Class 11 Physics: Oscillations | Physics Q&A

CBSE Class 11 › Unit X: Oscillations and Waves

Oscillations

NCERT Chapter 13 • Periodic Motion, Simple Harmonic Motion, Block-Spring Systems & Pendulums

NCERT 2025–26 Unit X • ~10 Marks JEE Main • 2 Questions

Artistic representation of a pendulum swinging in Simple Harmonic Motion. — Oscillations: The rhythmic back-and-forth motion that defines clocks, waves, and music.

1. Periodic and Oscillatory Motions

Periodic Motion: A motion that repeats itself at regular intervals of time (e.g., Earth orbiting the Sun).

Oscillatory Motion: A to-and-fro periodic motion about a fixed point called the mean position (e.g., a swinging pendulum).

Every oscillatory motion is periodic, but not every periodic motion is oscillatory.

2. Simple Harmonic Motion (SHM)

SHM is the simplest form of oscillatory motion where the restoring force is directly proportional to the displacement from the mean position and always directed towards it.

$F(x) = -kx$

Displacement equation:

$x(t) = A \cos(\omega t + \phi)$

Diagram showing Simple Harmonic Motion as the projection of Uniform Circular Motion. — SHM can be visualized as the projection of uniform circular motion on a diameter.

3. Velocity and Acceleration in SHM

Velocity (v)

$v = -\omega A \sin(\omega t + \phi) = \pm \omega \sqrt{A^2 - x^2}$

Max velocity $v_{max} = \omega A$ (at mean position).

Acceleration (a)

$a = -\omega^2A \cos(\omega t + \phi) = -\omega^2x$

Max acceleration $a_{max} = \omega^2 A$ (at extreme positions).

Problem 1: A body executes SHM with amplitude 5 cm and frequency 10 Hz. Find max velocity and acceleration.

Show Answer

A = 0.05 m, ω = 2π(10) = 20π rad/s.
$v_{max} = 20\pi(0.05) = \pi \approx 3.14$ m/s.
$a_{max} = (20\pi)^2(0.05) \approx 197.4$ m/s².

4. Energy in Simple Harmonic Motion

Total mechanical energy in SHM is conserved (ignoring friction).

Graph of Kinetic, Potential, and Total Energy vs Displacement for SHM. — Energy continuously transforms between Kinetic (K) and Potential (U), but Total Energy (E) remains constant.

$E = K + U =\dfrac{1}{2}kA^2$

$K = \dfrac{1}{2}m\omega^2(A^2 - x^2)$
$U = \dfrac{1}{2}m\omega^2 x^2$

5. Oscillations of a Spring (Block-Spring System)

A block of mass m attached to a massless spring of constant k executes SHM.

$T = 2\pi \sqrt{\dfrac{m}{k}}$

Combinations of Springs (JEE Essential)

Series Combination: Springs connected end-to-end.
$\dfrac{1}{k_{eq}} = \dfrac{1}{k_1} + \dfrac{1}{k_2}$ ⇒ $T = 2\pi \sqrt{\dfrac{m(k_1+k_2)}{k_1 k_2}}$
Parallel Combination: Springs connected side-by-side to same mass.
$k_{eq} = k_1 + k_2$ ⇒ $T = 2\pi \sqrt{\dfrac{m}{k_1+k_2}}$

Problem 2: Two springs of constants $k$ and $3k$ are connected in series to a mass $m$ . Find the time period.

Show Answer

$\frac{1}{k_{eq}} = \frac{1}{k} + \frac{1}{3k} = \frac{4}{3k}$ ⇒ $k_{eq} = \frac{3k}{4}$ .
$T = 2\pi \sqrt{\frac{m}{3k/4}} = 4\pi \sqrt{\frac{m}{3k}}$ .

6. The Simple Pendulum

Force diagram of a simple pendulum showing resolution of weight. — For small angles, the restoring force is proportional to displacement.

$T = 2\pi \sqrt{\dfrac{L}{g}$

The period is independent of the mass of the bob.

Problem 3: Length of a seconds pendulum (T=2s) on Earth?

Show Answer

$L = \dfrac{T^2 g}{4\pi^2} = \dfrac{4 \times 9.8}{4 \times 9.86} \approx 1$ meter.

7. Damped Simple Harmonic Motion

Real systems lose energy to friction/drag, causing amplitude to decay exponentially.

$x(t) = A e^{-bt/2m} \cos(\omega't + \phi)$

Where b is the damping constant and $\omega' = \sqrt{k/m - (b/2m)^2}$ .

8. Forced Oscillations and Resonance

When an external periodic force drives the system, it oscillates at the driving frequency $\omega_d$ .

Resonance

If the driving frequency equals the natural frequency ( $\omega_d = \omega$ ), the amplitude becomes very large. This is Resonance.

Example: Tacoma Narrows Bridge collapse was caused by wind-driven resonance.

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