Periodic and oscillatory motion
The needle of a sewing machine, a swing and the beating of the heart all repeat their motion again and again. Link to earlier classes : uniform circular motion of chapter 3 is also periodic.
A motion that repeats itself after equal intervals of time is periodic. If a body moves to and fro about a mean position, the motion is oscillatory (also called vibration).
Rule 1: Period, frequency and angular frequency — T = 1 ÷ ν and ω = 2π ÷ T = 2πν (T = time period in s, ν = frequency in Hz, ω = angular frequency in rad s⁻¹).
- Easy : ν = 50 Hz ⇒ T = 0.02 s.
- Medium : A pendulum makes 30 oscillations in 60 s ⇒ T = 2 s, ν = 0.5 Hz and ω = 2π ÷ 2 = 3.14 rad s⁻¹.
- Exam : Every oscillation is periodic, but not every periodic motion is oscillatory : the Earth going round the Sun is periodic, with no to-and-fro motion about a mean position.
T = 1 ÷ ν; ω = 2πν.
Confusing frequency ν with angular frequency ω.
Memory clue : "Omega is 2π times nu."
Spot the difference
| Quantity | Symbol | Unit |
|---|---|---|
| Frequency | ν | Hz |
| Angular frequency | ω | rad s⁻¹ |
| Time period | T | s |
Simple harmonic motion and its displacement
A weight hanging on a spring, pulled down and released, moves smoothly up and down. Link to earlier classes : the sine and cosine functions of mathematics now describe a motion.
Simple harmonic motion (SHM) is the periodic motion in which the restoring force is directly proportional to the displacement from the mean position and always points towards it.
Rule 1: Displacement — x = A cos(ωt + φ) (A = amplitude, the largest displacement; ωt + φ = phase; φ = initial phase). The motion is the projection of uniform circular motion on a diameter.
In the figure : The displacement varies like a cosine curve between +A and −A with the period T.
- Easy : A = 0.1 m, T = 1 s, φ = 0 ⇒ at t = 0.25 s, x = 0.1 cos(π ÷ 2) = 0 (the body is at the mean position).
- Medium : Two SHMs with φ = 0 and φ = π are exactly opposite : one is at +A when the other is at −A.
- Exam : Changing the amplitude does not change the period of SHM.
x = A cos(ωt + φ); amplitude A, angular frequency ω, phase constant φ.
Confusing the amplitude with the total distance between the two extremes (which is 2A).
Memory clue : "Cosine swing between plus A and minus A."
Spot the difference
| Quantity | Meaning |
|---|---|
| Amplitude A | largest displacement |
| Phase | angle (ωt + φ) |
Velocity, acceleration and the force law of SHM
A swing is fastest at the bottom and stops for an instant at each end. Link to earlier classes : velocity and acceleration of chapter 2.
In SHM the velocity is greatest at the mean position and zero at the extremes; the acceleration is greatest at the extremes and zero at the mean position and is always directed towards the mean position.
Rule 1: Formulas — v = ω √(A² − x²), v_max = A ω, a = −ω² x, a_max = A ω². The force is F = −m ω² x = −k x.
- Easy : A = 0.05 m and T = 0.5 s ⇒ ω = 4π = 12.57 rad s⁻¹; v_max = 0.63 m s⁻¹; a_max = 0.05 × 157.9 = 7.9 m s⁻².
- Medium : At x = 0.03 m, v = 12.57 × √(0.05² − 0.03²) = 12.57 × 0.04 = 0.50 m s⁻¹.
- Exam : At x = 1 cm the acceleration has the magnitude 4 m s⁻². Then ω² = 4 ÷ 0.01 = 400 ⇒ ω = 20 rad s⁻¹ ⇒ T = 2π ÷ 20 = 0.314 s.
a = −ω²x (negative sign : towards the mean); v is maximum at x = 0; a is maximum at x = ±A.
Saying that the acceleration is maximum at the mean position.
Memory clue : "Mean : Maximum speed, Minimum (zero) acceleration. Ends : the opposite."
Spot the difference
| Position | Speed | Acceleration |
|---|---|---|
| Mean (x = 0) | maximum | zero |
| Extreme (x = ±A) | zero | maximum |
Energy in simple harmonic motion
At the lowest point of a swing the energy is all kinetic; at the highest point it is all potential. Link to earlier classes : conservation of mechanical energy of chapter 5.
In SHM the kinetic energy and the potential energy keep changing into one another, but their sum, the total mechanical energy, stays constant.
Rule 1: Energies — K = ½ k (A² − x²), U = ½ k x², E = K + U = ½ k A² (k = m ω²). The average of both K and U over a period is ¼ k A².
- Easy : k = 200 N m⁻¹ and A = 0.1 m ⇒ E = ½ × 200 × 0.01 = 1.0 J.
