📝 Chapter Notes & Revision

Oscillations

🏫 MP BoardClass 11Physics

📐 Formula & Cheat Sheet (English)

MP Board Class 11 Physics Revision Notes

Chapter: Oscillations (दोलन)

1. Basic Concepts (मूल अवधारणाएँ)

  • Periodic Motion (आवर्ती गति): A motion that repeats itself at regular intervals of time.
    • Example: Motion of planets around the Sun, hands of a clock.
  • Oscillatory or Vibratory Motion (दोलनी गति): A periodic motion in which a body moves to-and-fro repeatedly about a fixed mean position.
    • Example: Motion of a simple pendulum, motion of a stretched string.
    • Note: All oscillatory motions are periodic, but all periodic motions are not oscillatory.
  • Time Period ($T$): The time taken to complete one full oscillation.
    • Unit: Seconds ($s$)
  • Frequency ($\nu$ or $f$): The number of oscillations completed per second.
    • Formula: $\nu = \frac{1}{T}$
    • Unit: Hertz ($Hz$) or $s^{-1}$
  • Angular Frequency ($\omega$):
    • Formula: $\omega = \frac{2\pi}{T} = 2\pi\nu$
    • Unit: $rad/s$

2. Simple Harmonic Motion - SHM (सरल आवर्त गति)

SHM is a special type of oscillatory motion in which the restoring force acting on the particle is directly proportional to its displacement from the mean position and is always directed towards the mean position.

Basic Condition for SHM:

$$F \propto -y \implies F = -k y$$

  • $F$ = Restoring force
  • $y$ = Displacement from mean position
  • $k$ = Force constant (or Spring constant), $k = m\omega^2$

Differential Equation of SHM:

$$\frac{d^2y}{dt^2} + \omega^2 y = 0$$

Displacement Equation:

$$y = A \sin(\omega t + \phi_0) \quad \text{or} \quad y = A \cos(\omega t + \phi_0)$$

  • $A$ = Amplitude (Maximum displacement)
  • $(\omega t + \phi_0)$ = Phase (कला)
  • $\phi_0$ = Initial phase or Epoch (प्रारंभिक कला)

3. Velocity and Acceleration in SHM

1. Displacement ($y$):

$$y = A \sin(\omega t)$$

2. Velocity ($v$):

$$v = \frac{dy}{dt} = A\omega \cos(\omega t) = \omega \sqrt{A^2 - y^2}$$

  • At Mean Position ($y = 0$): Velocity is MAXIMUM $\rightarrow v_{max} = A\omega$
  • At Extreme Position ($y = \pm A$): Velocity is MINIMUM $\rightarrow v_{min} = 0$

3. Acceleration ($a$):

$$a = \frac{dv}{dt} = -\omega^2 A \sin(\omega t) = -\omega^2 y$$

  • At Mean Position ($y = 0$): Acceleration is MINIMUM $\rightarrow a_{min} = 0$
  • At Extreme Position ($y = \pm A$): Acceleration is MAXIMUM $\rightarrow a_{max} = \omega^2 A$

4. Energy in Simple Harmonic Motion (SHM में ऊर्जा)

1. Kinetic Energy ($K$):

$$K = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - y^2)$$

  • Maximum at mean position ($y = 0$): $K_{max} = \frac{1}{2} m \omega^2 A^2$
  • Minimum at extreme position ($y = \pm A$): $K_{min} = 0$

2. Potential Energy ($U$):

$$U = \frac{1}{2} k y^2 = \frac{1}{2} m \omega^2 y^2$$

  • Minimum at mean position ($y = 0$): $U_{min} = 0$
  • Maximum at extreme position ($y = \pm A$): $U_{max} = \frac{1}{2} m \omega^2 A^2$

3. Total Energy ($E$):

$$E = K + U = \frac{1}{2} m \omega^2 A^2 = \frac{1}{2} k A^2$$

  • Total Energy is constant at all points during motion and independent of position $y$ or time $t$.

5. Time Period of Important Oscillatory Systems

1. Simple Pendulum (सरल लोलक):

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

  • $L$ = Effective length of the pendulum
  • $g$ = Acceleration due to gravity
  • Second's Pendulum (सेकण्ड लोलक): A pendulum whose time period is exactly $2\text{ seconds}$. Its effective length on Earth is approximately $0.993\text{ m} \approx 1\text{ m}$.

2. Mass-Spring System (कमानी-द्रव्यमान निकाय):

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

  • $m$ = Mass attached
  • $k$ = Spring constant

Combination of Springs:

  • Series Combination (श्रेणीक्रम): $$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} \implies T = 2\pi \sqrt{\frac{m(k_1 + k_2)}{k_1 k_2}}$$
  • Parallel Combination (समांतरक्रम): $$k_{eq} = k_1 + k_2 \implies T = 2\pi \sqrt{\frac{m}{k_1 + k_2}}$$

3. Oscillations of Liquid in a U-Tube:

$$T = 2\pi \sqrt{\frac{h}{g}} = 2\pi \sqrt{\frac{L}{2g}}$$

  • $h$ = Initial height of liquid column in one arm
  • $L$ = Total length of the liquid column ($L = 2h$)

6. Damped, Forced, and Resonant Oscillations

1. Damped Oscillations (मंदित दोलन):

  • Oscillations whose amplitude decreases exponentially with time due to dissipative forces (friction, air resistance).
  • Damping Force: $F_d = -b v$ (where $b$ is damping constant)
  • Differential Equation: $$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$$
  • Amplitude as a function of time: $$A(t) = A_0 e^{-\frac{bt}{2m}}$$

2. Forced Oscillations (प्रणोदित दोलन):

  • When a body oscillates under the influence of an external periodic force $F(t) = F_0 \sin(\omega_d t)$.

3. Resonance (अनुनाद):

  • A special case of forced oscillations where the frequency of the external driving force ($\omega_d$) matches the natural frequency ($\omega_0$) of the system.
  • Condition: $\omega_d = \omega_0$
  • Result: The amplitude of oscillation becomes maximum.

7. Important Quick Summary Table

ParameterMean Position ($y = 0$)Extreme Position ($y = \pm A$)
Displacement ($y$)$0$$\pm A$
Velocity ($v$)Maximum ($A\omega$)Zero ($0$)
Acceleration ($a$)Zero ($0$)Maximum ($\omega^2 A$)
Kinetic Energy ($K$)Maximum ($\frac{1}{2}m\omega^2 A^2$)Zero ($0$)
Potential Energy ($U$)Zero ($0$)Maximum ($\frac{1}{2}m\omega^2 A^2$)
Total Energy ($E$)Constant ($\frac{1}{2}m\omega^2 A^2$)Constant ($\frac{1}{2}m\omega^2 A^2$)