PERIODIC AND OSCILLATORY MOTIONS-

  • Periodic motion- It is a type of motion which repeats itself at regular intervals of time.
  • When frequency is small, we call it oscillatory where frequency is large, we call it vibration. That means both are equal.
  • Oscillation or vibrations- It is repeated back and forth motion or to and fro motion around a central position.
  • Each oscillatory motion is periodic but every periodic motion may be or not oscillatory. Also circular motion is periodic but not oscillatory.
  • The simplest form of oscillatory motion is simple harmonic motion(SHM).
  • Simple harmonic motion- It is a type of periodic motion where an object moves back and forth around an equilibrium position, with a force proportional to its displacement.

FREQUENCY-

  • Periodic Motion- It repeat itself at regular time intervals.
  • Period or Time Period-
    • It is minimum time taken which it repeat or shortest time interval where cycle complete of waves or motion.
    • Its SI unit is second(s).
    • It is denoted by T.
    • The time period of quartz crystal vibrations is expressed in units as microsecond (µs), 10-6.
  • Frequency-
    • It is number of complete cycles or waves that occur in one second.
      • It is measure and SI unit is hertz (Hz) or s-1 on the name of Heinrich Rudolph Hertz(1857-1894).
    • It is denoted by v.
    • v = 1/T.

THE DISPLACEMENT-

  • The measurement of displacement become easy when we measure the displacement of body from its equilibrium position.
  • Displacement(D)-
    • It is distance of an object form its equilibrium position (the rest position) at any given moment.
    • It show hoe far the object has moved during its oscillatory motion.
    • Displacement in oscillation can be positive or negative, depending the direction of motion.
    • The displacement can represented as periodic function of time, but in periodic motion, periodic in time is that function. It is represented by-
      • f(t) = A cos ωt, ωt is argument of this function f(t) and if it increased by integral multiple of 2π radian, then value of function remain same.
      • T = 2π/ω
    • D = (A2 + B2)1/2
    • ɸ = tan-1(A/B)
  • Any periodic function can be expressed as a superfunction of sine and cosine functions of different time periods with suitable coefficient. This statement is proved by French mathematician, Jean Baptiste Joseph Fourier (1768-1830).

SIMPLE HARMONIC MOTION-

Simple Harmonic Motion (SHM): Key Notes and Concepts

Definition of SHM

  • A type of oscillatory motion where a particle moves back and forth along an axis about an equilibrium position.
  • The displacement x(t) varies sinusoidally with time: x(t)=Acos⁡(ωt+ϕ) where:
    • A: Amplitude (maximum displacement from the origin)
    • ω: Angular frequency
    • ϕ: Phase constant (initial phase at t=0)

Characteristics of SHM

  1. Amplitude (A):
    • Maximum displacement from the equilibrium position.
    • Always positive and constant for a given SHM.
  2. Angular Frequency (ωω):
    • Related to the time period T of motion: ω=2π/T​
    • Unit: radians per second.
    • Frequency f=1/T​, so ω=2πf.
  3. Phase (ωt+ϕ):
    • Indicates the state of motion at any given time.
    • Initial phase (ϕ) determines the starting position.
  4. Periodicity:
    • Motion repeats after a time T: x(t)=x(t+T)
    • The cosine function ensures periodic motion.

Displacement, Velocity, and Acceleration

  1. Displacement (x):x(t)=Acos⁡(ωt+ϕ)
  2. Velocity (v):
    • Derivative of x(t): v(t)=−ωAsin⁡(ωt+ϕ)
    • Maximum speed: ωA at equilibrium position.
  3. Acceleration (aaa):
    • Derivative of v(t): a(t)=−ω2Acos(ωt+ϕ)=−ω2x(t)
    • Acceleration is always directed towards the equilibrium position (restoring force).

Key Graphs in SHM

  • Displacement vs. Time: Sinusoidal, varying between −A- and A.
  • Velocity vs. Time: Sinusoidal, shifted by π/2 relative to displacement.
  • Acceleration vs. Time: Sinusoidal, shifted by π relative to displacement.

Relation Between SHM and Uniform Circular Motion

  • Uniform circular motion projected on a diameter corresponds to SHM.
    • Consider a particle moving in a circle with radius A and angular speed ω\omegaω:
      • Position projection: x(t)= Acos⁡(ωt+ϕ).
      • Velocity projection: v(t)= −ωAsin⁡(ωt+ϕ).
      • Acceleration projection: a(t)= −ω2Acos⁡(ωt+ϕ).

