understand that simple harmonic motion occurs when acceleration is proportional to displacement from a fixed point and in the opposite direction
Simple Harmonic Motion (SHM) – Syllabus 17 (Oscillations)
Learning Objective (AO1 & AO2)
Understand that SHM occurs when the acceleration of a particle is directly proportional to its displacement from a fixed equilibrium point and acts in the opposite direction. Be able to:
- derive and manipulate the governing equations,
- interpret displacement‑time, velocity‑time and acceleration‑time graphs,
- analyse energy transformations, and
- apply the concepts to mass‑spring systems, simple pendulums, damped and forced oscillators (including resonance).
1. Fundamental Definitions
| Quantity | Symbol | Definition |
|---|---|---|
| Displacement from equilibrium | \(x\) | Signed distance from the fixed point (positive in the chosen direction). |
| Amplitude | \(A\) | Maximum magnitude of the displacement; \(|x|\le A\). |
| Angular frequency | \(\omega\) | Rate of oscillation in rad s\(^{-1}\); \(\omega = 2\pi f = 2\pi/T\). |
| Period | \(T\) | Time for one complete oscillation; \(T = 2\pi/\omega\). |
| Frequency | \(f\) | Number of oscillations per second; \(f = 1/T\). |
| Phase constant | \(\phi\) | Sets the starting point of the motion; fixed by the initial displacement and velocity. |
2. The Defining Condition for SHM
The motion is simple harmonic when the acceleration satisfies
a = -\omega^{2}x
\]
i.e. acceleration is proportional to displacement and directed towards the equilibrium point (restoring direction).
3. Equations of Motion (AO2)
Integrating the defining condition twice gives the standard sinusoidal solutions:
- Displacement: \(x(t)=A\cos(\omega t+\phi)\)
- Velocity: \(v(t)=\frac{dx}{dt}= -A\omega\sin(\omega t+\phi)\)
- Acceleration: \(a(t)=\frac{d^{2}x}{dt^{2}}= -A\omega^{2}\cos(\omega t+\phi)= -\omega^{2}x(t)\)
4. Phase Relationships
- Displacement and velocity are out of phase by \(\pi/2\) (90°); velocity leads displacement.
- Velocity and acceleration are also out of phase by \(\pi/2\); acceleration leads velocity (or lags displacement by another \(\pi/2\)).
5. Graphical Representation (AO2)
Key features to read from the graphs:
- Amplitude: vertical distance from the centre line ( \(A\) for \(x\), \(A\omega\) for \(v\), \(A\omega^{2}\) for \(a\) ).
- Period \(T\): horizontal distance between successive peaks of any curve.
- Phase difference: horizontal offset between corresponding points (e.g. a peak of \(x\) aligns with a zero‑crossing of \(v\)).
6. Derivation from Hooke’s Law – Mass‑Spring System (AO1 & AO2)
Hooke’s law: \(F = -k\,x\). Using Newton’s second law \(F = m a\):
m a = -k x \;\;\Longrightarrow\;\; a = -\frac{k}{m}x
\]
Comparing with the SHM condition gives \(\omega^{2}=k/m\) and therefore
\boxed{\;\omega = \sqrt{\frac{k}{m}}\;},\qquad
\boxed{\;T = 2\pi\sqrt{\frac{m}{k}}\;},\qquad
\boxed{\;f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\;}
\]
7. Simple Pendulum – Small‑Angle Approximation (AO1 & AO2)
For a pendulum of length \(L\) and bob mass \(m\):
\tau = -mgL\sin\theta
\]
When \(\theta\) is small (\(<\!10^{\circ}\)), \(\sin\theta\approx\theta\) (radians) and the equation of motion becomes
\frac{d^{2}\theta}{dt^{2}} + \frac{g}{L}\,\theta = 0
\]
Thus the pendulum executes SHM with
\boxed{\;\omega = \sqrt{\frac{g}{L}}\;},\qquad
\boxed{\;T = 2\pi\sqrt{\frac{L}{g}}\;}
\]
8. Energy in SHM (AO2)
- Kinetic Energy: \(K = \tfrac12 m v^{2}= \tfrac12 m\omega^{2}A^{2}\sin^{2}(\omega t+\phi)\)
- Potential Energy (elastic or gravitational): \(U = \tfrac12 k x^{2}= \tfrac12 kA^{2}\cos^{2}(\omega t+\phi)\)
- Total Mechanical Energy: \(E = K+U = \tfrac12 kA^{2}= \tfrac12 m\omega^{2}A^{2}\) (constant for an ideal undamped system).
Energy exchange:
- At \(x=\pm A\) (extremes) the energy is purely potential.
- At \(x=0\) (equilibrium) the energy is purely kinetic.
