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Simple Harmonic Motion (SHM): A-Level Science Guide with Examples
Simple Harmonic Motion: A Complete Guide for A-Level Science
What Is Simple Harmonic Motion (SHM)?
Simple Harmonic Motion (SHM) describes periodic oscillatory motion where the restoring force is proportional to displacement and directed towards equilibrium.
Key Characteristics of SHM
- Motion repeats over time (periodic).
- Restoring force is proportional to displacement: \( F = -kx \)
Key Equations for SHM
Displacement (\( x \))
Displacement from the equilibrium position at time \( t \):
\[ x = A \cos(\omega t + \phi) \]
Where:
- \( A \): Amplitude.
- \( \omega \): Angular frequency (\( \omega = \frac{2\pi}{T} \)).
- \( \phi \): Phase constant.
Velocity (\( v \))
Velocity changes with displacement:
\[ v = -\omega A \sin(\omega t + \phi) \]
Acceleration (\( a \))
Acceleration is proportional to displacement:
\[ a = -\omega^2 x \]
Energy in SHM
Kinetic Energy (\( KE \))
\[ KE = \frac{1}{2} m \omega^2 (A^2 – x^2) \]
Potential Energy (\( PE \))
\[ PE = \frac{1}{2} m \omega^2 x^2 \]
Example Problem
A \( 2 \, \text{kg} \) mass oscillates on a spring with a spring constant of \( 50 \, \text{N/m} \) and an amplitude of \( 0.1 \, \text{m} \). Find the period and maximum velocity.
- Angular Frequency (\( \omega \)): \( \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{2}} = 5 \, \text{rad/s} \)
- Period (\( T \)): \( T = \frac{2\pi}{\omega} = \frac{2\pi}{5} \approx 1.26 \, \text{s} \)
- Maximum Velocity (\( v_{\text{max}} \)): \( v_{\text{max}} = \omega A = 5 \times 0.1 = 0.5 \, \text{m/s} \)
Applications of SHM
- Clocks: Pendulum motion in grandfather clocks.
- Engineering: Suspension systems in vehicles.
- Seismology: Understanding Earth’s oscillations during earthquakes.
Common Mistakes in SHM Calculations
- Forgetting the negative sign in the restoring force equation.
- Mixing up displacement and amplitude.
- Using incorrect units for angular frequency.
Practice Questions
- A mass oscillates with \( A = 0.2 \, \text{m} \) and \( \omega = 10 \, \text{rad/s} \). Find the maximum acceleration.
- Calculate the energy stored in a spring with \( k = 100 \, \text{N/m} \) and \( A = 0.05 \, \text{m} \).
Explain why the total energy in SHM is conserved.
