Forced vibrations and resonance (A-level only) — AQA A-Level Physics

Forced vibrations and resonance are key topics in A-level Physics that explore how systems respond to external periodic forces. Unlike free vibrations, where a system oscillates at its natural frequency, forced vibrations occur when an external driving force dictates the frequency of oscillation. This topic is crucial for understanding real-world phenomena such as musical instruments, engineering structures, and electrical circuits.

Resonance, a special case of forced vibration, occurs when the driving frequency matches the natural frequency, leading to a dramatic increase in amplitude. This concept is fundamental in physics and engineering, with applications ranging from tuning radios to designing earthquake-resistant buildings. Mastering this topic requires a solid grasp of energy transfer, damping, and the mathematical relationships governing oscillatory motion.

• →Forced vibrations: The system oscillates at the driving frequency, not its natural frequency. The amplitude depends on the driving frequency and the amount of damping.
• →Resonance: Occurs when driving frequency equals natural frequency. The amplitude reaches a maximum, and energy transfer from the driver to the oscillator is most efficient.
• →Damping: Reduces the amplitude of resonance and broadens the resonance peak. Heavier damping results in a lower and wider peak; lighter damping gives a sharper, taller peak.
• →Phase relationship: At resonance, the driving force is in phase with the velocity of the oscillator (phase difference between force and displacement is 90 degrees).
• →Natural frequency: The frequency at which a system oscillates freely without any external force, determined by its physical properties (e.g., mass and stiffness).

• Distinguishes free vibration, where the system oscillates at its natural frequency, from forced vibration, where a periodic driving force determines the frequency.
• States that the amplitude of a forced vibration depends on the difference between the driving frequency and the natural frequency.
• Describes resonance as the condition where driving frequency equals natural frequency and amplitude reaches a maximum.
• Explains that increased damping reduces the maximum amplitude and broadens the resonance peak.
• Applies the ideas to a named example, such as a swing pushed at its natural frequency or a building vibrating in an earthquake.
• Resonance is the condition where the driving frequency equals a natural frequency of the system, giving maximum energy transfer per cycle and a maximum steady-state amplitude.
• At resonance the phase relationship between driver and oscillator is such that the driver does work on the system throughout the cycle, so energy input is greatest.
• Damping dissipates mechanical energy, usually as thermal energy, so the resonant amplitude is reduced compared with the undamped or lightly damped case.
• Increasing damping broadens the resonance curve, reducing the sharpness of resonance; decreasing damping narrows the curve and raises the peak.
• The sharpness of resonance can be compared by measuring the width of the resonance peak at a stated level, for example at half the peak amplitude, and relating a narrower width to sharper tuning.
• Practical examples include a mass on a spring driven by a vibrating support, a pendulum driven by a motor, or a loudspeaker cone near its natural frequency.
• A mechanical example must identify the driver, the oscillating system, and the natural frequency that is matched or approached.
• Damping in a mechanical system removes energy from the oscillations, often as thermal energy, so amplitudes decay or steady-state amplitudes are reduced.
• In stationary waves on a string fixed at both ends, resonance occurs at frequencies where the string length equals a whole number of half-wavelengths.
• In tubes, the resonant frequencies depend on whether the ends are open or closed, because the boundary conditions determine the allowed stationary-wave patterns.
• Damping broadens each resonance, so the stationary-wave response is less sharply tuned to a single driving frequency.
• A suitable comparison is a lightly damped and heavily damped system driven through the same frequency range, noting the change in peak height and width.

• 💡Use the terms driving frequency and natural frequency precisely in your answer.
• 💡Sketch or describe the amplitude–frequency graph, marking the peak at the natural frequency.
• 💡Link each statement to a familiar example so the qualitative explanation is clear.
• 💡Sketch two resonance curves on the same axes, one lightly damped and one heavily damped, and label peak height and width clearly.
• 💡Use the phrase sharpness of resonance only after defining what is being compared, such as the width of the peak at a stated amplitude level.
• 💡When explaining energy transfer, refer to the driver doing work on the oscillating system each cycle rather than saying energy is created.
• 💡Link any practical example back to the driving frequency and the natural frequency of the system, not just to large vibrations.
• 💡For each example, state the driver, the oscillating object, and why the driving frequency is near a natural frequency.
• 💡When discussing stationary waves, sketch the mode shape and mark nodes and antinodes before doing any calculation.
• 💡Use the same axes for comparing damped and undamped cases so the change in peak height and width is visible.
• 💡Keep explanations of damping tied to energy removal rather than to vague ideas of friction slowing everything down.
• 💡Always specify that in forced vibrations, the frequency of oscillation equals the driving frequency, not the natural frequency. This is a common mark lost in definitions.
• 💡When describing resonance, mention that the amplitude is maximum and that this occurs when driving frequency equals natural frequency. Include the effect of damping on the resonance peak.
• 💡For calculation questions, ensure you use the correct formula for amplitude at resonance, such as A = F0 / (m * gamma * omega0) or A = 2Q F0 / (m omega0^2), and show all steps clearly with units.

