Energy in simple harmonic motion Cambridge International AS & A Level Physics revision
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In plain words
A swing at the top of its arc is still for an instant, with all its energy stored as height. At the bottom it is at its fastest, with all its energy as movement. Simple harmonic motion is a constant trade between the two, and the total never changes.
4 things to know
- At the ends the energy is all potential, and the kinetic energy is zero. At the centre it is all kinetic. In between it is a mixture of the two.
- The total energy is E = ½mω²A², which is ½mv² at the centre. It stays constant if there is no damping, and it is proportional to the amplitude squared.
- Conservation of energy at any point gives the speed: v = ±ω√(A² − x²).
- In one complete oscillation the kinetic energy reaches its maximum twice, and so does the potential energy.
Worked example
A 0.20 kg mass oscillates with an angular frequency of 10 rad/s and an amplitude of 0.050 m. Find its total energy, and its speed when its displacement is 0.030 m.
- E = ½mω²A² = ½ × 0.20 × 10² × 0.050² = 0.025 J.
- v = ω√(A² − x²) = 10 × √(0.050² − 0.030²) = 10 × √0.0016.
- = 10 × 0.040 = 0.40 m/s.
Tips and tricks
- Double the amplitude and the total energy is four times bigger.
- On a graph of energy against displacement, kinetic energy is an upside-down curve, potential energy is a U-shaped curve, and the total is a level line.
It lands in your notebook with its questions as flashcards.
Energy in simple harmonic motion: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
Where is the kinetic energy of a simple harmonic oscillator greatest?
The speed is greatest there.
Where is the potential energy of a simple harmonic oscillator greatest?
The oscillator is momentarily at rest there.
How does the total energy of a simple harmonic oscillator depend on its amplitude A?
E = ½mω²A².
What is the speed of a simple harmonic oscillator when its displacement equals its amplitude?
It is at an end, about to turn back.
When the displacement is half the amplitude, what fraction of the total energy is potential?Stretch
Potential energy is proportional to x², and (½)² is a quarter.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 7 marks. Write your answers on paper, then check them.
Energy in simple harmonic motion
Cambridge International AS & A Level Physics 9702 · 7 marks · papermunch.org
Name ______________________________ Date ______________
Describe the energy changes of a pendulum bob as it swings from one end to the other.[2]
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At the end it has only potential energy. As it moves towards the centre, potential energy is transferred to kinetic energy, which is greatest at the centre. As it rises to the other end, the kinetic energy is transferred back to potential energy.
The amplitude of an oscillation is halved. State what happens to its total energy.[2]
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It falls to a quarter.
A 0.50 kg mass oscillates with an amplitude of 0.10 m and a frequency of 2.0 Hz. Find its total energy.[3]
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0.39 J. ω = 2π × 2.0 = 12.6 rad/s, and E = ½ × 0.50 × 12.6² × 0.10².
Answers: Energy in simple harmonic motion
- 1. At the end it has only potential energy. As it moves towards the centre, potential energy is transferred to kinetic energy, which is greatest at the centre. As it rises to the other end, the kinetic energy is transferred back to potential energy.
- 2. It falls to a quarter.
- 3. 0.39 J. ω = 2π × 2.0 = 12.6 rad/s, and E = ½ × 0.50 × 12.6² × 0.10².