periodic and oscillatory motion-1

Periodic and Oscillatory Motion

Definitions

• Periodic Motion: Any motion repeating itself over regular intervals of time.

  • Examples:

    • Motion of a planet around the sun.

    • Motion of hands of a clock.

• Oscillatory Motion: Motion back and forth repeatedly about a mean position.

  • Examples:

    • Motion of a swing.

    • Motion of a pendulum in a wall clock.

  • Confined to well-defined limits on either side of the mean position.

  • All oscillatory motions are periodic; not all periodic motions are oscillatory.

  • Difference between oscillation and vibration:

    • Oscillation: occurs with small frequency.

    • Vibration: occurs with high frequency (e.g., guitar string).

Mathematical Representation of Oscillatory Motion

• Harmonic Functions:

  • Only sine and cosine functions describe simple periodic and bounded motion.

  • Period: 2𝜋 radians.

  • Oscillatory motion can be expressed in terms of sine and cosine functions:

    • Example:

      • s(t) = a sin(𝜔t)

      • c(t) = a cos(𝜔t)

Terms Related to Periodic Motion

Time Period (T)

• Smallest interval after which the motion repeats.

• SI Unit: second.

Frequency (𝜈)

• Number of periodic motions per second.

• Formula: 𝜈 = 1/T.

• SI Unit: Hertz (Hz).

Angular Frequency (𝜔)

• Formula: 𝜔 = 2𝜋/T = 2𝜋𝜈.

• SI Unit: rad/s.

Periodic Functions

• Used to represent periodic motion.

  • A function f(t) is periodic with period T if:

    • f(T) = f(t+T) = f(t+2T)

  • a sin 𝜔(𝑡 + 𝑇) = a sin 𝜔𝑡 + 2𝜋

  • 𝜔(𝑡 + 𝑇) = 𝜔𝑡 + 2𝜋

  • 𝜔 = 2𝜋 𝑇 = 2𝜋𝜈

Phase of Oscillation

• Phase: Describes position and direction of motion of a particle with respect to the mean position.

  • displacement : y = a sin(𝜔t + 𝜙0).

  • here, phase = 𝜔t + 𝜙0, (𝜔 = angular frequency, t = time, 𝜙0 = initial phase angle that defines the starting position of the oscillating particle. )

• Initial Phase (Epoch): Phase of the vibrating particle at time t = 0.

Displacement

• Physical quantity representing deviation from the mean position.

  • it is measured as a function of time.

  • Example: Displacement in a spring is its deviation from mean position.

  • Displacement function: F(t) = a sin(𝜔t). (a=max displacement of oscillation)

Phase Difference

• Indicates the lack of harmony between two vibrating particles at a given instant.

  • Measured as the difference in phase angles.

  • Example cases:

    1. Zero phase difference when particles cross mean position at same time, in the same direction.

    2. Phase difference of π when they go in opposite directions.

    3. Phase difference of π/2 indicates that one particle is ahead while the other is at the mean position.

Simple Harmonic Motion (SHM)

• Definition: A particle executes SHM if it moves about a mean position under a restoring force directly proportional to its displacement.

• Force, F ∝ x (displacement).

  • F = -kx (k = force constant, negative means force acts opposite to displacement).

Motion Characteristics

• Acceleration in SHM is directly proportional to displacement: a ∝ x.

• ( acc to newtons 2nd law, F=ma that implies ma=-kx »» a ∝ x

• Examples of SHM:

  • Vibration of a tuning fork, oscillation of a loaded spring.

SHM and Circular Motion

• SHM is defined as the projection of uniform circular motion on any diameter of a circle of reference

Characteristics of SHM

1. Displacement: Distance from the mean position at any instant.

2. Amplitude: Maximum displacement on either side of the mean position.

3. Velocity: Rate of change of displacement, derived from displacement function.

4. Acceleration: Rate of change of velocity, connected with displacement by A = -𝜔²y.

5. Time Period: Time taken for one complete vibration; formulas relate to acceleration and displacement.

Graphical Representation

• Energy considerations in SHM include potential and kinetic energy variations.

Energy in Simple Harmonic Motion

Potential Energy (PE)

• due to displacement; PE = 1/2 k y² (with k as the force constant).

Kinetic Energy (KE)

• due to velocity, KE formula: 1/2 m v².

Total Energy in SHM

• Total energy remains constant and is the sum of KE and PE.

• Maximum energy at the mean position is kinetic, while at extreme positions, it is potential.

Force Law for SHM

• The equation F = -ky defines the relationship between restoring force and displacement.

  • k is the force constant; 𝚔 = m𝜔².

Time Period Expressions

• Time period (T) can be expressed as T = 2𝜋 √(m/k).

• Specific systems in SHM include pendulums and spring systems.

Types of Oscillations

Undamped Simple Harmonic Oscillation

• Consistent amplitude with constant total energy.

Damped Simple Harmonic Oscillation

• Amplitude decreases over time, energy dissipated due to friction and similar forces.

Free Oscillations

• Occur naturally after initial displacement without external forces.

Forced Oscillations

• Result from an external periodic force applied at a different frequency.

Resonant Oscillations

• Occur when external frequency matches natural frequency, leading to large amplitude.

  • Examples include breaking glass with sound frequency matching resonance.