Section 6.1: Simple Harmonic Motion Basics
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction:
SHM Definition:
\( F = -kx \) or \( a = -\omega^2 x \)
Key characteristics:
- Displacement \(x\) oscillates about equilibrium.
- Acceleration \(a\) is always directed toward equilibrium.
- Period \(T\) and frequency \(f\) define time characteristics: \( T = \frac{2\pi}{\omega}, f = \frac{1}{T} \)
Example 1
A mass-spring system with \(k = 50 \text{ N/m}\) and \(m = 2 \text{ kg}\) is displaced 0.1 m from equilibrium. Find the angular frequency, period, and maximum acceleration.
\( \omega = \sqrt{k/m} = \sqrt{50/2} = 5 \text{ rad/s} \)
\( T = 2\pi/\omega = 2\pi/5 \approx 1.26 \text{ s} \)
\( a_\text{max} = \omega^2 x_\text{max} = 5^2 \cdot 0.1 = 2.5 \text{ m/s²} \)
Practice Problems
- A 0.5 kg mass oscillates on a spring with \(k = 20 \text{ N/m}\). Find the period and frequency.
- A spring stretches 0.2 m under a 10 N force. Find the angular frequency for 0.5 kg mass.
- A particle moves in SHM: \(x = 0.1 \cos(4t)\). Find maximum velocity and acceleration.
- Determine the period of a 3 kg mass with \(k = 75 \text{ N/m}\).
- A mass on a spring has maximum speed 2 m/s at amplitude 0.1 m. Find angular frequency.