Section 8.1: Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.
\[
F = -k x
\]
where:
\( F \) = restoring force (N)
\( k \) = force constant (N/m)
\( x \) = displacement from equilibrium (m)
\( F \) = restoring force (N)
\( k \) = force constant (N/m)
\( x \) = displacement from equilibrium (m)
Key quantities in SHM:
- Angular frequency: \( \omega = \sqrt{\frac{k}{m}} \) (rad/s)
- Period: \( T = \frac{2\pi}{\omega} \) (s)
- Frequency: \( f = \frac{1}{T} \) (Hz)
- Displacement as a function of time: \( x(t) = A \cos(\omega t + \phi) \)
Example: Mass-Spring System
A 0.5 kg mass is attached to a spring with \( k = 200 \, \text{N/m} \). Find the period and frequency of oscillation.
Angular frequency: \( \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{200}{0.5}} = 20 \, \text{rad/s} \)
Period: \( T = \frac{2\pi}{\omega} = \frac{2\pi}{20} \approx 0.314 \, \text{s} \)
Frequency: \( f = \frac{1}{T} \approx 3.18 \, \text{Hz} \)
Practice Problems
- A mass of 2 kg is attached to a spring with \( k = 50 \, \text{N/m} \). Find the period and frequency of oscillation.
- A mass-spring system oscillates with amplitude 0.1 m and angular frequency 10 rad/s. Find the maximum speed.
- For a 0.3 kg mass on a spring with period 0.5 s, calculate the spring constant.
- Write the equation of motion for a SHM system with amplitude 0.2 m and phase angle 0, angular frequency 15 rad/s.
- A spring stretches 0.05 m under a force of 25 N. Find the angular frequency for a 1 kg mass attached to this spring.