Periodic Motion: SHM, Pendulums, Springs & Waves

TAKEAWAYS:

• SHM is a special kind of periodic motion where acceleration is proportional to displacement and directed toward equilibrium.
• Amplitude, period, and frequency define the size and speed of oscillations.
• Energy in SHM is conserved and switches between kinetic and potential forms.
• Pendulums and springs follow SHM under ideal conditions, but real systems have damping and resonance.
• Damped oscillations lose energy over time, while forced oscillations can be amplified near resonance.
• NEB exams test definitions, equations, graphs, and real-world applications (e.g., clocks, bridges, musical instruments).

What is Periodic Motion?

Periodic motion is any motion that repeats itself at regular time intervals. Examples include:

• A swinging pendulum (clock).
• A vibrating guitar string.
• The Earth’s orbit around the Sun.
• A bouncing spring.

Key terms:

• Period (T): Time taken to complete one full cycle (unit: seconds).
• Frequency (f): Number of cycles per second (unit: Hertz, Hz). $f=\frac{1}{T}$.
• Amplitude (A): Maximum displacement from equilibrium.

graph LR
    A["Periodic Motion"] --> B["Repeats at regular intervals"]
    A --> C["Examples: Pendulum, Spring, Waves"]
    B --> D["Period (T) = Time for one cycle"]
    B --> E["Frequency (f) = 1/T"]

Simple Harmonic Motion (SHM)

SHM is a special type of periodic motion where:

1. The restoring force is directly proportional to displacement and acts toward equilibrium.
2. The acceleration is constant in magnitude but changes direction.

Mathematical Definition: If $x$ is displacement from equilibrium, then: $$F=-kx$$ $$a=-{\omega }^{2}x$$ where:

• $k$ = spring constant (for springs) or $mg/L$ (for pendulums).
• $\omega$ = angular frequency ($\omega =2\pi f$).

Equation of SHM: $$x(t)=A\cos (\omega t+\varphi )$$ where:

• $A$ = amplitude,
• $\omega$ = angular frequency,
• $\varphi$ = phase angle.

Velocity and Acceleration in SHM

Differentiate $x(t)$ to find velocity ($v$) and acceleration ($a$): $$v(t)=\frac{dx}{dt}=-A\omega \sin (\omega t+\varphi )$$ $$a(t)=\frac{dv}{dt}=-A{\omega }^2\cos (\omega t+\varphi )=-{\omega }^{2}x$$

Key Observations:

• Maximum velocity occurs at equilibrium ($x=0$): $v_{max}=A\omega$.
• Maximum acceleration occurs at extreme positions ($x=\pm A$): $a_{max}=A{\omega }^2$.

Energy in SHM

Total energy ($E$) is constant and switches between:

1. Kinetic Energy (KE): Maximum at equilibrium. $$KE=\frac{1}{2}m{v}^2=\frac{1}{2}m{\omega }^2(A^2-x^2)$$
2. Potential Energy (PE): Maximum at extreme positions. $$PE=\frac{1}{2}k{x}^2$$

At any point: $$E=KE+PE=\frac{1}{2}k{A}^2$$

Graph of Energy vs. Displacement:

graph TD
    A["Total Energy (E)"] --> B["Constant"]
    B --> C["KE = 1/2 mω²(A² - x²)"]
    B --> D["PE = 1/2 kx²"]
    C --> E["Max at x=0"]
    D --> F["Max at x=±A"]

Examples of SHM

1. Mass-Spring System

A block attached to a spring oscillates when displaced.

• Restoring force: $F=-kx$.
• Angular frequency: $\omega =\sqrt{\frac{k}{m}}$.
• Period: $T=2\pi \sqrt{\frac{m}{k}}$.

Example: A 2 kg mass is attached to a spring with $k=100\text{N/m}$. Find:

1. Angular frequency ($\omega$).
2. Period ($T$).
3. Maximum velocity if amplitude is 0.1 m.

Solution:

1. $\omega =\sqrt{\frac{k}{m}}=\sqrt{\frac{100}{2}}=7.07\text{rad/s}$.
2. $T=\frac{2\pi }{\omega }=\frac{6.28}{7.07}\approx 0.89\text{s}$.
3. $v_{max}=A\omega =0.1\times 7.07=0.707\text{m/s}$.

2. Simple Pendulum

A small mass (bob) suspended by a string/rod.

• Restoring force: $F=-mg\sin \theta \approx -mg\theta$ (for small angles, $\theta$ in radians).
• Angular frequency: $\omega =\sqrt{\frac{g}{L}}$.
• Period: $T=2\pi \sqrt{\frac{L}{g}}$.

Example: A pendulum has length $L=1\text{m}$. Find its period.

Solution: $$T=2\pi \sqrt{\frac{1}{9.8}}\approx 2.0\text{s}$$

Note: For large angles, the period increases slightly (non-linear SHM).

Comparison: Spring vs. Pendulum

Damped and Forced Oscillations

1. Damped Oscillations

Real systems lose energy due to friction/air resistance, causing oscillations to die out.

• Under-damped: Oscillations decrease slowly (e.g., swinging door).
• Over-damped: No oscillations (e.g., car shock absorbers).
• Critically damped: Fastest return to equilibrium without oscillation.

