Physics: Periodic Motion

Introduction to Periodic Motion

1. Definition:

   • Periodic motion is a type of motion that repeats itself at regular intervals of time.
   • Examples:
     • Motion of a pendulum.
     • Oscillations of a spring.
     • Revolution of the Earth around the Sun.

2. Key Parameters:

   • Time Period ($T$):
     • The time taken for one complete cycle of motion.
     • SI Unit: Seconds ($s$).
   • Frequency ($f$):
     • The number of cycles completed in one second.
     • Relation to time period: $f = \frac{1}{T}$
     • SI Unit: Hertz ($Hz$).
   • Angular Frequency ($\omega$):
     • The rate of change of angular displacement.
     • Relation to frequency: $\omega = 2\pi f = \frac{2\pi}{T}$
     • SI Unit: Radians per second ($rad/s$).

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Types of Periodic Motion

1. Oscillatory Motion:

   • A type of periodic motion in which an object moves back and forth about a mean position.
   • Example: Simple pendulum.

2. Rotational Motion:

   • A type of periodic motion in which an object rotates about a fixed axis.
   • Example: Spinning wheel.

3. Simple Harmonic Motion (SHM):

   • A special type of oscillatory motion where the restoring force is directly proportional to displacement and acts in the opposite direction.
   • Equation of motion: $F = -kx \quad \text{or} \quad a = -\omega^2 x$

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Simple Harmonic Motion (SHM)

1. Characteristics:

   • The motion is sinusoidal in nature.
   • The restoring force $F$ is proportional to displacement $x$.

2. Equation of SHM:

   • Displacement as a function of time: $x(t) = A \cos(\omega t + \phi)$
     • $A$: Amplitude.
     • $\omega$: Angular frequency.
     • $\phi$: Phase constant.

3. Velocity in SHM:

   • Velocity is the time derivative of displacement: $v(t) = -A \omega \sin(\omega t + \phi)$
   • Maximum velocity: $v_{\text{max}} = A \omega$

4. Acceleration in SHM:

   • Acceleration is the time derivative of velocity: $a(t) = -A \omega^2 \cos(\omega t + \phi)$
   • Maximum acceleration: $a_{\text{max}} = A \omega^2$

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Energy in SHM

1. Kinetic Energy (KE):

   • Energy due to motion: $KE = \frac{1}{2} mv^2$
     • Substituting $v$: $KE = \frac{1}{2} m \omega^2 (A^2 - x^2)$

2. Potential Energy (PE):

   • Energy due to position: $PE = \frac{1}{2} k x^2$

3. Total Energy (TE):

   • Sum of kinetic and potential energy: $TE = \frac{1}{2} k A^2$
   • Remains constant throughout the motion.

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Numerical Example

1. Example 1: A body of mass $2 \, kg$ is executing SHM with an amplitude of $0.1 \, m$ and angular frequency $5 \, rad/s$. Find the total energy of the system.

   • Total Energy: $TE = \frac{1}{2} m \omega^2 A^2$
   • Substituting values: $TE = \frac{1}{2} \cdot 2 \cdot (5)^2 \cdot (0.1)^2$ $TE = 1 \cdot 25 \cdot 0.01 = 0.25 \, J$

2. Example 2: A particle in SHM has a maximum velocity of $3 \, m/s$ and an angular frequency of $2 \, rad/s$. Find its amplitude.

   • Formula for maximum velocity: $v_{\text{max}} = A \omega$
     • Solving for $A$: $A = \frac{v_{\text{max}}}{\omega} = \frac{3}{2} = 1.5 \, m$

Examples of Simple Harmonic Motion (SHM)

1. Spring-Mass System:

   • A mass attached to a spring exhibits SHM when displaced from its equilibrium position.
   • Time Period: $T = 2\pi \sqrt{\frac{m}{k}}$
     • $m$: Mass of the object.
     • $k$: Spring constant.
   • Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$

2. Simple Pendulum:

   • A pendulum consisting of a small bob of mass $m$ suspended by a string of length $L$ performs SHM for small angular displacements.
   • Time Period: $T = 2\pi \sqrt{\frac{L}{g}}$
     • $L$: Length of the pendulum.
     • $g$: Acceleration due to gravity.
   • Angular Frequency: $\omega = \sqrt{\frac{g}{L}}$

3. Oscillations of a Liquid in a U-Tube:

   • The liquid column oscillates when displaced from its equilibrium level.
   • Time Period: $T = 2\pi \sqrt{\frac{l}{2g}}$
     • $l$: Length of the liquid column.

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Damped Harmonic Motion

1. Definition:

   • When the amplitude of oscillation decreases over time due to resistive forces (like friction or air resistance), the motion is called damped harmonic motion.

2. Types of Damping:

   • Under-damped: Oscillations occur with gradually decreasing amplitude.
   • Critically Damped: The system returns to equilibrium without oscillating.
   • Over-damped: The system returns to equilibrium slower than the critically damped case.

3. Equation of Motion: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$

   • $b$: Damping coefficient.

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Forced Oscillations and Resonance

1. Forced Oscillations:

   • When a periodic external force acts on a system, it is called forced oscillation.
   • Equation of motion: $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0 \cos(\omega t)$
     • $F_0$: Amplitude of the external force.
     • $\omega$: Angular frequency of the external force.

