Chapter 2: Oscillation Practice Flashcards

Periodic and Oscillatory Motion

Periodic Motion

• Definition: A motion which repeats itself over and over again after a fixed interval of time is called a periodic motion.
• Path: The motion can occur along any path.
• Condition: The period must be fixed for the motion to be considered periodic.
• Examples:
  • Earth revolving around the sun ($T = 1\,\text{year}$).
  • Rotation of Earth about its axis ($T = 1\,\text{day}$).
  • Motion of a bob of a simple pendulum swinging to and fro.
  • Motion of the hands of a clock.
  • Motion of electrons around the nucleus.
  • Motion of a spiral spring.
  • Motion of a needle in a sewing machine.
  • Vibration of a guitar string.
  • Vibration of a tuning fork.

Oscillatory Motion

• Definition: A periodic motion in which a body moves back and forth or to and fro, or up and down repeatedly about a fixed point (known as the mean position) in a definite interval of time.
• Path: The path taken by the body must be the same during the repetition.
• Examples:
  • Motion of a bob of a simple pendulum.
  • Motion of a spiral spring.
  • An alternating current.
  • Motion of a needle in a sewing machine.
  • Vibration of a guitar string.
  • Vibration of a tuning fork.

Key Relationship

• All oscillatory motions are periodic, but all periodic motions are not oscillatory.
• Case Study: The revolution of the Earth around the sun is periodic but not oscillatory because it does not move back and forth about a mean position. Conversely, the motion of a clock pendulum is both periodic and oscillatory.

Characteristics and Properties of Oscillatory Motion

• Equilibrium/Mean Position: If a body in a fixed position is disturbed, it has a natural tendency to return to its mean or equilibrium position.
• Restoring Force ($F_R$): The force responsible for oscillatory motion. It always acts towards the mean/equilibrium position.
• Relationship between Force and Displacement: The restoring force acting on the body is directly proportional to its displacement but is directed in the opposite direction.
  • Mathematically: $F_R \propto -x$ (where $x$ is displacement).
  • General form: $F_R = -kx^n$, where $n = 1, 3, 5, 7, 9, \dots$ (all odd numbers).
• Conservation of Energy: Total energy is always conserved in a system executing oscillatory motion.

Simple Harmonic Motion (SHM)

Definition

• Simple Harmonic Motion is a special type of oscillatory motion in which a particle moves to and fro repeatedly about a mean position under the influence of a restoring force.
• In SHM, acceleration ($a$) is directly proportional to the displacement ($x$) of the particle from the mean position and is directed toward that mean position (opposite to the displacement).
• It is considered the simplest form of oscillatory motion and occurs along a straight line.

Essential Points

• Not every oscillatory motion is SHM, but all SHM is oscillatory.
• The magnitude of the restoring force at any instant is directly proportional to the displacement at that instant, though they are oppositely directed.
• Equation: $F_R = -kx^1$.

Characteristics of SHM

1. The particle moves in one dimension.
2. The particle moves to and fro about a fixed mean position where the net force ($F_{net}$) is zero.
3. The net force is always directed towards the mean position.
4. The magnitude of the net force is proportional to the displacement from the mean position.

Classification of Oscillations

• Harmonic Oscillation: An oscillation that can be expressed as a single harmonic function (either sine or cosine).
  • Examples: $y = A \sin(\omega t)$ or $y = A \cos(\omega t)$.
• Non-harmonic Oscillation: A combination of two or more harmonic oscillations.
  • Example: $y = A \sin(\omega t) + B \sin(2\omega t)$.

Terms Related to SHM

• Time Period ($T$): The smallest time interval after which a periodic motion is repeated; the time taken to complete one full oscillation.
• Angular Velocity / Angular Frequency ($\omega$): The rate of change of angular displacement.
  • Calculation: $\omega = \frac{2\pi}{T}$.
  • Therefore: $T = \frac{2\pi}{\omega}$.
  • SI Unit: $\text{rad/sec}$.
• Frequency ($f$): The number of oscillations or vibrations made by a body in one second.
  • Calculation: $f = \frac{1}{T} = \frac{\omega}{2\pi}$.
  • Relationship: $T \times f = 1$.
  • SI Unit: $\text{per second } (s^{-1})$ or $\text{hertz (Hz)}$. $1\,\text{Hz} = 1\,\text{oscillation per second}$.
• Displacement ($x$ or $y$): The distance of the particle from its mean position at any instant. In general equations, $x$ is often used for cosine functions and $y$ for sine functions.
  • SI Unit: $\text{meter (m)}$.
• Amplitude ($A$ or $a$): The maximum displacement of the particle on either side of the mean position.
  • SI Unit: $\text{meter (m)}$.
• Phase: A physical quantity that completely expresses the position and direction of motion of the particle at any instant with respect to its mean position. Mathematically, it is the argument of the sine or cosine function: $\theta = (\omega t + \phi)$.
  • Initial Phase (Epoch): The phase at time $t = 0$. In $\theta = \omega t + \phi$, the epoch is $\phi$.
  • Same Phase: Two particles are in the same phase if their phase difference is an even integral multiple of $\pi$ or their path difference is an even integral multiple of $\frac{\lambda}{2}$.
  • Opposite Phase (Out of Phase): Occurs if the phase difference is an odd integral multiple of $\pi$ or the path difference is an odd integral multiple of $\frac{\lambda}{2}$.
  • Phase Difference (\Delta \phi): If $y_1 = a \sin(\omega t + \phi_1)$ and $y_2 = a \sin(\omega t + \phi_2)$, then $\Delta \phi = (\omega t + \phi_2) - (\omega t + \phi_1) = \phi_2 - \phi_1$.

