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Oscillations: Class 11 Physics
Periodic and simple harmonic motion, displacement, velocity and acceleration in SHM, the spring mass system, the simple pendulum, energy in SHM, damped and forced oscillations, and resonance.
Key points
An oscillation (vibration) is a to and fro motion about a mean position. Motion that repeats itself at regular intervals is periodic. Important terms: displacement x is the distance from the mean position; amplitude A is the maximum displacement; one cycle is one complete to and fro motion; the period T is the time for one cycle; the frequency f = 1/T is the number of cycles per second, in hertz (Hz).
Simple harmonic motion (SHM) is a periodic motion in which the acceleration is directly proportional to the displacement from the mean position and is always directed towards it: a = −ω²x. The minus sign shows that the acceleration is opposite to the displacement. Equivalently, the restoring force is F = −kx. Here ω = 2πf is the angular frequency in rad/s.
SHM is closely related to uniform circular motion. If a particle moves round a circle of radius A with angular speed ω, the shadow (projection) of the particle on a diameter performs SHM with amplitude A and angular frequency ω. This is the reference circle method. It gives the displacement as x = A sin ωt when the time is counted from the mean position.
Equations of SHM (starting from the mean position):
- Displacement: x = A sin ωt
- Velocity: v = Aω cos ωt = ω√(A² − x²)
- Acceleration: a = −ω²x
- Maximum speed ω A occurs at the mean position (x = 0). The speed is zero at the extreme positions (x = ±A).
- Maximum acceleration ω²A occurs at the extreme positions. It is zero at the mean position.
A mass m attached to a spring of spring constant k oscillates on a smooth surface. By Hooke's law the restoring force is F = −kx, so ma = −kx and ω² = k/m. The period is T = 2π√(m/k). A stiffer spring (larger k) gives a shorter period and a heavier mass gives a longer period. The period does not depend on the amplitude. For a vertical spring the same formula holds, because gravity only shifts the equilibrium position.
A simple pendulum is a small heavy bob on a light inextensible string of length L. For small angles (less than about 10°) the restoring force on the bob is mg sin θ, which is close to mgθ, so the motion is SHM with T = 2π√(L/g). The period does not depend on the mass or on the amplitude (for small swings). It increases with the length, and it is longer at a place where g is smaller. A pendulum with a period of 2 s is the seconds pendulum, and its length is nearly 1 m. The pendulum can be used to measure g.
Energy in SHM. The total mechanical energy of an oscillating system is constant if there is no friction:
- Kinetic energy: KE = ½mv² = ½k(A² − x²)
- Potential energy: PE = ½kx²
- Total energy: E = ½kA² (constant)
- At the mean position all the energy is kinetic. At the extreme position it is all potential.
- The energy is proportional to the square of the amplitude.
Free oscillations occur at the natural frequency of the system, when no outside force acts. In real systems, friction and air resistance take energy away, and the amplitude falls with time. These are damped oscillations. Damping is useful in the shock absorbers of vehicles and in the pointers of instruments, which should settle quickly. Damping can be light, critical (the system returns to rest in the shortest time without oscillating) or heavy.
Forced oscillations are produced when a periodic driving force acts on a system. After some time the system oscillates at the frequency of the driving force, not at its own natural frequency. The amplitude is largest when the driving frequency equals the natural frequency. This is called resonance. Examples are a child on a swing pushed at the right moments, tuning a radio receiver, and a singer breaking a glass with a note of its natural frequency. Resonance can also be harmful: soldiers break step when crossing a bridge so that their marching does not match its natural frequency, and buildings in earthquake zones are designed to avoid resonance.
Graphs: the displacement time graph of SHM is a sine (or cosine) curve. The graph of acceleration against displacement is a straight line through the origin with a negative slope equal to −ω². The graph of energy against displacement shows PE as a parabola and KE as an inverted parabola, and their sum is a horizontal line.
Study tip: Learn the three starting equations of SHM and the relations for speed, acceleration and energy at the mean and extreme positions. In calculations, keep the angle in radians, and take care to use ω in rad/s, not the frequency in Hz, in the formulas for speed and acceleration.
Definitions and flashcards
Say the meaning first, then open the card to check.
Amplitude
The maximum displacement from the mean position.
Period
The time taken for one complete oscillation.
Frequency
The number of oscillations per second, in hertz.
Simple harmonic motion
Motion in which acceleration is proportional to displacement and directed towards the mean position.
Restoring force
The force that always acts towards the equilibrium position.
Damping
The loss of amplitude due to friction or resistance.
Resonance
The large rise in amplitude when the driving frequency equals the natural frequency.
