Simple Harmonic Motion (SHM)

• Oscillation is a periodic motion around an equilibrium position.

Simple Harmonic Motion (SHM)

• SHM is a special type of oscillation in which the restoring force is directly proportional to the displacement and acts in the opposite direction.
• Represented by: F = -kx, where F is the restoring force, k is the force constant, and x is the displacement from equilibrium.
• The motion is sinusoidal.
• A particle executing SHM moves to and fro repeatedly about the mean position.
• Examples include a mass-spring system and a simple pendulum (for small angles).

Key Quantities in SHM

• Amplitude (A): The maximum displacement from the equilibrium position.
• Period (T): The time taken to complete one full oscillation.
• Frequency (f): The number of oscillations per unit time; f = 1/T.
• Angular Frequency (ω): ω = 2πf = 2π/T.
• Phase (φ): Specifies the position and direction of motion of the oscillator at a particular time.
• Epoch: initial phase angle

Equations of Motion for SHM

• Displacement: x(t) = A cos(ωt + φ), where x(t) is the displacement at time t.
• Velocity: v(t) = -Aω sin(ωt + φ).
• Acceleration: a(t) = -Aω² cos(ωt + φ) = -ω²x(t).

Energy in SHM

• Potential Energy (U): U = (1/2)kx² = (1/2)mω²x².
• Kinetic Energy (K): K = (1/2)mv² = (1/2)mA²ω²sin²(ωt + φ).
• Total Mechanical Energy (E): E = U + K = (1/2)kA² = (1/2)mω²A², which is constant.
• The total energy is proportional to the square of the amplitude.

Mass-Spring System

• A mass (m) attached to a spring with spring constant (k).
• Angular frequency: ω = √(k/m).
• Period: T = 2π√(m/k).
• Frequency: f = (1/2π)√(k/m).

Simple Pendulum

• A point mass (m) suspended from a fixed point by a string of length (L).
• For small angles (θ), SHM is approximated.
• Angular frequency: ω = √(g/L), where g is the acceleration due to gravity.
• Period: T = 2π√(L/g).
• Frequency: f = (1/2π)√(g/L).
• The period is independent of the mass of the bob.

Damped Oscillations

• Oscillations where the amplitude decreases with time due to energy loss.
• Energy loss is typically due to friction or air resistance.
• Damping force is often proportional to velocity: F_d = -bv, where b is the damping coefficient.
• The equation of motion becomes more complex, involving exponential decay.
• Amplitude decreases exponentially with time: A(t) = A₀e^(-bt/2m).
• Damping affects the frequency of oscillation.
• Examples include shock absorbers in cars.

Types of Damping

• Underdamped: The system oscillates with gradually decreasing amplitude.
• Critically Damped: The system returns to equilibrium as quickly as possible without oscillating.
• Overdamped: The system returns to equilibrium slowly without oscillating.

Forced Oscillations

• Oscillations driven by an external periodic force.
• The system oscillates at the driving frequency, not its natural frequency.
• Resonance occurs when the driving frequency is close to the natural frequency of the system.
• At resonance, the amplitude of oscillation is maximized.
• Examples include a child being pushed on a swing.

Resonance

• Occurs when the driving frequency matches the natural frequency of the system.
• Results in a large amplitude of oscillation.
• Sharpness of resonance depends on the damping. Less damping leads to a sharper resonance.
• Examples include tuning a radio to a specific frequency.

Mathematical Representation of Damped SHM

• The equation of motion for damped SHM is m(d²x/dt²) + b(dx/dt) + kx = 0.
• The solution depends on the relative values of m, b, and k, determining whether the system is underdamped, critically damped, or overdamped.

Mathematical Representation of Forced SHM

• The equation of motion for forced SHM is m(d²x/dt²) + b(dx/dt) + kx = F₀cos(ω_d t), where F₀ is the amplitude of the driving force and ω_d is the driving frequency.
• The steady-state solution involves oscillations at the driving frequency.