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Simple harmonic motion and waves: Class 10 Physics
Vibrations, simple harmonic motion, the mass on a spring and the simple pendulum, wave motion, transverse and longitudinal waves and the wave equation.
Key points
A vibration is a to and fro motion about a fixed point. One complete vibration is the motion from a point, through the extremes, and back to the same point moving in the same direction. The time for one vibration is the time period T, and the number of vibrations in one second is the frequency f. They are related by f = 1/T. Frequency is measured in hertz (Hz).
The amplitude is the maximum distance of the vibrating body from its mean (equilibrium) position. The displacement is its distance from the mean position at a given instant.
A body moves with simple harmonic motion (SHM) when:
- its acceleration is always directed towards the mean position,
- its acceleration is directly proportional to its displacement from the mean position.
In SHM the speed is greatest at the mean position and zero at the extreme positions. The acceleration is greatest at the extreme positions and zero at the mean position. A mass on a spring and a simple pendulum with a small angle of swing are examples of SHM.
Mass and spring. If a mass m hangs on a spring of stiffness constant k (the force needed per metre of stretch), the time period is T = 2π√(m/k). A stiffer spring or a smaller mass gives a shorter time period. The time period does not depend on the amplitude.
Simple pendulum. A small heavy bob hangs on a light thread of length l. For small swings the time period is T = 2π√(l/g), where g is the acceleration of free fall. The time period depends on the length and on g. It does not depend on the mass of the bob or on the amplitude, as long as the amplitude is small. A pendulum with a time period of 2 s is called a seconds pendulum, because it takes 1 s for each swing from one side to the other.
A wave is a disturbance that carries energy from one place to another without carrying the material of the medium with it. Mechanical waves need a medium. Light and other electromagnetic waves do not.
Two kinds of mechanical wave:
- In a transverse wave the particles of the medium vibrate at right angles to the direction in which the wave travels. A wave on a rope is an example. It has crests and troughs.
- In a longitudinal wave the particles vibrate along the direction in which the wave travels. Sound in air is an example. It has compressions and rarefactions.
The wavelength λ is the distance between two consecutive crests (or two consecutive troughs) of a transverse wave, or between two consecutive compressions of a longitudinal wave. The wave speed, frequency and wavelength are related by v = fλ. The speed of a wave depends on the medium, and the frequency is fixed by the source.
A ripple tank shows the behaviour of water waves. When a wave meets a barrier it is reflected, with the angle of reflection equal to the angle of incidence. When a wave goes from deep water to shallow water, its speed and wavelength decrease while its frequency stays the same. This change of direction is refraction.
Study tip: For a pendulum or spring numerical, write the formula, convert every quantity to SI units, then substitute. Check the answer: a 1 m pendulum should have a time period of about 2 s.
Definitions and flashcards
Say the meaning first, then open the card to check.
Time period
The time taken for one complete vibration.
Frequency
The number of complete vibrations in one second, in hertz.
Amplitude
The maximum displacement of a vibrating body from its mean position.
Simple harmonic motion
Motion in which the acceleration is directly proportional to the displacement and is directed towards the mean position.
Spring constant
The force needed to produce a stretch of one metre in a spring, in newtons per metre.
Seconds pendulum
A simple pendulum with a time period of 2 seconds.
Wave
A disturbance that transfers energy from one place to another.
Transverse wave
A wave in which the particles vibrate at right angles to the direction of travel.
Longitudinal wave
A wave in which the particles vibrate parallel to the direction of travel.
Wavelength
The distance between two consecutive points that are in the same state of vibration, for example two crests.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Frequency and time period | f = 1 / T | Frequency in hertz when T is in seconds. |
| Mass on a spring | T = 2π√(m / k) | m in kg and k in N/m give T in seconds. |
| Simple pendulum | T = 2π√(l / g) | l in metres and g = 9.8 m/s² give T in seconds. |
| Wave equation | v = f λ | Speed in m/s, frequency in Hz and wavelength in metres. |
| Wave speed from the period | v = λ / T | The same equation, written with the time period. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
Define time period and frequency. (2 marks)
The time period is the time for one complete vibration. The frequency is the number of complete vibrations in one second. They are related by f = 1/T.
Does the time period of a simple pendulum depend on the mass of the bob? (2 marks)
No. For small swings T = 2π√(l/g), which contains only the length and g. A heavier bob has the same time period as a lighter one.
State the two conditions for simple harmonic motion. (3 marks)
The acceleration of the body must be directed towards the mean position, and it must be directly proportional to the displacement of the body from the mean position.
Distinguish between a transverse and a longitudinal wave. Give an example of each. (3 marks)
In a transverse wave the particles vibrate at right angles to the direction of travel, and the wave has crests and troughs. A wave on a rope is an example. In a longitudinal wave the particles vibrate along the direction of travel, and the wave has compressions and rarefactions. Sound in air is an example.
Why does a simple pendulum take longer for each swing on the Moon than on the Earth? (3 marks)
The time period is T = 2π√(l/g). The value of g on the Moon is smaller than on the Earth, so the time period is larger and each swing takes longer.
Describe what happens to the speed, wavelength and frequency of a water wave when it passes from deep water into shallow water. (5 marks)
In a ripple tank the speed of the wave decreases when it goes from deep to shallow water. The frequency stays the same because it is fixed by the source. Since v = fλ and f is constant, a smaller speed means a smaller wavelength, so the waves in the shallow water are closer together. If the waves meet the boundary at an angle, their direction also changes. This is refraction.
Explain how the speed and the acceleration of a body in SHM change from the mean position to an extreme position. (5 marks)
At the mean position the displacement is zero, so the acceleration is zero and the speed is at its maximum. As the body moves outwards, the displacement increases, so the acceleration increases and is directed back towards the mean position. This slows the body, so its speed falls. At the extreme position the displacement equals the amplitude, the acceleration is at its maximum and the speed is zero for an instant before the body returns.
Numericals with worked steps
A pendulum completes 20 vibrations in 10 s. Find its time period and frequency.
- Time period T = total time ÷ number of vibrations = 10 ÷ 20 = 0.5 s.
- Frequency f = 1 / T = 1 / 0.5 = 2 Hz.
Answer: T = 0.5 s and f = 2 Hz
Find the time period of a simple pendulum of length 1.0 m. Take g = 9.8 m/s².
- T = 2π√(l/g).
- = 2 × 3.142 × √(1.0 / 9.8).
- √(0.10204) = 0.3194.
- T = 6.284 × 0.3194 = 2.007 s.
Answer: About 2.0 s
A mass of 0.5 kg hangs on a spring of constant 200 N/m. Find the time period of its vibrations.
- T = 2π√(m/k).
- = 2 × 3.142 × √(0.5 / 200).
- √(0.0025) = 0.05.
- T = 6.284 × 0.05 = 0.314 s.
Answer: About 0.31 s
A wave has a frequency of 50 Hz and a wavelength of 6.8 m. Find its speed.
- v = f λ.
- = 50 × 6.8 = 340 m/s.
Answer: 340 m/s
A sound wave travels at 340 m/s and has a frequency of 170 Hz. Find its wavelength.
- λ = v / f.
- = 340 / 170 = 2 m.
Answer: 2 m
Common mistakes
- Saying that a heavier pendulum bob gives a longer time period. The mass does not appear in the formula.
- Using the length of the pendulum in centimetres in T = 2π√(l/g) with g in m/s². Convert to metres first.
- Mixing up the particle motion in transverse and longitudinal waves.
- Saying that the frequency of a wave changes when it enters a new medium. The speed and the wavelength change, and the frequency stays the same.