- Medium : At x = A ÷ 2 : U = ¼ E = 0.25 J and K = ¾ E = 0.75 J.
- Exam : K = U at x = A ÷ √2 = 0.0707 m (for A = 0.1 m).
E = ½kA² is constant; K is maximum at the mean position and U at the extremes.
Using the displacement x instead of the amplitude A for the total energy.
Memory clue : "Total is Half k A-squared : fixed; K and U just swap."
Spot the difference
| Position | K | U |
|---|---|---|
| x = 0 | maximum (E) | zero |
| x = ±A | zero | maximum (E) |
The spring–mass system
The suspension of a bike uses springs which keep bobbing when the bike goes over a bump. Link to earlier classes : Hooke's law of chapters 4 and 8.
A body of mass m attached to a spring of constant k and displaced from its rest position performs SHM because the spring force F = −kx is a restoring force.
Rule 1: Period of a spring–mass system — T = 2π √(m ÷ k). The period depends on m and k, not on the amplitude, and (for a vertical spring) not on g. For springs in parallel k = k₁ + k₂; in series 1 ÷ k = 1 ÷ k₁ + 1 ÷ k₂.
- Easy : m = 0.5 kg and k = 200 N m⁻¹ ⇒ T = 2π√(0.5 ÷ 200) = 0.314 s.
- Medium : If the mass is made four times as large, the period doubles.
- Exam : Two identical springs of k = 100 N m⁻¹ in series have k_eff = 50 N m⁻¹. With m = 0.5 kg, T = 2π√(0.5 ÷ 50) = 0.628 s.
T = 2π√(m ÷ k); stiffer spring, shorter period; heavier mass, longer period.
Putting k on top of the fraction.
Memory clue : "Mass over k inside the root, then two pi."
Spot the difference
| Spring combination | Effective k |
|---|---|
| Parallel | k₁ + k₂ |
| Series | k₁k₂ ÷ (k₁ + k₂) |
The simple pendulum
The pendulum of a wall clock swings with a steady beat. Link to earlier classes : gravity and the component of weight from chapter 4.
A simple pendulum is a small heavy bob hanging on a long light inextensible string. For small angles (below about 10°) its motion is SHM.
Rule 1: Period of a simple pendulum — T = 2π √(L ÷ g) (L = length from the point of suspension to the centre of the bob, g = acceleration due to gravity). The period does not depend on the mass of the bob nor on the amplitude (for small swings).
- Easy : L = 1 m ⇒ T = 2π√(1 ÷ 9.8) = 2.0 s; a "seconds pendulum" (T = 2 s) is about 0.99 m long.
- Medium : On the Moon (g = 1.7 m s⁻²) the same pendulum has T = 2π√(1 ÷ 1.7) = 4.8 s.
- Exam : In a lift accelerating upwards at 2.0 m s⁻², g_eff = 9.8 + 2.0 = 11.8 m s⁻² ⇒ T = 2π√(1 ÷ 11.8) = 1.83 s, shorter than in a still lift.
T = 2π√(L ÷ g); T ∝ √L; independent of the mass.
Including the mass of the bob in the formula.
Memory clue : "Longer swings take longer: T goes with the square root of L."
Spot the difference
| Change | Effect on T |
|---|---|
| Longer string | larger |
| Heavier bob | none |
| Larger g | smaller |
Damped and forced oscillations, resonance
A swing slows down and stops unless you keep pushing it, and a push in time with the swing makes it go higher. Link to earlier classes : friction and energy loss from chapters 4 and 5.
In damped oscillations friction or air drag removes energy, so the amplitude decreases with time. In forced oscillations an outside periodic force drives the system. Resonance occurs when the driving frequency is equal to the natural frequency of the system and the amplitude becomes very large.
Rule 1: Damping — The amplitude decays as A(t) = A₀ e^(−bt ÷ 2m) (b = damping constant) and the energy is lost as heat.
- Easy : A pendulum left alone slowly comes to rest.
- Medium : If the amplitude falls to half in 10 s, it falls to a quarter in 20 s.
- Exam : A shock absorber in a vehicle is a deliberate damping device.
Rule 2: Resonance — The amplitude is greatest when ω_driving = ω₀, the natural angular frequency.
- Easy : A swing with a natural frequency of 0.5 Hz rises highest when pushed every 2 s.
- Medium : Soldiers break step on a bridge so that their marching frequency does not match the bridge's natural frequency.
- Exam : Tuning a radio set to a station is resonance of an electric circuit.
Damping reduces the amplitude; resonance occurs when the driving frequency equals the natural frequency.
Thinking that a stronger push always gives the larger amplitude : the timing (frequency) matters most.
Memory clue : "Push in tune : resonance."
Spot the difference
| Oscillation | Driven by | Amplitude |
|---|---|---|
| Free | none (once started) | constant (ideal) |
| Damped | none, with drag | decreases |
| Forced | outside force | depends on the frequency |