Equation of damped SHM: $$x(t)=A{e}^{-\gamma t}\cos ({\omega }^{\prime }t+\varphi )$$ where:

• $\gamma$ = damping coefficient,
• ${\omega }^{\prime }=\sqrt{{\omega }^2-{\gamma }^2}$.

Example: A damped oscillator has $\omega =10\text{rad/s}$ and $\gamma =2\text{rad/s}$. Find the new angular frequency (${\omega }^{\prime }$).

Solution: $${\omega }^{\prime }=\sqrt{10^2-2^2}=\sqrt{96}\approx 9.8\text{rad/s}$$

2. Forced Oscillations and Resonance

When an external periodic force is applied, the system oscillates at the driver’s frequency.

• Resonance: Occurs when the driver’s frequency matches the natural frequency ($\omega ={\omega }_0$).
  • Amplitude becomes very large (can cause damage).
  • Example: Bridge collapse (Tacoma Narrows Bridge, 1940).

Applications of Resonance:

• Musical instruments (guitars, pianos).
• Radio tuning (LC circuits).
• MRI machines (nuclear magnetic resonance).

Graph of Amplitude vs. Frequency:

graph TD
    A["Amplitude"] --> B["Increases with frequency"]
    B --> C["Peaks at resonance (ω = ω₀)"]
    B --> D["Drops off at high/low frequencies"]

NEB-Style Questions and Solutions

Short Answer Questions

1. Define simple harmonic motion (SHM). Answer: SHM is a type of periodic motion where the restoring force is directly proportional to displacement and acts toward equilibrium.

2. What is the period of a simple pendulum? Derive it. Answer: For small angles, $T=2\pi \sqrt{\frac{L}{g}}$. Derivation:

   • Restoring force: $F=-mg\sin \theta \approx -mg\theta$.
   • Torque: $\tau =-mgL\sin \theta \approx -mgL\theta$.
   • Angular acceleration: $\alpha =\frac{\tau }{I}=\frac{-mgL\theta }{m{L}^2}=-\frac{g}{L}\theta$.
   • Compare with SHM: $\alpha =-{\omega }^2\theta$, so $\omega =\sqrt{\frac{g}{L}}$.
   • Period: $T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{L}{g}}$.

Numerical Problems

3. A mass-spring system has $k=50\text{N/m}$ and $m=0.5\text{kg}$. If amplitude is 0.2 m, find: a) Angular frequency. b) Maximum velocity. c) Total energy.

   Solution: a) $\omega =\sqrt{\frac{k}{m}}=\sqrt{\frac{50}{0.5}}=10\text{rad/s}$. b) $v_{max}=A\omega =0.2\times 10=2\text{m/s}$. c) $E=\frac{1}{2}k{A}^2=\frac{1}{2}\times 50\times (0.2{)}^2=1\text{J}$.

Conceptual Questions

4. Why does a pendulum clock keep time accurately? Answer: The period of a simple pendulum ($T=2\pi \sqrt{L/g}$) is independent of amplitude (for small angles) and depends only on length ($L$) and gravity ($g$). Thus, it provides a consistent time interval.

5. What happens to the period of a spring-mass system if the mass is doubled? Answer: The period increases by a factor of $\sqrt{2}$, because $T\propto \sqrt{m}$.

Exam Tip: How to Score Full Marks

1. Understand Definitions:

   • Know the difference between periodic motion, SHM, and damped/forced oscillations.
   • Memorize key equations (e.g., $T=2\pi \sqrt{m/k}$ for springs, $T=2\pi \sqrt{L/g}$ for pendulums).

2. Graphs are Important:

   • Draw displacement-time, velocity-time, and acceleration-time graphs for SHM.
   • Label axes, amplitude, period, and equilibrium correctly.

3. Units Matter:

   • Always include units (e.g., $\text{rad/s}$, $\text{s}$, $\text{m/s}$).
   • NEB deducts marks for missing units!

4. Real-World Applications:

   • Relate SHM to clocks, seismometers, musical instruments, and resonance disasters (e.g., bridges).
   • Example: "A metronome uses a pendulum because its period is independent of amplitude."

5. Damping and Resonance:

   • Explain why damping reduces amplitude and how resonance causes large amplitudes.
   • Example: "A radio tuner uses resonance to select specific frequencies."

6. Common Mistakes to Avoid:

   • Assuming SHM for large angles (pendulum period changes).
   • Forgetting the negative sign in $F=-kx$.
   • Mixing up angular frequency ($\omega$) and frequency ($f$).

Summary Table for Quick Revision

Final Advice:

• Practice deriving equations (e.g., pendulum period).
• Solve numerical problems from past NEB papers.
• Visualize SHM using graphs and animations (YouTube helps!).
• Relate theory to real life—NEB loves applications!

A block-spring system demonstrating SHM with displacement, velocity, and acceleration graphs. (Image: Mazemaster, Public domain, via Wikimedia Commons) A pendulum bob swinging with small angles, showing restoring force and equilibrium. (Image: Chetvorno, Public domain, via Wikimedia Commons)