Section 6.9: Review Examples
This section brings together the major concepts of Simple Harmonic Motion (SHM) studied so far — including oscillations of springs, pendulums, energy in SHM, damping, resonance, and real-world applications.
These review problems will help consolidate understanding and prepare for assessments.
Example 1
A 0.5 kg mass attached to a spring oscillates with amplitude 0.2 m and maximum speed 2.0 m/s. Find the spring constant \(k\).
Maximum speed in SHM: \( v_{\text{max}} = \omega A \)
\(\omega = \frac{v_{\text{max}}}{A} = \frac{2.0}{0.2} = 10 \,\text{rad/s}\)
\(\omega = \sqrt{\tfrac{k}{m}} \Rightarrow k = m\omega^2 = 0.5 \times 10^2 = 50 \,\text{N/m}\)
Example 2
A pendulum of length 1.2 m swings with small amplitude. Find its period.
Formula: \(T = 2\pi \sqrt{\tfrac{L}{g}} = 2\pi \sqrt{\tfrac{1.2}{9.8}} \approx 2.20 \,\text{s}\)
Example 3
A damped oscillator loses 5% of its mechanical energy each cycle. If the initial energy is 20 J, how much energy remains after 10 cycles?
Energy decay: \( E_n = E_0 (0.95)^n \)
\( E_{10} = 20 (0.95)^{10} \approx 12.0 \,\text{J} \)