 100 m.

§4. Potential energy

Potential energy depends on the relative positions of bodies (or of parts of a body).

• Potential energy is defined only for conservative forces; their work equals the decrease in potential energy: A = E_p1 − E_p2.
• The zero level is chosen freely (the ground, a table, the lowest point); only changes in potential energy have physical meaning. So E_p can even be negative.
• h is the height of the centre of mass; x is the spring’s deformation (extension or compression), not its length.

Worked example

A plane flies at 7200 m with a speed of 216 km/h. How many times is its potential energy greater than its kinetic energy?

1. Convert the speed: v = 216/3.6 = 60 m/s.
2. E_p/E_k = mgh/(mv²/2) = 2gh/v² = 2·10·7200/3600.
3. E_p/E_k = 144 000/3600 = 40.

§5. Conservation of mechanical energy

Total mechanical energy is the sum of kinetic and potential energy.

As a body falls, potential energy turns into kinetic; when it is thrown up, the reverse happens. A normal reaction or string tension perpendicular to the motion does no work and does not break the law.

With friction or air resistance, mechanical energy decreases and turns into internal energy (heat):

The general law of conservation of energy always holds: energy only changes form.

Worked example

A body falls from rest from a height of 20 m. What is its speed at 5 m?

1. mgh₁ = mgh₂ + mv²/2.
2. v² = 2g(h₁ − h₂) = 2·10·15 = 300.
3. v ≈ 17.3 m/s; the result does not depend on mass.

Worked example

A spring gun with a 400 N/m spring compressed by 5 cm fires a 10 g ball vertically upwards. Find the ball’s speed and the height it reaches.

1. kx²/2 = mv²/2 → v = x·√(k/m) = 0.05·√(400/0.01) = 0.05·200 = 10 m/s.
2. mv²/2 = mgh → h = v²/(2g) = 100/20 = 5 m.

§6. Using the energy method

Method for solving a problem with energy:

1. Choose the system and two states (initial and final).
2. Fix the zero level of potential energy.
3. Write E_k and E_p for each state.
4. Without friction, set them equal; with friction, set the difference equal to the heat produced or the work of friction.

For a pendulum: a bob on a string of length l released from angle α rises by h = l(1 − cos α), and its speed at the bottom is v = √(2gl(1 − cos α)).

Worked example

A 50 kg sledge slides down a 10 m high hill and reaches the bottom at 10 m/s. How much energy was lost to friction?

1. Initial energy: E₁ = mgh = 50·10·10 = 5000 J.
2. Final energy: E₂ = mv²/2 = 50·100/2 = 2500 J.
3. Q = E₁ − E₂ = 2500 J; half the energy became heat.

Check yourself

1. A box is pulled 4 m by a 50 N force at 60° to the horizontal. Work done?

2. Kinetic energy of a 2 kg body moving at 3 m/s?

3. A 400 N/m spring is compressed by 5 cm. Potential energy?

4. A body is thrown up at 10 m/s. At what height is its speed halved?

5. Power of a locomotive pulling with 30 kN at 72 km/h?

6. A 1.6 m pendulum is pulled to 60° and released. Speed at the bottom?

7. The speed triples. By what factor does kinetic energy grow?

8. How much work does the centripetal force do in uniform circular motion?

Common mistakes

• Forgetting cos α or measuring the angle from the wrong direction.
• Leaving speed in km/h: energy depends on v², so the error is about 13 times.
• Assuming mechanical energy is conserved when there is friction.
• Swapping useful and input work in efficiency: an efficiency above 100 % means an error.
• Using the spring’s whole length instead of its deformation in kx²/2.