9. Damped Oscillations (17.3 – Damping) (AO1 & AO2)
A resistive force proportional to velocity is introduced:
F_{d} = -b\,v \qquad (b>0)
\]
The equation of motion becomes
m\frac{d^{2}x}{dt^{2}} + b\frac{dx}{dt} + kx = 0
\]
Defining the damping coefficient \(\gamma = b/(2m)\) and the natural angular frequency \(\omega_{0}= \sqrt{k/m}\), the solution is
x(t)=A\,e^{-\gamma t}\cos(\omega' t+\phi)
\]
where \(\displaystyle \omega' = \sqrt{\omega_{0}^{2}-\gamma^{2}}\) (underdamped case). Three regimes are distinguished:
| Regime | Condition | Behaviour of \(x(t)\) |
|---|---|---|
| Underdamped | \(\gamma < \omega_{0}\) | Oscillatory with exponentially decreasing amplitude. |
| Critically damped | \(\gamma = \omega_{0}\) | Returns to equilibrium as quickly as possible without oscillation. |
| Overdamped | \(\gamma > \omega_{0}\) | Returns to equilibrium slowly, no oscillation. |
10. Forced Oscillations and Resonance (17.3 – Forced Motion) (AO1 & AO2)
If an external periodic force \(F(t)=F{0}\cos(\omega{d}t)\) acts on the oscillator, the equation of motion is
m\frac{d^{2}x}{dt^{2}} + b\frac{dx}{dt} + kx = F{0}\cos(\omega{d}t)
\]
In the steady‑state (after transients die out) the solution is
x(t)=X\cos(\omega_{d}t-\delta)
\]
with amplitude
X = \frac{F{0}}{\sqrt{(k-m\omega{d}^{2})^{2}+(b\omega_{d})^{2}}}
\]
and phase lag \(\displaystyle \tan\delta = \frac{b\omega{d}}{k-m\omega{d}^{2}}\).
Resonance occurs when the driving frequency \(\omega{d}\) is close to the natural frequency \(\omega{0}\) and the damping is small. The maximum amplitude is
X{\max} \approx \frac{F{0}}{2b\omega{0}} \quad (\text{for } b\ll m\omega{0})
\]
The quality factor \(Q\) quantifies the sharpness of resonance:
Q = \frac{\omega{0}}{2\gamma} = \frac{m\omega{0}}{b}
\]
11. Conditions for Simple Harmonic Motion (AO1)
- The restoring force (or torque) is directly proportional to the displacement (Hooke’s law or its angular analogue).
- The motion is one‑dimensional (or rotational about a fixed axis) with a fixed equilibrium position.
- No non‑conservative forces remove energy – the ideal system is undamped. (Real systems may be approximated as SHM if damping is weak.)
- For pendulums, the angular displacement must be small enough that \(\sin\theta\approx\theta\).
12. Summary Table of Key Quantities (AO2)
| Quantity | Symbol | Expression | Units |
|---|---|---|---|
| Angular frequency (mass‑spring) | \(\omega\) | \(\sqrt{k/m}\) | rad s\(^{-1}\) |
| Angular frequency (pendulum) | \(\omega\) | \(\sqrt{g/L}\) | rad s\(^{-1}\) |
| Period | \(T\) | \(2\pi/\omega\) | s |
| Frequency | \(f\) | \(1/T\) | Hz |
| Maximum speed | \(v_{\max}\) | \(A\omega\) | m s\(^{-1}\) |
| Maximum acceleration | \(a_{\max}\) | \(A\omega^{2}\) | m s\(^{-2}\) |
| Total mechanical energy | \(E\) | \(\tfrac12 kA^{2}= \tfrac12 m\omega^{2}A^{2}\) | J |
| Damping coefficient | \(\gamma\) | \(b/(2m)\) | s\(^{-1}\) |
| Quality factor | \(Q\) | \(\omega{0}/(2\gamma)=m\omega{0}/b\) | – |
13. Practical Skills (AO3 & AO4)
- Using a motion sensor (or video analysis) to record \(x(t)\) for a mass‑spring or pendulum system.
- Extracting period \(T\) from successive peaks and calculating \(\omega\) and \(k\) (or \(g\)) via the formulas above.
- Introducing damping (e.g., a dash‑pot) and measuring the decay constant \(\gamma\) from an exponential fit to the envelope of the \(x\)‑\(t\) curve.
- Driving a system with a sinusoidal force (e.g., a mechanical vibrator) and constructing a resonance curve by varying the driving frequency.
- Evaluating uncertainties and discussing the limits of the small‑angle approximation.
14. Quick‑Check Questions (AO2 & AO3)
- Derive the period formula \(T = 2\pi\sqrt{m/k}\) for a mass‑spring system starting from \(F = -kx\) and Newton’s second law.
- Explain why a simple pendulum follows SHM only for small angular displacements, and show how \(\sin\theta\approx\theta\) leads to \(\omega = \sqrt{g/L}\).
- A 0.5 kg mass is attached to a spring with \(k = 200\ \text{N m}^{-1}\). If the amplitude is 0.10 m, calculate:
- \(\omega\),
- the period \(T\),
- the maximum speed \(v_{\max}\).
- For a damped oscillator with \(m=0.2\) kg, \(k=50\) N m\(^{-1}\) and damping constant \(b=0.5\) kg s\(^{-1}\):
- Determine whether the motion is under‑, critically‑ or over‑damped.
- Calculate the damped angular frequency \(\omega'\) (if under‑damped).
- A forced oscillator is driven at \(\omega{d}=0.98\omega{0}\) with a small damping \(b=0.05\) kg s\(^{-1}\). Sketch the expected amplitude‑frequency curve and indicate the resonance peak and the quality factor \(Q\).
15. Suggested Diagrams (Place‑holders)
- Mass‑spring illustration – shows block, spring constant \(k\), displacement \(x\), restoring force \(F=-kx\) and acceleration \(a=-\omega^{2}x\).
- Simple pendulum – bob of mass \(m\), length \(L\), angle \(\theta\) and torque \(\tau=-mgL\theta\).
- Damping schematic – dash‑pot attached to a mass‑spring system.
- Forced‑oscillation set‑up – driver applying a sinusoidal force to a mass‑spring.
16. Further Reading (AO4)
Cambridge International AS & A Level Physics (9702) – Chapter 17 (Oscillations). Past paper questions (Paper 1, Paper 2, Paper 5) provide practice on derivations, graph interpretation, energy analysis and experimental design for damped and forced oscillations.