Oscillation in One Shot - Class 11 Maharashtra Board Physics

Oscillations and Simple Harmonic Motion

Oscillations involve repetitive to-and-fro motion about a mean position. Simple Harmonic Motion (SHM) is a special type of oscillatory motion where the displacement follows a sine or cosine function with respect to time. Not all oscillatory motions are SHM, but all SHM are both oscillatory and periodic.

Fundamental Concepts: Oscillation, Periodicity, and SHM

Oscillation is defined as a repetitive motion that returns to its initial state after a fixed interval. Periodic motion repeats after equal intervals of time, but not all periodic motions are oscillatory. SHM is a periodic oscillation where displacement varies…

Oscillations and Simple Harmonic Motion

Oscillations involve repetitive to-and-fro motion about a mean position. Simple Harmonic Motion (SHM) is a special type of oscillatory motion where the displacement follows a sine or cosine function with respect to time. Not all oscillatory motions are SHM, but all SHM are both oscillatory and periodic.

Fundamental Concepts: Oscillation, Periodicity, and SHM

Oscillation is defined as a repetitive motion that returns to its initial state after a fixed interval. Periodic motion repeats after equal intervals of time, but not all periodic motions are oscillatory. SHM is a periodic oscillation where displacement varies sinusoidally with time.

• Oscillation: To-and-fro motion about a mean position.
• Periodicity: Motion repeats after a fixed time interval.
• SHM: Displacement $x$ follows $x = a \sin(\omega t + \phi)$ or $x = a \cos(\omega t + \phi)$.
• Time Period ($T$): Time taken for one complete oscillation.
• Frequency ($f$): Number of oscillations per unit time, $f = 1/T$ (measured in Hz).
• Angular Frequency ($\omega$): $\omega = 2\pi f = 2\pi/T$ (measured in rad/s).

Key Terms in SHM

In SHM, several terms are essential for describing the motion. The mean position is the equilibrium point, while extreme positions are the farthest points reached during oscillation. Amplitude is the maximum displacement from the mean position.

• Mean Position: Central equilibrium point.
• Extreme Position: Farthest points from the mean.
• Amplitude ($a$): Maximum displacement from mean to extreme.
• Angular Frequency ($\omega$): Rate of change of phase with time.

Spring-Mass System: Equations and Analysis

A mass attached to a spring exhibits SHM when displaced and released. The restoring force is proportional to displacement and acts in the opposite direction.

• Restoring Force: $F = -kx$, where $k$ is the spring constant.
• Spring Constant ($k$): $k = \frac{mg}{x}$ (from equilibrium analysis).
• Units: $k$ in N/m.
• Differential Equation: $m\frac{d^2x}{dt^2} + kx = 0$.
• General Solution: $x = a \sin(\omega t + \phi)$, where $\omega = \sqrt{k/m}$.

Position, Velocity, and Acceleration in SHM

The displacement, velocity, and acceleration in SHM are interrelated and can be expressed as functions of time or position.

• Displacement: $x = a \sin(\omega t + \phi)$
• Velocity: $v = a\omega \cos(\omega t + \phi)$
• Acceleration: $a = -a\omega^2 \sin(\omega t + \phi)$ or $a = -\omega^2 x$
• Maximum Velocity: $v_{max} = a\omega$ (at mean position)
• Maximum Acceleration: $a_{max} = a\omega^2$ (at extreme positions)
• Velocity as a function of position: $v = \pm \omega \sqrt{a^2 - x^2}$

Graphical Interpretation and Special Graphs

The time variation of displacement, velocity, and acceleration in SHM produces characteristic graphs. The force vs. displacement graph is a straight line with negative slope.

• Displacement-Time: Sine curve.
• Velocity-Time: Cosine curve.
• Acceleration-Time: Negative sine curve.
• Force-Displacement: Straight line, negative slope ($F = -kx$).
• Acceleration-Displacement: Straight line, negative slope ($a = -\omega^2 x$).
• Velocity-Displacement: Elliptical curve.

Energy in SHM

The total mechanical energy in SHM is conserved and is the sum of kinetic and potential energies, which interchange during oscillation.

