Energy in simple harmonic motion

Section: Oscillations  |  Syllabus: Cambridge A Level Physics (9702)

Energy Interchange in SHM

During SHM, energy is continually swapped between kinetic energy (E_k) and potential energy (E_p).

In a pendulum, E_p is gravitational. In a mass-spring system, E_p is elastic.

Total Energy in SHM

For an ideal system with no resistive forces, the total energy (E_total) remains constant throughout the oscillation.

E_total = (1/2)mω^2 x_0^2

Where m is mass, ω is angular frequency, and x_0 is amplitude.

Energy as a Function of Displacement

E_k = (1/2)mω^2(x_0^2 - x^2)

E_p = (1/2)mω^2 x^2

Figure 17.3: Energy-Displacement Graph

A graph of Energy vs. Displacement (x).
- Total Energy: Horizontal line.
- Potential Energy: U-shaped parabola (y = x^2).
- Kinetic Energy: Inverted parabola (y = -x^2).

Worked Examples

Worked Example: Pendulum Energy

Question: A 50 g pendulum bob is displaced 12 cm and released. The period is 2.9 s. Calculate the maximum kinetic energy.

Solution

1. ω = 2π / T = 2π / 2.9 = 2.167 rad s^-1.

2. E_max = (1/2)mω^2 x_0^2.

3. E_max = 0.5 × 0.050 × (2.167)^2 × (0.12)^2 = 1.69 × 10^-3 J (or 1.69 mJ).

Worked Example: Tuned Mass Damper

Question: A 6.6 × 10^5 kg mass oscillates with a period of 6 s and amplitude 0.8 m. Calculate the energy transferred.

Solution

1. ω = 2π / 6 = 1.047 rad s^-1.

2. E = 0.5 × (6.6 × 10^5) × (1.047)^2 × (0.8)^2 = 2.31 × 10^5 J (or 231 kJ).

Interactive revision notes, videos and practice questions load below.

Related: Past Papers Topical Questions AS Physics A2 Chemistry Grade Boundaries Command Words