Soldiers are told to break step when they cross a bridge. A swing pushed at just the right moments climbs higher and higher. Both come down to one idea: what happens when an oscillating body is pushed at its own natural frequency.
Chapter 14 of the PECTAA Physics 12 book is Simple Harmonic Motion. After the graphs of SHM and conservation of energy (sections 14.6 and 14.7), section 14.8 compares natural and forced oscillations, section 14.9 covers damped oscillation, section 14.10 resonance and section 14.11 sharpness of resonance. This note is for FSc Part 2 (2nd year, Class 12) students in Lahore, Faisalabad, Sargodha, Rawalpindi, Multan, Gujranwala and the other Punjab boards. Gujrat district comes under the Gujranwala board. PECTAA writes the books. Your own board sets your paper.
What to know
Natural (unforced) oscillations (14.8). A body that oscillates at its own natural frequency, with no external force interfering, is vibrating on its own (unforced). A simple pendulum slightly displaced and released is the book's example: its natural frequency depends only on its length. With no resistance, total energy is conserved and the amplitude stays constant. The book's side box lists a guitar string, a ringing bell, atomic oscillation, bridge vibration and building sway as examples.
Forced oscillations. If an external force acts on such a system, forced vibrations take place: a swing struck repeatedly, a factory floor shaken by heavy machinery, the wooden boards of string instruments, a vehicle vibrating from its running engine.
Damped oscillations (14.9). In practice, resistive forces such as air resistance and friction remove energy. Oscillations whose amplitude decreases with time are damped oscillations; the resistive forces are damping forces. A swing left alone, a pendulum slowly stopping, and a tuning fork touched with a finger are examples. Fig. 14.15 shows a displacement–time graph whose peaks shrink steadily.
| Type | What happens | Book example |
|---|---|---|
| Light damping | energy and amplitude fall gradually | playground swing |
| Heavy damping | takes a long time to come to rest | pendulum in thick oil |
| Critical damping | returns to equilibrium in the shortest time | shock absorbers in a car |
Shock absorbers in a car's suspension damp the oscillations from a bumpy road so the ride stays comfortable.
Resonance (14.10). A periodic external force (the driving force) can keep the amplitude constant; its frequency is the driving frequency. The amplitude of a forced oscillation depends on how the driving frequency compares with the natural frequency. When they are equal, the amplitude grows very large: this is resonance. The book adds that resonance also occurs at whole-number multiples: f = 2f₀, 3f₀, 4f₀, …, nf₀.
Book demonstration (Fig. 14.17). Pendulums A and B have length ℓ₁; C and D have length ℓ₂; all hang from one horizontal rod. A fifth pendulum P of adjustable length is set swinging. If P has length ℓ₁, A and B pick up large oscillations through the rod while C and D stay almost at rest. Make P's length ℓ₂ and the roles swap.
Useful resonance: microwave ovens (the waves make water molecules in food oscillate, and their collisions produce heat; dry plastic or glass containers stay cool); radio tuning (turning the knob changes the circuit's natural frequency until it matches a station's frequency).
Resonance to avoid: marching soldiers on a bridge (their step frequency could match the bridge's natural frequency); aeroplane wings (aerodynamic forces, turbulence or engine vibration matching the wing's natural frequency). The book also recalls the Tacoma Narrows Bridge in 1940, set into resonant vibration by a mild wind a few months after opening.
Sharpness of resonance (14.11). Amplitude and sharpness at resonance both depend on damping: the smaller the damping, the bigger the peak and the sharper the resonance. Heavy damping gives a fairly flat amplitude–frequency curve (Fig. 14.20). In the book's test, a pith-ball bob and a lead bob of equal size and length are driven by a third pendulum of the same length: the lead bob swings with much larger amplitude because air resistance damps the light pith ball far more.
Key points
- Natural oscillation: own frequency, constant amplitude (no resistance).
- Damped: amplitude falls with time.
- Resonance: driving frequency = natural frequency.
- Less damping → sharper, taller resonance peak.
Worked examples
Example 1 (resonant frequencies). A body has natural frequency 50 Hz. At which driving frequencies does the book say resonance occurs?
- f = nf₀ with n = 1, 2, 3, …
- So 50 Hz, 100 Hz, 150 Hz, …
Example 2 (pendulum demonstration). P has length ℓ₂. Which pendulums swing strongly?
- Same length means same natural frequency.
- C and D (length ℓ₂) resonate; A and B stay nearly at rest.
Example 3 (match the damping). Classify: car shock absorber; swing slowly stopping; pendulum in thick oil.
- Critical; light; heavy.
Common mistakes in the exam
- Saying amplitude stays constant in real oscillations. Damping reduces it.
- Confusing heavy and critical damping. Critical returns fastest.
- Saying any driving force causes resonance. The frequencies must match.
- Thinking more damping gives a sharper peak. It is the opposite.
- Forgetting why soldiers break step: to avoid matching the bridge's natural frequency.
Exam technique
- Define, then give one book example for each type of oscillation.
- Sketch the decaying displacement–time graph and the amplitude–frequency curves.
- Describe the pendulum demonstration in three steps: setup, P = ℓ₁, P = ℓ₂.
- Check your board's notice for the sections in your paper.
Practice questions
- Define damping forces. Answer: Resistive forces such as air resistance and friction that reduce the amplitude of an oscillation.
- Give one useful and one harmful case of resonance. Answer: Useful: radio tuning or microwave heating. Harmful: soldiers marching on a bridge.
- Which bob in the pith-ball and lead test swings more, and why? Answer: The lead bob; air resistance damps the light pith ball more.
- What shape is the resonance curve of a heavily damped system? Answer: Fairly flat.
- Why does a microwave oven heat food but not a dry glass dish? Answer: It works on water molecules, which the glass does not contain.
Ready to practise this? Continue in 2nd Year Physics.