Section 3.1: Simple Harmonic Motion Review
Simple harmonic motion (SHM) occurs when the restoring force is directly proportional to the displacement from equilibrium. It is characterized by oscillatory motion.
Key Equations:
- Displacement: \( x(t) = A \cos(\omega t + \phi) \)
- Velocity: \( v(t) = -\omega A \sin(\omega t + \phi) \)
- Acceleration: \( a(t) = -\omega^2 x(t) \)
- Angular frequency: \( \omega = 2\pi f = \sqrt{k/m} \)
- Period: \( T = 2\pi/\omega \), Frequency: \( f = 1/T \)
Example 1
A mass-spring system with mass 0.5 kg and spring constant 200 N/m is displaced by 0.1 m. Find the period and maximum velocity.
Angular frequency: \( \omega = \sqrt{k/m} = \sqrt{200/0.5} = \sqrt{400} = 20 \text{ rad/s} \)
Period: \( T = 2\pi/\omega = 2\pi/20 \approx 0.314 \text{ s} \)
Maximum velocity: \( v_\text{max} = \omega A = 20 * 0.1 = 2 \text{ m/s} \)
Practice Problems
- A 0.2 kg mass oscillates on a spring with k = 50 N/m. Find its period and frequency.
- For a SHM with amplitude 0.15 m and angular frequency 10 rad/s, find maximum velocity and acceleration.
- A pendulum of length 1 m is displaced by 5°. Compute its period.
- Determine the displacement after 0.25 s if a 0.3 kg mass-spring system with ω = 15 rad/s starts from rest at maximum displacement 0.1 m.
- A mass-spring system oscillates with 3 Hz. Find angular frequency and period.