• Potential Energy (Spring): $U = \frac{1}{2}kx^2$
• Kinetic Energy: $K = \frac{1}{2}mv^2$
• Total Energy: $E = \frac{1}{2}ka^2 = \frac{1}{2}m\omega^2 a^2$ (constant)
• Energy-Amplitude Relation: $E \propto a^2$
• At $x = \pm a/\sqrt{2}$, $K = U$

Combination of Springs

Springs can be combined in series or parallel, affecting the equivalent spring constant and the time period of oscillation.

• Series Combination: $\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}$
• Parallel Combination: $k_{eq} = k_1 + k_2$
• Cutting a spring into $n$ parts: Each part has $k' = n k$
• Time Period with Equivalent $k$: $T = 2\pi \sqrt{m/k_{eq}}$

SHM as Projection of Uniform Circular Motion

SHM can be visualized as the projection of uniform circular motion onto a diameter. The phase constant represents the initial angular position in the reference circle.

• Projection Principle: The shadow of a particle in uniform circular motion on a diameter performs SHM.
• Phase ($\phi$): Indicates initial position; phase difference between two SHMs gives their relative state.
• Phasor Diagram: Represents SHM as a rotating vector (phasor) with angular speed $\omega$.

Superposition and Combination of SHMs

When two SHMs of the same frequency are combined, the resultant is also an SHM with amplitude found by vector addition of individual amplitudes.

• Resultant Amplitude: $A_{net} = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\Delta\phi)}$
• Constructive Combination: Phase difference $0^\circ$, $A_{max} = A_1 + A_2$
• Destructive Combination: Phase difference $180^\circ$, $A_{min} = |A_1 - A_2|$
• General Resultant: $x_{net} = A_{net} \sin(\omega t + \phi_{net})$

Angular SHM and Torsional Oscillations

Angular SHM involves rotational analogs of linear SHM quantities. The restoring torque is proportional to angular displacement.

• Moment of Inertia ($I$): Rotational analog of mass.
• Angular Displacement ($\theta$): Analog of $x$.
• Restoring Torque: $\tau = -C\theta$, $C$ is torsional constant.
• Differential Equation: $I\frac{d^2\theta}{dt^2} + C\theta = 0$
• Angular Frequency: $\omega = \sqrt{C/I}$
• Time Period: $T = 2\pi \sqrt{I/C}$
• Maximum Angular Velocity: $\theta_0 \omega$
• Maximum Angular Acceleration: $\theta_0 \omega^2$

Simple Pendulum and Special Cases

A simple pendulum is a mass suspended by a string, oscillating with small angular displacement. For small angles, $\sin\theta \approx \theta$.

• Restoring Torque: $-lmg\theta$
• Time Period: $T = 2\pi \sqrt{l/g}$
• Second's Pendulum: $T = 2$ s, $l \approx 1$ m (since $l = g/\pi^2$ for $T=2$ s)

Damped Oscillations

In real systems, energy is lost due to resistive forces, causing amplitude to decrease over time. This is called damping.

• Damping Force: $F_{damp} = -bv$, $b$ is damping constant.
• Differential Equation: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$
• Amplitude Decay: $a = a_0 e^{-bt/2m}$
• Damped Angular Frequency: $\omega' = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2}$
• Energy decreases exponentially with time.

Forced Oscillations and Resonance

When an external periodic force acts on an oscillator, it is called forced oscillation. If the frequency of the external force matches the natural frequency, resonance occurs, resulting in maximum amplitude.

• Natural Frequency ($f_n$): Frequency at which system oscillates freely.
• Forced Frequency ($f$): Frequency of external periodic force.
• Resonance: Occurs when $f = f_n$, amplitude becomes maximum.
• Amplitude vs. Frequency Graph: Shows a peak at resonance.
• Practical Example: Two pendulums of equal length coupled together; the one with matching natural frequency exhibits maximum amplitude when forced.

Exam-Focused Takeaways

• SHM is characterized by sinusoidal variation of displacement, velocity, and acceleration.
• The restoring force (or torque) is always proportional and opposite to displacement (or angular displacement).
• Energy in SHM is conserved in ideal cases; in real systems, damping causes energy loss.
• Combination of SHMs with the same frequency results in another SHM; amplitude is found using vector addition.
• Resonance leads to large amplitude oscillations when forced and natural frequencies match.
• For all derivations and problem-solving, always start from the fundamental differential equation and apply boundary/initial conditions as required.