Section 1.7: Oscillations (Mechanics)

This section introduces simple harmonic motion (SHM), its equations, energy, and characteristics in mechanical systems such as springs and pendulums.

Key Equations & Concepts:
  • Displacement in SHM: \( x(t) = A \cos(\omega t + \phi) \)
  • Velocity: \( v(t) = -\omega A \sin(\omega t + \phi) \)
  • Acceleration: \( a(t) = -\omega^2 x(t) \)
  • Angular frequency: \( \omega = 2\pi f = \sqrt{\frac{k}{m}} \) (spring) or \( \sqrt{\frac{g}{L}} \) (pendulum)
  • Period: \( T = \frac{2\pi}{\omega} \)
  • Mechanical Energy: \( E = \frac{1}{2} k A^2 = \frac{1}{2} m \omega^2 A^2 \)

Example 1

A mass of 0.5 kg is attached to a spring with k = 200 N/m. Find the period and angular frequency of oscillation.

Angular frequency: \( \omega = \sqrt{k/m} = \sqrt{200/0.5} = \sqrt{400} = 20 \text{ rad/s} \)

Period: \( T = 2\pi / \omega = 2\pi / 20 \approx 0.314 \text{ s} \)

Example 2

A simple pendulum of length 1.5 m is displaced slightly. Compute its period (take g = 9.8 m/s²).

Period: \( T = 2\pi \sqrt{L/g} = 2\pi \sqrt{1.5 / 9.8} \approx 2.46 \text{ s} \)

Practice Problems

  1. A 0.2 kg mass on a spring with k = 50 N/m is displaced by 0.1 m. Find maximum velocity.
  2. A pendulum of length 2 m is oscillating. Find its period and angular frequency.
  3. A spring-mass system oscillates with period 0.5 s. Find its angular frequency.
  4. Compute total energy of a 0.3 kg mass oscillating with amplitude 0.05 m on a spring of k = 80 N/m.
  5. A mass on a spring moves with maximum velocity 2 m/s and amplitude 0.1 m. Find angular frequency.
  6. Period of oscillation for a 1 kg mass on a spring with k = 100 N/m.
  7. A 0.4 kg mass is oscillating on a spring with k = 64 N/m. Find max acceleration if amplitude is 0.05 m.
  8. A pendulum has a period of 3 s. Find its length.
  9. Amplitude of SHM if max velocity is 5 m/s and ω = 10 rad/s.
  10. A mass-spring system has energy 0.5 J and amplitude 0.1 m. Find spring constant k.
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