Section 3.5: Energy in Oscillations
Energy in oscillating systems alternates between kinetic and potential forms. Total mechanical energy remains constant in ideal (undamped) oscillations.
Key Formulas:
- Displacement in SHM: \( x = A \cos(\omega t + \phi) \)
- Kinetic Energy: \( KE = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2) \)
- Potential Energy: \( PE = \frac{1}{2} k x^2 \)
- Total Energy: \( E = KE + PE = \frac{1}{2} k A^2 \)
Example 1
A mass-spring system has m = 0.5 kg, k = 200 N/m, amplitude A = 0.1 m. Find total energy and kinetic energy when x = 0.05 m.
Total Energy: \( E = \frac{1}{2} k A^2 = 0.5 * 200 * 0.1^2 = 1 \text{ J} \)
Kinetic Energy: \( KE = E - PE = 1 - 0.5 * 200 * 0.05^2 = 1 - 0.25 = 0.75 \text{ J} \)
Practice Problems
- A 0.2 kg mass on a spring with k = 50 N/m oscillates with amplitude 0.15 m. Compute total energy.
- Determine kinetic energy when displacement is half the amplitude in above system.
- A pendulum of length 1 m has amplitude 0.2 m. Compute maximum potential energy.
- Show that in SHM, maximum KE = maximum PE at equilibrium and extreme positions, respectively.
- A spring-mass system oscillates with frequency 2 Hz and amplitude 0.1 m. Calculate total energy.