Edexcel A Level (IAL) Physics-5.20 Equations for Simple Harmonic Motion- Study Notes- New Syllabus
Edexcel A Level (IAL) Physics -5.20 Equations for Simple Harmonic Motion- Study Notes- New syllabus
Edexcel A Level (IAL) Physics -5.20 Equations for Simple Harmonic Motion- Study Notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- be able to use the equations a = –ω²x, x = A cos ωt, v = –A ω sin ωt, a = –A ω² cos ωt, and T = 1/f = 2π/ω and ω = 2πf as applied to a simple harmonic oscillator
Equations of Simple Harmonic Motion and Their Application
In simple harmonic motion (SHM), the displacement, velocity and acceleration of a particle vary sinusoidally with time. These quantities are linked through the angular frequency \( \omega \).
Acceleration–Displacement Relationship
The defining equation of SHM is:
\( a = -\omega^2 x \)
- \( a \) = acceleration (m s⁻²)
- \( x \) = displacement from equilibrium (m)
- \( \omega \) = angular frequency (rad s⁻¹)
Meaning:
- Acceleration is proportional to displacement.
- Acceleration is always directed towards equilibrium.
- The negative sign confirms SHM.
Displacement Equation
The displacement of a particle in SHM is given by:
\( x = A \sin \omega t \)
- \( A \) = amplitude (maximum displacement)
- \( t \) = time (s)
Key points:
- Displacement varies sinusoidally.
- At \( t = 0 \), displacement is maximum.
Velocity Equation
Velocity is the rate of change of displacement:
\( v = A\omega \cos \omega t \)
- Velocity is zero at maximum displacement.
- Velocity is maximum at equilibrium.
- Velocity is \( 90^\circ \) out of phase with displacement.
Maximum velocity:
\( v_{\text{max}} = A\omega \)
Acceleration Equation
Acceleration is the rate of change of velocity:
\( a = -A\omega^2 \sin \omega t \)
- Acceleration is maximum at extreme positions.
- Acceleration is zero at equilibrium.
- Acceleration is always opposite in direction to displacement.
Maximum acceleration:
\( a_{\text{max}} = A\omega^2 \)
Period, Frequency and Angular Frequency
The time period \( T \) and frequency \( f \) are related by:
\( T = \dfrac{1}{f} \)
The angular frequency is defined as:
\( \omega = 2\pi f \)
Hence:
\( T = \dfrac{2\pi}{\omega} \)
- \( T \) = period (s)
- \( f \) = frequency (Hz)
- \( \omega \) = angular frequency (rad s⁻¹)
Phase Relationships in SHM
- Velocity leads displacement by \( 90^\circ \).
- Acceleration leads velocity by \( 90^\circ \).
- Acceleration is \( 180^\circ \) out of phase with displacement.
Example (Easy)
A particle performs SHM with amplitude \( 0.20\,\mathrm{m} \) and angular frequency \( 5\,\mathrm{rad\,s^{-1}} \). Find its maximum speed.
▶️ Answer / Explanation
\( v_{\text{max}} = A\omega = 0.20 \times 5 = 1.0\,\mathrm{m\,s^{-1}} \)
Example (Medium)
An oscillator has frequency \( 2.0\,\mathrm{Hz} \). Calculate its angular frequency and period.
▶️ Answer / Explanation
\( \omega = 2\pi f = 4\pi \approx 12.6\,\mathrm{rad\,s^{-1}} \)
\( T = \dfrac{1}{f} = 0.50\,\mathrm{s} \)
Example (Hard)
At a certain time, the displacement of a particle in SHM is zero. Describe the velocity and acceleration at this instant.
▶️ Answer / Explanation
- Velocity is maximum.
- Acceleration is zero.
- The particle is passing through equilibrium.