• Saying that a forced oscillator always vibrates at its natural frequency; correct by stating that it vibrates at the driving frequency, with maximum amplitude when this equals the natural frequency.
• Confusing resonance with any large vibration; correct by linking resonance specifically to driving frequency matching natural frequency.
• Ignoring damping when describing the resonance peak; correct by explaining that damping lowers and broadens the peak.
• Thinking resonance creates energy: correct this by stating that the driver supplies the energy and resonance maximises the energy transferred per cycle.
• Confusing amplitude with frequency: correct this by keeping the axes distinct, with driving frequency on the horizontal axis and steady-state amplitude on the vertical axis.
• Assuming damping always destroys resonance: correct this by explaining that damping lowers and broadens the peak but the driven system still shows a maximum near the natural frequency.
• Treating the natural frequency as changing with damping: correct this by noting that damping mainly changes peak height and width, while the natural frequency is set by the system's inertia and stiffness.
• Describing a stationary wave as travelling along the string: correct this by stating that the wave appears to stand still because incident and reflected waves superpose.
• Ignoring boundary conditions in tubes: correct this by identifying nodes and antinodes at closed and open ends before choosing the resonant frequencies.
• Claiming damping changes the natural frequency dramatically: correct this by explaining that damping mainly reduces and broadens the resonance peak.
• Using a mechanical example without naming the driver: correct this by stating what supplies the periodic driving force, such as a motor or a vibrating support.
• Misconception: At resonance, the amplitude becomes infinite. Correction: In reality, damping always limits the amplitude to a finite maximum. Without damping, the amplitude would grow without bound, but some damping is always present.
• Misconception: The system vibrates at its natural frequency during forced vibrations. Correction: The system vibrates at the driving frequency. Only at resonance do the driving and natural frequencies coincide.
• Misconception: Resonance always leads to large amplitudes. Correction: The amplitude at resonance depends on the damping; heavy damping can significantly reduce the maximum amplitude.

1. 1Day 1-2: Review the basics of simple harmonic motion and damping. Ensure you can define natural frequency and understand energy transformations in oscillating systems.
2. 2Day 3-4: Study forced vibrations and resonance. Learn the definitions, the conditions for resonance, and how damping affects the resonance curve. Practice sketching amplitude-frequency graphs for different damping levels.
3. 3Day 5-6: Work through calculations involving resonance, such as finding maximum amplitude using quality factor or damping coefficient. Solve past paper questions on this topic.
4. 4Day 7-8: Focus on applications and examples of resonance (e.g., Tacoma Narrows Bridge, microwave oven, musical instruments) and practice explaining them in terms of energy transfer.
5. 5Day 9-10: Complete a full topic test under timed conditions. Review any mistakes and revisit weak areas. Use flashcards for key definitions and formulas.

• 📋Definition and explanation questions: e.g., 'Define resonance and explain why the amplitude is maximum at resonance.' Advice: Use precise terminology and refer to energy transfer and phase relationship.
• 📋Graph sketching and interpretation: e.g., 'Sketch the amplitude-frequency graph for a forced oscillator with light and heavy damping.' Advice: Label axes correctly, show resonance peak shifting and broadening with increased damping.
• 📋Calculation questions: e.g., 'Calculate the maximum amplitude of a driven oscillator at resonance given the driving force, mass, and quality factor.' Advice: Use the correct formula, show substitution, and give units.
• 📋Practical application questions: e.g., 'Explain how resonance is used in a microwave oven or why it is dangerous for bridges.' Advice: Link the phenomenon to the specific context, mentioning energy transfer and damping.