2. Resonance:

   • When the frequency of the external force matches the natural frequency of the system, the amplitude of oscillation becomes maximum.
   • Example: Swing pushed at regular intervals matching its natural frequency.

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Energy Dissipation in Damped Motion

1. Energy Loss:
   • In damped motion, energy is dissipated over time due to resistive forces.
   • The total energy decreases exponentially: $E = E_0 e^{-bt/m}$
     • $E_0$: Initial energy.
     • $b$: Damping coefficient.
     • $m$: Mass.

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Practical Applications of SHM

1. Clocks:

   • Pendulum clocks use the periodic motion of a pendulum for timekeeping.

2. Seismographs:

   • Used to measure ground motion during earthquakes, based on the oscillation of a mass.

3. Tuning Forks:

   • Vibrations of a tuning fork produce sound waves with a specific frequency.

4. Shock Absorbers:

   • Use damped oscillations to minimize vibrations in vehicles.

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Numerical Examples

1. Example 1: A spring with a constant $k = 200 \, N/m$ is attached to a $5 \, kg$ mass. Find the time period of the oscillations.

   • Formula: $T = 2\pi \sqrt{\frac{m}{k}}$
   • Substituting values: $T = 2\pi \sqrt{\frac{5}{200}} = 2\pi \sqrt{0.025} = 2\pi \cdot 0.158 \approx 0.99 \, s$

2. Example 2: A pendulum of length $1.5 \, m$ is oscillating. Find its time period. (Take $g = 9.8 \, m/s^2$).

   • Formula: $T = 2\pi \sqrt{\frac{L}{g}}$
   • Substituting values: $T = 2\pi \sqrt{\frac{1.5}{9.8}} = 2\pi \cdot 0.39 \approx 2.45 \, s$

Wave Motion and SHM

1. Relation Between SHM and Wave Motion:
   • SHM is the basis of wave motion, where particles oscillate about their equilibrium position in a periodic manner.
   • A sinusoidal wave can be represented as: $y(x, t) = A \sin(kx - \omega t + \phi)$
     • $A$: Amplitude.
     • $k$: Wave number ($k = \frac{2\pi}{\lambda}$).
     • $\omega$: Angular frequency.
     • $\phi$: Phase constant.
     • $\lambda$: Wavelength.

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Combined SHM

1. Superposition of Two SHMs:

   • When two SHMs of the same frequency are combined: $x(t) = A_1 \cos(\omega t + \phi_1) + A_2 \cos(\omega t + \phi_2)$
   • The resultant amplitude is: $A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi_2 - \phi_1)}$

2. Lissajous Figures:

   • When two SHMs act perpendicular to each other, their superposition can create patterns known as Lissajous figures.

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Practical Applications of Periodic Motion

1. Mechanical Systems:

   • Springs and pendulums in timekeeping devices (e.g., clocks).
   • Vibration isolation in machinery using dampers.

2. Acoustic Applications:

   • Tuning forks for generating sound waves.
   • Resonance in musical instruments.

3. Electromagnetic Applications:

   • Oscillations in LC circuits for generating radio waves.
   • Resonance in antennas for signal reception.

4. Medical Applications:

   • Ultrasonic imaging uses wave motion for diagnostic purposes.

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Damped Oscillations in Real Life

1. Car Suspension:

   • Shock absorbers in vehicles dampen oscillations for a smoother ride.

2. Bridge Vibrations:

   • Damping is used to minimize oscillations caused by wind or traffic.

3. Earthquake Engineering:

   • Seismic dampers in buildings reduce oscillations during earthquakes.

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Resonance in Practical Situations

1. Musical Instruments:

   • String instruments use resonance to amplify sound.

2. Structural Engineering:

   • Resonance can cause structural failures (e.g., Tacoma Narrows Bridge collapse).
   • Structures are designed to avoid natural frequencies matching external vibrations.

3. Microwave Ovens:

   • Utilize resonance to heat water molecules efficiently.

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Numerical Examples

1. Example 1: Two SHMs with amplitudes $3 \, cm$ and $4 \, cm$ are superposed at a phase difference of $90^\circ$. Find the resultant amplitude.

   • Formula: $A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi)}$
   • Substituting values: $A = \sqrt{3^2 + 4^2 + 2 \cdot 3 \cdot 4 \cdot \cos 90^\circ}$ $A = \sqrt{9 + 16 + 0} = \sqrt{25} = 5 \, cm$

2. Example 2: A pendulum with a length of $2.25 \, m$ oscillates with a period of $3 \, s$. Find the acceleration due to gravity.

   • Formula: $T = 2\pi \sqrt{\frac{L}{g}} \implies g = \frac{4\pi^2L}{T^2}$
   • Substituting values: $g = \frac{4\pi^2 \cdot 2.25}{3^2} = \frac{88.2}{9} \approx 9.8 \, m/s^2$

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Recap: Key Points to Remember

• Periodic motion repeats after regular intervals, with SHM being a key example.
• Damped and forced oscillations are extensions of SHM, with real-life applications in vehicles, structures, and electronics.
• Resonance occurs when the natural frequency of a system matches an external periodic force, leading to large oscillations.