Oscillations Class 11 Physics Notes

Key Definitions

Oscillatory Motion

Repetitive back-and-forth motion of a body about a mean (equilibrium) position.

Periodic Motion

Any motion that repeats itself after a fixed interval of time. All oscillatory motions are periodic, but not all periodic motions are oscillatory (e.g., uniform circular motion).

Simple Harmonic Motion (SHM)

A special type of oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and always directed towards it (F = -kx).

Amplitude (A)

The maximum displacement or distance moved by an oscillating particle from its equilibrium position.

Time Period (T)

The time taken to complete one full oscillation or cycle. T = 1/f.

Frequency (f)

The number of oscillations completed per unit time. f = 1/T.

Angular Frequency (ω)

Related to frequency by ω = 2πf = 2π/T. It is crucial in the mathematical description of SHM.

Phase (φ)

Describes the state of oscillation (position and direction of velocity) at any instant relative to a reference point or another oscillation.

Restoring Force

The force that always tends to bring the oscillating body back to its equilibrium position. In SHM, this force is F = -kx.

Simple Harmonic Motion (SHM) - The Heart of Oscillations

Simple Harmonic Motion (SHM) is the most fundamental and crucial type of oscillatory motion you'll encounter. It is defined by a specific condition: the restoring force acting on the oscillating body is directly proportional to its displacement from the equilibrium (mean) position and always directed opposite to the displacement. Mathematically, this is expressed as F = -kx, where 'F' is the restoring force, 'x' is the displacement, and 'k' is the force constant. The negative sign signifies that the restoring force always acts towards the equilibrium position, attempting to bring the system back to its original state.

The key to understanding SHM lies in this linear restoring force. Any system obeying this force law will undergo SHM. From Newton's second law, F = ma, so for SHM, we have ma = -kx, which leads to the differential equation of SHM: d²x/dt² + (k/m)x = 0. This equation's solution gives us the displacement of the particle as a function of time: x(t) = A sin(ωt + φ) or x(t) = A cos(ωt + φ). Here, 'A' is the amplitude (maximum displacement), 'ω' is the angular frequency, 't' is time, and 'φ' is the initial phase constant.

From the displacement equation, we can derive the velocity and acceleration of the particle in SHM.
Velocity v(t) = dx/dt = Aω cos(ωt + φ). The maximum velocity occurs at the mean position (x=0) and is v_max = Aω.
Acceleration a(t) = dv/dt = -Aω² sin(ωt + φ) = -ω²x(t). The maximum acceleration occurs at the extreme positions (x = ±A) and is a_max = Aω². Notice that acceleration is directly proportional to displacement and opposite in direction, confirming the condition for SHM.

The energy of a particle in SHM is continuously transformed between kinetic energy (KE) and potential energy (PE), but its total mechanical energy (E) remains constant (in the absence of damping).
Potential Energy (PE) = ½ kx² = ½ mω²x². It is maximum at extreme positions.
Kinetic Energy (KE) = ½ mv² = ½ m(A²ω² - ω²x²) = ½ mω²(A² - x²). It is maximum at the mean position.
Total Energy (E) = KE + PE = ½ mω²A². This demonstrates the conservation of energy in SHM and shows that the total energy is proportional to the square of the amplitude and angular frequency. Understanding these energy transformations is vital for solving problems related to SHM.

Key Formulas and Must Remember Concepts

• Condition for SHM: F = -kx (Restoring force proportional to displacement and opposite in direction).
• Differential Equation of SHM: d²x/dt² + ω²x = 0, where angular frequency ω = √(k/m).
• Displacement: x(t) = A sin(ωt + φ) or A cos(ωt + φ).
• Velocity: v(t) = Aω cos(ωt + φ) (or -Aω sin(ωt + φ)). Maximum velocity v_max = Aω (at mean position).
• Acceleration: a(t) = -Aω² sin(ωt + φ) = -ω²x(t). Maximum acceleration a_max = Aω² (at extreme positions).
• Time Period of a Simple Pendulum: T = 2π√(L/g) (valid for small angles, L is length).
• Time Period of a Spring-Mass System: T = 2π√(m/k) (m is mass, k is spring constant).
• Total Mechanical Energy in SHM: E = KE + PE = ½ kA² = ½ mω²A². This energy is conserved in an undamped oscillator.

SHM as Projection of Uniform Circular Motion (UCM)

Beyond Ideal SHM: Damped and Forced Oscillations

Solved Examples

• A particle executes SHM with an amplitude of 5 cm and a time period of 4 s. What is its maximum velocity? Given A = 5 cm = 0.05 m, T = 4 s. Angular frequency ω = 2π/T = 2π/4 = π/2 rad/s. Maximum velocity v_max = Aω = (0.05 m)(π/2 rad/s) = 0.025π m/s ≈ 0.0785 m/s.
• A mass of 0.2 kg attached to a spring oscillates with a frequency of 2.5 Hz. What is the spring constant? Given m = 0.2 kg, f = 2.5 Hz. Angular frequency ω = 2πf = 2π(2.5) = 5π rad/s. For a spring-mass system, ω = √(k/m). Squaring both sides: ω² = k/m => k = mω². k = (0.2 kg)(5π rad/s)² = 0.2 * 25π² = 5π² N/m ≈ 49.3 N/m.

Exam Tip: Mastering Oscillations

To score well in this chapter, focus on:

• Graphical Analysis: Be prepared to interpret and draw graphs for displacement, velocity, and acceleration vs. time, and energy vs. displacement. Understand where Kinetic Energy (KE) and Potential Energy (PE) are maximum/minimum.
• Formula Derivations: Practice deriving the time period formulas for simple pendulum (T = 2π√(L/g)) and spring-mass system (T = 2π√(m/k)). Pay attention to the assumptions made (e.g., small angles for pendulum).
• Conditions for SHM: Clearly state the conditions for a motion to be Simple Harmonic Motion (F = -kx). This is a frequent theoretical question.
• Energy Conservation: Apply the principle of conservation of mechanical energy in SHM problems, especially when finding velocity at a particular displacement or relating amplitude to total energy.
• Conceptual Understanding: Differentiate between periodic, oscillatory, and SHM. Understand damping, forced oscillations, and resonance with practical examples.

Practice Questions with Solutions

• Q1: What is the primary condition for a motion to be classified as Simple Harmonic Motion? A1: The restoring force acting on the body must be directly proportional to its displacement from the equilibrium position and always directed towards the equilibrium (F = -kx).
• Q2: How does the total mechanical energy of an undamped simple harmonic oscillator change over time? A2: For an undamped simple harmonic oscillator, the total mechanical energy remains constant over time. It continuously converts between kinetic and potential energy.
• Q3: If the amplitude of a simple harmonic oscillator is doubled, how does its maximum acceleration change? A3: The maximum acceleration (a_max = Aω²) will also double, as it is directly proportional to the amplitude.
• Q4: What is the phase difference between the displacement and velocity of a particle in SHM? A4: The velocity leads the displacement by a phase of π/2 radians (90 degrees). When displacement is maximum, velocity is zero, and vice versa.

Frequently Asked Questions