Section 3.8: Problem-Solving Strategies
Oscillations and waves often require multi-step reasoning. The following strategies help systematically tackle problems in this topic.
Problem-Solving Approach:
- Identify the type of oscillation (simple harmonic, damped, forced).
- Write down the governing equations, e.g., \( x = A \cos(\omega t + \phi) \), \( a = -\omega^2 x \).
- Determine known and unknown quantities.
- Use energy methods if applicable: \( E = \frac{1}{2} k A^2 \), \( E = \frac{1}{2} m v^2 + \frac{1}{2} k x^2 \).
- Check boundary conditions and initial conditions.
- For waves: identify frequency, wavelength, speed relation \( v = f \lambda \).
- Apply superposition principle if multiple waves interact.
- Verify units and reasonability of answers.
Example 1
A mass-spring system has amplitude 0.1 m and frequency 2 Hz. Find the maximum velocity of the mass.
Maximum velocity: \( v_\text{max} = \omega A = 2\pi f A = 2\pi (2)(0.1) \approx 1.257 \text{ m/s} \)
Example 2
A wave has wavelength 0.5 m and speed 2 m/s. Find the period and frequency.
Frequency: \( f = v / \lambda = 2 / 0.5 = 4 \text{ Hz} \)
Period: \( T = 1/f = 1/4 = 0.25 \text{ s} \)
Practice Problems
- A pendulum of length 1 m has small oscillations. Determine its period.
- A mass-spring system has k = 200 N/m and m = 2 kg. Find its angular frequency and period.
- Two waves of same amplitude and frequency interfere. Determine resultant amplitude at constructive and destructive interference.
- A wave travels 30 m in 6 s. Find its speed, frequency, and wavelength if \( f = 5 \) Hz.
- A damped oscillator loses 10% of its energy each cycle. Compute energy after 3 cycles if initial energy is 5 J.