Oscillations and Simple Harmonic Motion Solved Examples (Class 11 Physics)
Oscillations numericals cover simple harmonic motion equations, energy in SHM, and pendulum problems. These concepts are fundamental to understanding vibra
TL;DR: Oscillations numericals cover simple harmonic motion equations, energy in SHM, and pendulum problems. These concepts are fundamental to understanding…
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Oscillations numericals cover simple harmonic motion equations, energy in SHM, and pendulum problems. These concepts are fundamental to understanding vibra
Oscillations and Simple Harmonic Motion — Solved Numerical Examples (Step by Step)
Example 1: A particle undergoes SHM with amplitude 0.05 m and frequency 10 Hz. Calculate its maximum velocity.
Solution: Amplitude A = 0.05 m, Frequency f = 10 Hz. Angular frequency ω = 2πf = 2π × 10 = 20π rad/s. Maximum velocity vmax = A × ω = 0.05 × 20π = π m/s = 3.14 m/s.
Example 2: A spring-mass system has mass 0.5 kg and spring constant 200 N/m. Find the period of oscillation.
Solution: Mass m = 0.5 kg, Spring constant k = 200 N/m. Period T = 2π × sqrt(m/k) = 2π × sqrt(0.5/200) = 2π × sqrt(0.0025) = 2π × 0.05 = 0.1π = 0.314 s.
Example 3: A simple pendulum has length 1 m and oscillates at a place where g = 10 m/s². Calculate its period.
Solution: Length L = 1 m, g = 10 m/s². Period T = 2π × sqrt(L/g) = 2π × sqrt(1/10) = 2π × sqrt(0.1) = 2π × 0.316 = 1.99 s.
Example 4: A mass undergoing SHM has equation x = 0.1 sin(4πt) m. Find amplitude, frequency, and velocity at t = 0.25 s.
Solution: From equation x = 0.1 sin(4πt), amplitude A = 0.1 m. Angular frequency ω = 4π rad/s, so frequency f = ω/(2π) = 2 Hz. Velocity v = dx/dt = 0.1 × 4π × cos(4πt) = 0.4π cos(4πt). At t = 0.25 s: v = 0.4π × cos(π) = 0.4π × (-1) = -0.4π = -1.256 m/s.
Example 5: A mass of 0.2 kg attached to a spring oscillates with amplitude 0.1 m. If the spring constant is 50 N/m, calculate total mechanical energy.
Solution: Mass m = 0.2 kg, Amplitude A = 0.1 m, k = 50 N/m. Total energy E = 0.5 × k × A² = 0.5 × 50 × 0.01 = 0.25 J.
Tips
- In SHM, maximum velocity occurs at equilibrium position; velocity is zero at maximum displacement.
- Total mechanical energy in SHM is conserved: E = 0.5kA² = constant.
- Period of simple pendulum depends only on length and gravity, not on mass or amplitude.
Frequently Asked Questions
What is simple harmonic motion?
SHM is motion in which an object oscillates back and forth about an equilibrium position, with a restoring force proportional to displacement. Examples include pendulums and masses on springs.
Why does a pendulum clock run slower at higher altitudes?
At higher altitudes, g is smaller. Since period T = 2π√(L/g), as g decreases, period increases, making the clock run slower.
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