Simple Harmonic Motion (SHM)

• Oscillatory motion is characterized by a body moving to and fro about a mean position.
• When oscillatory motion repeats itself after equal intervals of time, it is called periodic motion.
• Typical vibrating bodies include a mass suspended from a spring, the bob of a simple pendulum, a steel ruler clamped at one end, and a steel ball rolling in a curved dish.
• Oscillations occur when a body is pulled away from its rest position and released, resulting in a restoring force that causes the body to accelerate and overshoot the rest position due to inertia.

Simple Harmonic Motion (SHM)

• Involves a restoring force that is directly proportional to the displacement of the system from its equilibrium position.
• If a mass is attached to a spring on a frictionless surface:
  • Then the restoring force is given by F = -kx, where k is the spring constant and x is that displacement.
  • The acceleration of the mass is proportional to the displacement and directed towards the mean position.

Instantaneous Displacement and Amplitude of Vibration

• Instantaneous displacement refers to the distance of a vibrating body from its mean position at any given time.
• Amplitude is the maximum value of displacement.

Vibration

• Describes one complete round trip of the body in motion, from its mean position to the upper extreme, then to the lower extreme, and back to the mean position.

Time Period

• The time, T, required to complete one vibration.

Frequency

• The number of vibrations, f, executed by a body in one second and is measured in hertz (Hz).
• f = 1/T, showing the inverse relationship between frequency and time period.

Angular Frequency

• Given by ω = 2π/T = 2πf.
• Allows for easy computation of instantaneous displacement and velocity in SHM.

SHM and Uniform Circular Motion

• SHM can be related to uniform circular motion by considering a point P moving on a circle.
• The projection of point P on the diameter of the circle (point N) oscillates in a manner that replicates SHM
• The radius of this circle corresponds to the amplitude of the SHM.

Displacement

• Given by x = xo sin(ωt), with x being the displacement of point N at time t, xo the amplitude, and ωt the phase.

Instantaneous Velocity

• The velocity of point N is v = xoω cos(ωt).
• The relationship for velocity can also be expressed as v = ω√(xo² - x²).

Acceleration in Terms of ω

• The acceleration is given by a = -ω²x.
• It's proportional to the displacement and directed towards the mean position, confirming SHM.

Phase

• The angle θ = ωt which defines the displacement and direction of motion.

Phase Angle

• The angle which the rotating radius makes with the reference direction at any instant.
• Displacement can be generalized as x = xo sin(ωt + φ), where φ represents an initial phase.

Horizontal Mass Spring System

• A simple harmonic system can involve a vibrating mass attached to a spring.
  • Angular frequency: ω = √(k/m)
  • Time period: T = 2π√(m/k)
  • Displacement: x = xo sin(ωt)
  • Velocity: v = √(k/m)√(xo² - x²) = xo√(k/m)√(1 - x²/xo²)

Simple Pendulum

• Consists of a mass m suspended by a light string of length l.
• Restoring force: F = -mg sin θ
  • With the small angle approximation, F ≈ -mgθ.
• Acceleration: a = -gx/l
• Time period: T = 2π√(l/g). Period depends on length and gravitational acceleration.
  • It is independent of mass.

Energy Conservation in SHM

• Total energy is the sum of potential energy (P.E.) and kinetic energy (K.E.).
• Potential Energy: P.E. = (1/2)kx²
• Total energy for Hooke's Law: (1/2)kx₀²
• Kinetic Energy: K.E. = (1/2)k(x₀² - x²)
  • The energy continuously interchanges between potential and kinetic forms.

Free and Forced Oscillations

• Free oscillations occur without external interference, vibrating at the natural frequency.
• Forced oscillations occur when an external periodic force is applied.

Resonance

• A phenomenon where the amplitude of the motion becomes extraordinarily large when the external driving force is periodic and comparable to the natural period of the oscillator.
• At resonance, energy transfer is maximum.

Damped Oscillations

• Oscillations where the amplitude decreases gradually with time due to energy dissipation from friction and air resistance.

Sharpness of Resonance

• The amplitude and how rapidly it decreases at frequencies slightly different from the resonant frequency, depend on damping.
• Smaller damping results in a greater amplitude and a sharper resonance.

Oscillatory motion involves a body moving back and forth around a central point, repeating at regular intervals in periodic motion. SHM features a restoring force proportional to displacement from equilibrium. For example, a mass attached to a spring demonstrates SHM.