Class 11 Physics: Oscillations Notes

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1. Introduction

• Oscillation: Repetitive back-and-forth motion of a body about a mean position.
• Examples: Pendulum, spring-mass system, vibrating strings.
• Simple Harmonic Motion (SHM): Special type of oscillation where restoring force is proportional to displacement and directed towards equilibrium.

$$F = -kx$$F=−kx

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2. Simple Harmonic Motion (SHM)

• Equation of Motion:

$$m \frac{d^2x}{dt^2} + kx = 0$$mdt2d2x​+kx=0

• Solution:

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

Where:

• $A$A = amplitude
• $\omega = \sqrt{\frac{k}{m}}$ω=mk​​ = angular frequency
• $\phi$ϕ = phase constant
• $T = \frac{2\pi}{\omega}$T=ω2π​ = time period
• $f = \frac{1}{T}$f=T1​ = frequency

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3. Key Quantities in SHM

1. Amplitude (A): Maximum displacement from equilibrium.
2. Time Period (T): Time for one complete oscillation.
3. Frequency (f): Number of oscillations per second.
4. Angular Frequency (ω): Rate of change of phase,

$$\omega = 2\pi f$$ω=2πf

5. Phase (φ): Determines initial position and direction.

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4. Velocity and Acceleration in SHM

• Velocity:

$$v = \frac{dx}{dt} = -\omega A \sin(\omega t + \phi)$$v=dtdx​=−ωAsin(ωt+ϕ)

• Acceleration:

$$a = \frac{d^2x}{dt^2} = -\omega^2 x$$a=dt2d2x​=−ω2x

• Acceleration is proportional to displacement and opposite in direction.

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5. Energy in SHM

1. Kinetic Energy:

$$K.E. = \frac{1}{2} m \omega^2 (A^2 – x^2)$$K.E.=21​mω2(A2−x2)

2. Potential Energy:

$$P.E. = \frac{1}{2} m \omega^2 x^2$$P.E.=21​mω2×2

3. Total Energy:

$$E = K.E. + P.E. = \frac{1}{2} m \omega^2 A^2 \quad (\text{constant})$$E=K.E.+P.E.=21​mω2A2(constant)

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6. Simple Pendulum

• Time period:

$$T = 2\pi \sqrt{\frac{l}{g}}$$T=2πgl​​

Where:

• $l$l = length of the pendulum
• $g$g = acceleration due to gravity
• Approximation: Small angles ($\theta \leq 10^\circ$θ≤10∘)

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7. Physical Pendulum

• Time period:

$$T = 2\pi \sqrt{\frac{I}{mgd}}$$T=2πmgdI​​

Where:

• $I$I = moment of inertia about pivot
• $d$d = distance from pivot to center of mass

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8. Damped and Forced Oscillations (NCERT Basics)

• Damped Oscillation: Amplitude decreases with time due to friction/resistance.
• Forced Oscillation: External periodic force applied.
• Resonance: Maximum amplitude occurs when driving frequency = natural frequency.

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9. Key Points

• SHM is a special case of periodic motion.
• Energy oscillates between kinetic and potential.
• Time period of SHM is independent of amplitude (small oscillations).
• Simple pendulum → example of SHM for small angles.