Example Problems

  1. Identify SHM:
    Given functions, check if they satisfy SHM: x(t)=Acos⁡(ωt+ϕ).
  2. Find Period and Frequency:
    • Use T=2π/ω​.
    • Frequency: f=1/T​.
  3. Velocity and Acceleration:
    • Find using derivatives of displacement or directly substitute into formulas.

Extra Insights

  • Phase Difference:
    • Different SHMs can have the same amplitude and frequency but differ by a phase constant.
    • Example: x1​= Acos (ωt) and x2= Acos (ωt+ϕ).
  • Energy in SHM:
    • Kinetic Energy: KE=1/2 ​mω2(A2−x2).
    • Potential Energy: PE= ½ mω2x2
    • Total Energy: E=1/2 ​mω2A2 (constant).

Key Formulas for Competitive Exams

  1. Displacement: x(t)= Acos⁡(ωt+ϕ).
  2. Velocity: v(t)= −ωAsin⁡(ωt+ϕ).
  3. Acceleration: a(t)= −ω2x(t).
  4. Time Period: T=2π/ ω ​.
  5. Frequency: f=ω/2π​.
  6. Relation between KE and PE: KE+PE=constant.

Force Law for Simple Harmonic Motion (SHM)

  1. Force in SHM:
    Using Newton’s Second Law, the force acting on a particle of mass m in SHM is:F(t)=ma=−mω2x(t)=−kx(t)F(t) = ma = -m\omega^2 x(t) = -kx(t)F(t)=ma=−mω2x(t)=−kx(t)
    • k=mω2: This is the force constant.
    • ω= (k​/m)1/2​​: Angular frequency.
  2. Restoring Force:
    The force is always directed toward the mean position, acting to restore the particle to equilibrium.
  3. Linear and Nonlinear Oscillators:
    • In SHM, F∝x(t), making it a linear harmonic oscillator.
    • If additional terms likex2, x3 appear in the force equation, the system becomes a nonlinear oscillator.
  4. Deriving Relationships:
    • SHM can be described by displacement x(t)= Acos(ωt+ϕ) or the force law F(t)=−kx(t).
    • Differentiating x(t) twice gives acceleration and force.
    • Integrating the force law twice recovers x(t).

Energy in SHM

  1. Kinetic Energy (KE): K= 1/2 ​mv2= 1/2 ​kA2sin2(ωt+ϕ)
    • KE is zero at the extreme positions (max displacement).
    • KE is maximum at the mean position (x=0x).
  2. Potential Energy (PE): U= 1/2 ​kx2= 1/2 ​kA2cos2(ωt+ϕ)
    • PE is zero at the mean position.
    • PE is maximum at the extreme positions.
  3. Total Energy (E):
    • The total mechanical energy is constant: E= K+U= ½ ​kA2
    • Energy oscillates between KE and PE but remains conserved.

Key Properties of SHM

  1. Period and Frequency:
    • Period (T): Time for one complete oscillation.T= 2π​/ω= 2π (m/k)1/2​​​​
    • Frequency (ν): Number of oscillations per second. ν=1/T​
  2. Velocity and Acceleration:
    • Velocity: v(t)= −ωAsin⁡(ωt+ϕ)
    • Acceleration: a(t)= −ω2x(t)= −ω2Acos(ωt+ϕ)
    • Both are periodic and phase-shifted from displacement.

Simple Pendulum

  1. System: A small bob of mass mmm attached to a massless string of length L.
  2. Restoring Torque:
    • τ=−mgLsin⁡θ
    • For small angles (sin⁡θ≈θ):
      • α=−(g/L)θ
  3. Time Period: T= 2π (L​/g)1/2
    • Independent of mass or amplitude (for small angles).

Examples

  1. Two Springs with Constant kkk:
    • A block of mass mmm attached to two springs has a net restoring force: F= −2kx
    • Time period :T= 2π (m/2k)1/2​​​​
  2. Energy Conservation:
    • Total energy at any displacement (x): E= K.E.+P.E.

Important Observations

  1. Periodic motion is not always SHM. SHM requires force proportional to displacement (F=−kx).
  2. Energy oscillation between KE and PE explains smooth motion transitions.
  3. SHM is a projection of circular motion, which helps understand phase relationships.
  4. Amplitude does not affect the period of SHM, but it influences the total energy.

Competitive Exam Tips

  1. Memorize formulas for ω, T, and energy expressions.
  2. Understand how energy conservation applies to oscillatory systems.
  3. Practice problems involving combinations of springs and pendulums.
  4. Recognize SHM in different physical systems (pendulum, springs, molecules).

This comprehensive understanding will help in solving problems and mastering SHM concepts for exams.

These all are the notes of chapter 13 in physics. And after some time you get important questions and NCERT solutions HERE. *#THANKS FOR VISITING, VISIT AGAIN#* 😊