Section 6.5: Pendulums
A pendulum is another classic system that exhibits simple harmonic motion (SHM) under small-angle approximation. It consists of a mass (called a bob) attached to a string or rod, swinging about a pivot point.
Key Equations:
- Restoring force (small angle): \( F \approx -mg\theta \)
- Angular frequency: \( \omega = \sqrt{\tfrac{g}{L}} \)
- Period: \( T = 2\pi \sqrt{\tfrac{L}{g}} \)
- Valid for \( \theta \lesssim 15^\circ \)
Example 1
A pendulum of length \( L = 1.0 \, \text{m} \) swings with small oscillations. Find its period.
\( T = 2\pi \sqrt{\tfrac{L}{g}} = 2\pi \sqrt{\tfrac{1.0}{9.8}} \approx 2.01 \, \text{s} \)
Practice Problems
- A pendulum of length 2.25 m is set in motion. Find its period.
- Find the frequency of oscillation for a pendulum of length 0.5 m.
- A pendulum clock is designed for Earth’s gravity. How would its period change on the Moon (\( g = 1.6 \, \text{m/s}^2 \))?
- If a pendulum makes 30 oscillations in 60 s, what is its length?
- Explain why the small-angle approximation is necessary for SHM in pendulums.