Seconds pendulum
A simple pendulum with a time period of 2 s.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Angular frequency | ω = 2πf = 2π/T | In rad/s. |
| SHM acceleration | a = −ω²x | Directed towards the mean position. |
| Velocity | v = ω√(A² − x²) | Maximum value ωA at x = 0. |
| Spring period | T = 2π√(m/k) | k is the spring constant. |
| Pendulum period | T = 2π√(L/g) | For small angles. |
| Total energy | E = ½kA² | Constant in SHM. |
| Kinetic energy | KE = ½k(A² − x²) | At displacement x. |
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 simple harmonic motion. (2 marks)
Simple harmonic motion is a periodic motion in which the acceleration of the body is directly proportional to its displacement from the mean position and is always directed towards the mean position, a = −ω²x.
At what positions in SHM is the speed maximum and the acceleration maximum? (2 marks)
The speed is maximum (ωA) at the mean position, where the acceleration is zero. The acceleration is maximum (ω²A) at the extreme positions, where the speed is zero.
Derive the time period of a mass spring system. (5 marks)
A mass m on a spring of constant k is displaced by x. The restoring force is F = −kx. By Newton's second law, ma = −kx, so a = −(k/m)x. Comparing this with the defining equation of SHM, a = −ω²x, we get ω² = k/m, so ω = √(k/m). The period is T = 2π/ω = 2π√(m/k). The period depends on the mass and the spring constant only, and not on the amplitude.
Why should the angle of swing of a simple pendulum be small? (2 marks)
The restoring force is mg sin θ. It is proportional to the displacement only when sin θ is nearly equal to θ in radians, which is true for small angles. For large angles the motion is not simple harmonic.
What is resonance? Give two examples. (3 marks)
Resonance occurs when the frequency of an external periodic force equals the natural frequency of a system, so the system oscillates with a very large amplitude. Examples are pushing a swing at its natural frequency and tuning a radio set to the frequency of a station.
Show that the total energy in SHM is constant. (3 marks)
At displacement x, KE = ½k(A² − x²) and PE = ½kx². Adding them gives E = ½kA² − ½kx² + ½kx² = ½kA². This does not depend on x, so the total energy is constant and proportional to the square of the amplitude.
Numericals with worked steps
A 0.5 kg mass hangs on a spring of constant 200 N/m. Find the period and frequency of oscillation.
- T = 2π√(m/k) = 2π√(0.5/200) = 2π√0.0025.
- √0.0025 = 0.05, so T = 2π × 0.05 = 0.314 s.
- f = 1/T = 1/0.314 = 3.18 Hz.
Answer: T = 0.314 s and f = 3.18 Hz.
Find the length of a simple pendulum that has a period of 2 s. Take g = 9.8 m/s².
- T² = 4π²L/g, so L = gT²/4π².
- L = 9.8 × 4/39.48 = 39.2/39.48 = 0.993 m.
Answer: About 0.99 m.
A particle performs SHM with amplitude 0.1 m and frequency 2 Hz. Find the maximum speed and maximum acceleration.
- ω = 2πf = 2π × 2 = 12.57 rad/s.
- v_max = ωA = 12.57 × 0.1 = 1.257 m/s.
- a_max = ω²A = (12.57)² × 0.1 = 157.9 × 0.1 = 15.8 m/s².
Answer: 1.26 m/s and 15.8 m/s².
A spring of constant 100 N/m is stretched by 0.2 m and released. Find the total energy and the kinetic energy at x = 0.1 m.
- E = ½kA² = ½ × 100 × 0.04 = 2 J.
- KE = ½k(A² − x²) = ½ × 100 × (0.04 − 0.01) = 50 × 0.03 = 1.5 J.
- Check: PE = ½ × 100 × 0.01 = 0.5 J and 1.5 + 0.5 = 2 J.
Answer: E = 2 J and KE = 1.5 J.
A particle in SHM has amplitude 5 cm and angular frequency 4 rad/s. Find its speed at a displacement of 3 cm.
- v = ω√(A² − x²).
- = 4 × √(25 − 9) = 4 × √16 = 4 × 4 = 16 cm/s.
Answer: 16 cm/s (0.16 m/s).
Common mistakes
- Saying that the period of a pendulum depends on the mass of the bob. It does not.
- Using the formula T = 2π√(L/g) for a large angle of swing.
- Taking the acceleration as maximum at the mean position. It is zero there.
- Forgetting that x in the formulas is the displacement from the mean position, not from the starting point.
- Mixing up the amplitude with the peak to peak distance, which is twice the amplitude.