Learn Work & energy from scratch
Before this: Newton’s laws
1What energy is
Energy is the ability to make something move or change. It is measured in joules (J). One joule is the energy needed to push with 1 N over 1 m.
Energy is never created or destroyed. It only changes form. A falling ball turns height into speed. A car that brakes turns speed into heat in its brakes.
This gives you a powerful shortcut. Instead of following the motion second by second with forces, you compare the energy at the start with the energy at the end.
In this topic you meet three forms of mechanical energy: energy of motion, energy of height and energy stored in a spring. Friction takes energy out and turns it into heat.
Self-check A box slides across a rough floor and stops. What happened to its kinetic energy?
- Friction turned it into heat
- It became potential energy
- It is still stored in the box as speed
- It was destroyed
Answer: Friction turned it into heat. Energy cannot disappear. On a level floor the height does not change, so all the kinetic energy went into heat through friction.
2Work and kinetic energy
A force does work when it moves something. Work is force times the distance moved in the direction of the force:
W = F · d · cos α
Here α is the angle between the force and the motion. A force along the motion (α = 0) does full work. A force at right angles (α = 90°) does no work at all.
The energy of motion is kinetic energy. It grows with the square of the speed, so doubling the speed gives four times the energy:
Ek = ½ · m · v²
The work done by all forces together equals the change in kinetic energy. This is the work–energy theorem.
Self-check A rope pulls a crate 4 m along a level floor with a force of 50 N, at 60° above the horizontal. How much work does the rope do?
- 100 J
- 12.5 J
- 173 J
- 200 J
Answer: 100 J. W = 50 · 4 · cos 60° = 100 J. Leaving out the cos α gives 200 J, and using sin 60° gives 173 J.
3Potential energy: height and springs
An object that is higher up can fall and pick up speed. The energy it has because of its height is the gravitational potential energy:
Ep = m · g · h
Only the change in height matters, so you choose the zero level yourself. The lowest point of the motion is usually the easiest choice.
A spring that is squeezed or stretched by a distance x stores elastic energy. Here k is the spring stiffness in N/m:
Es = ½ · k · x²
Put everything together in one energy balance. Wf is the work done against friction, which leaves as heat:
Ek1 + Ep1 + Es1 = Ek2 + Ep2 + Es2 + Wf
For a free fall from rest, with no friction, this gives m·g·h = ½·m·v², so v = √(2gh). The mass cancels.
Self-check A stone is dropped from rest from a height of 20 m. Ignoring air resistance, how fast is it moving when it reaches the ground? (g = 9.81 m/s²)
- 14.0 m/s
- 19.8 m/s
- 196 m/s
- 392 m/s
Answer: 19.8 m/s. v = √(2 · 9.81 · 20) = √392.4 = 19.8 m/s. 392 m/s forgets the square root, and 14.0 m/s forgets the factor 2 from the ½ in ½·m·v².
4The method step by step
- Draw the situation. Mark the start point 1 and the end point 2, and choose the zero level for height.
- Write the energy at point 1: Ek1 = ½·m·v1², Ep1 = m·g·h1, and ½·k·x1² if a spring is squeezed.
- Write the energy at point 2 in the same way. Leave the unknown as a symbol.
- Find the work against friction. On a slope the normal force is N = m·g·cos θ, so Wf = μ·m·g·cos θ·d, with d the distance slid.
- Fill in the energy balance and solve for the unknown.
Draw the forces too, even when you only use energy. The drawing shows which forces do work and which do not.
The normal force N is at right angles to the motion, so it does no work. It only matters because friction depends on it.
5A worked example
Follow every step. Try to predict the next one before you open it.
Worked exampleA 2 kg block slides from rest down a slope at 30° to the horizontal. The coefficient of kinetic friction is 0.2. Use g = 9.81 m/s². Find its speed after it has slid 5 m along the slope.
1Point 1 is the start, point 2 is 5 m further down the slope. Put the zero level for height at point 2.
2The block slides 5 m along the slope, but it only drops the vertical part of that:
h = d · sin θ = 5 · sin 30° = 2.5 m
Ep1 = m · g · h = 2 · 9.81 · 2.5 = 49.05 J
3On the slope the normal force is smaller than the weight:
N = m · g · cos θ = 2 · 9.81 · cos 30° = 16.99 N
Wf = μ · N · d = 0.2 · 16.99 · 5 = 16.99 J
4The block starts at rest, so Ek1 = 0. At point 2 the height is zero:
Ep1 = Ek2 + Wf
Ek2 = 49.05 − 16.99 = 32.06 J
5½ · 2 · v² = 32.06
v = √32.06 = 5.66 m/s
6Without friction the block would reach √(2 · 9.81 · 2.5) = 7.00 m/s. Friction must make it slower, and 5.66 m/s is slower. The answer is sensible.
The mass cancels in the end: v² = 2·g·d·(sin θ − μ·cos θ) = 32.06, which gives the same 5.66 m/s.
Answer The block reaches v ≈ 5.66 m/s after sliding 5 m.
6Common mistakes
- Using the weight m·g as the normal force on a slope. On a slope N = m·g·cos θ.
- Using the distance along the slope as the height. The height drop is d·sin θ.
- Forgetting the square root at the end: ½·m·v² gives v², not v.
- Adding friction work to the energy at the end with the wrong sign. Friction always takes energy out of the motion.
- Writing a spring’s energy as k·x instead of ½·k·x², or leaving x in cm.
Self-check A 1 kg block slides 2 m down a 30° slope with μ = 0.5. How much work is done against friction? (g = 9.81 m/s²)
- 19.6 J
- 4.91 J
- 8.50 J
- 9.81 J
Answer: 8.50 J. Wf = μ · m · g · cos θ · d = 0.5 · 1 · 9.81 · cos 30° · 2 = 8.50 J. Using m·g as the normal force gives 9.81 J, and using sin θ instead of cos θ gives 4.91 J.
7How exam questions look
- Easy: free fall and other single conversions. A dropped ball (find v or Ek), a pendulum released from an angle, a block running into a spring, the work of a pulling rope, or the power of a hoist (P = m·g·h / t).
- Medium: a block on a slope with friction: find the speed after a distance or the work against friction. Variations include a car skidding to a stop, a spring launcher and the power needed to drive up a hill.
- Hard: several stages in one question. For example a smooth ramp, then a rough strip, then a spring: find the speed at the spring or the maximum compression. Others combine a pulled block, a loop or two connected blocks.
In the hard questions, write one energy balance from the very start to the very end. Add the friction work of every rough part.
For the maximum spring compression, the block stops for a moment, so all its kinetic energy is in the spring: ½·m·v² = ½·k·x².
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