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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œ‹ radians.

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

      • Example:

        • s(t) = a sin(πœ”t)

        • c(t) = a cos(πœ”t)

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 (πœ”)

  • Formula: πœ” = 2πœ‹/T = 2πœ‹πœˆ.

  • 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 πœ”(𝑑 + 𝑇) = a sin πœ”π‘‘ + 2πœ‹

    • πœ”(𝑑 + 𝑇) = πœ”π‘‘ + 2πœ‹

    • πœ” = 2πœ‹ 𝑇 = 2πœ‹πœˆ

Phase of Oscillation

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

    • displacement : y = a sin(πœ”t + πœ™0).

    • here, phase = πœ”t + πœ™0, (πœ” = angular frequency, t = time, πœ™0 = 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(πœ”t). (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 Ο€ when they go in opposite directions.

      3. Phase difference of Ο€/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 = -πœ”Β²y.

  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 yΒ² (with k as the force constant).

Kinetic Energy (KE)

  • due to velocity, KE formula: 1/2 m vΒ².

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; πš” = mπœ”Β².

Time Period Expressions

  • Time period (T) can be expressed as T = 2πœ‹ √(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.