Displacement in SHM

SHM as a Projection of Uniform Circular Motion

• SHM can be viewed as the one-dimensional projection of a particle moving with uniform circular motion onto a diameter of the reference circle.
• If a particle moves along a circle of radius $A$ with angular velocity $\omega$, its projection $N$ on a diameter moves to and fro in SHM.

Displacement Equations

1. Starting from Mean Position: When time is noted from the instant the particle is at the mean position ($t=0$, $y=0$), we use the sine function:
   • $y = A \sin(\omega t)$
2. Starting from Extreme Position: When time is noted from the extreme position ($t=0$, $x=A$), we use the cosine function:
   • $x = A \cos(\omega t)$
3. General Equation with Initial Phase:
   • $y = A \sin(\omega t \pm \phi)$
   • $x = A \cos(\omega t \pm \phi)$

Displacement-Time Values (for $y = A \sin(\omega t)$):

• $t = 0$: $y = 0$
• $t = \frac{T}{4}$: $y = A \sin\left(\frac{2\pi}{T} \times \frac{T}{4}\right) = A$
• $t = \frac{T}{2}$: $y = 0$
• $t = \frac{3T}{4}$: $y = -A$
• $t = T$: $y = 0$

Velocity and Acceleration in SHM

Velocity Equation

• Definition: The time rate of change of displacement.
• Derivation: Given $y = A \sin(\omega t)$, velocity $v = \frac{dy}{dt} = A \omega \cos(\omega t)$.
• In terms of Displacement: Using $\cos(\omega t) = \sqrt{1 - \sin^2(\omega t)}$ and substituting $\sin(\omega t) = \frac{y}{A}$:
  • $v = \omega \sqrt{A^2 - y^2}$
• Maximum Velocity: Occurs at the mean position ($y = 0$).
  • $v_{max} = \omega A$ or $v_{max} = 2\pi f A$.
  • Relationship: $v_{max} \propto f$.
• Minimum Velocity: Velocity is zero at the extreme positions ($y = \pm A$).

Acceleration Equation

• Definition: The time rate of change of velocity.
• Derivation: Given $v = A \omega \cos(\omega t)$, acceleration $a = \frac{dv}{dt} = -A \omega^2 \sin(\omega t)$.
• In terms of Displacement: Substituting $y = A \sin(\omega t)$:
  • $a = -\omega^2 y$
• Significance of Negative Sign: Acceleration and displacement are always in opposite directions; acceleration is always directed toward the mean position.
• Maximum Acceleration: Occurs at the extreme positions ($y = A$).
  • $a_{max} = -\omega^2 A$.
  • In terms of frequency: $a = 4\pi^2 f^2 A$.
  • Relationship: $a \propto f^2$.
• Minimum Acceleration: Zero at the mean position ($y = 0$).

Comparison Table: Mean vs. Extreme Position

Energy in Simple Harmonic Motion

Total Mechanical Energy

• A particle in SHM possesses both Potential Energy (due to displacement) and Kinetic Energy (due to velocity).
• Potential Energy (PE): Work done against the restoring force.
  • $PE = \frac{1}{2} m \omega^2 y^2$ or $PE = \frac{1}{2} m \omega^2 A^2 \sin^2(\omega t)$.
  • Max at extreme positions: $PE_{max} = \frac{1}{2} m \omega^2 A^2$.
  • Min at mean position: $PE = 0$.
• Kinetic Energy (KE):
  • $KE = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - y^2)$ or $KE = \frac{1}{2} m \omega^2 A^2 \cos^2(\omega t)$.
  • Max at mean position: $KE_{max} = \frac{1}{2} m \omega^2 A^2$.
  • Min at extreme positions: $KE = 0$.
• Total Energy (E):
  • $E = PE + KE = \frac{1}{2} m \omega^2 y^2 + \frac{1}{2} m \omega^2 (A^2 - y^2) = \frac{1}{2} m \omega^2 A^2$.
  • Invariance: Total energy is independent of both displacement and time. It remains constant throughout the oscillation.

Points to Remember

• At $y = \pm \frac{A}{\sqrt{2}}$, Potential Energy equals Kinetic Energy.
• The graph for PE and KE vs. displacement is parabolic.
• In terms of frequency: $E = 2\pi^2 m f^2 A^2$.
  • $E \propto f^2$ and $E \propto A^2$.

Simple Pendulum

Definition and Components

• Ideal Simple Pendulum: A heavy point-mass body suspended by a weightless, inextensible, and perfectly flexible string from a rigid support.
• Practical Simple Pendulum: A small metallic solid sphere (bob) suspended by a cotton thread.
• Effective Length ($l$): The distance from the point of suspension to the center of gravity of the bob ($l = \text{length of thread} + \text{radius of bob}$).

Derivation of Time Period

• When the bob is displaced by an angle $\theta$, gravity $mg$ is resolved into components:
  1. $mg \cos(\theta)$: Balances the tension $T$.
  2. $mg \sin(\theta)$: Acts as the restoring force.
• Restoring force $F = -mg \sin(\theta) \approx -mg \theta$ (for small angles).
• Since $\theta = \frac{x}{l}$, $F = -mg \left(\frac{x}{l}\right)$.
• Using Newton's Second Law ($F = ma$): $ma = -\frac{mgx}{l} \implies a = -\frac{gx}{l}$.
• Since $a \propto -x$, its motion is SHM.
• Time period $T = 2\pi \sqrt{\frac{\text{displacement}}{\text{acceleration}}} = 2\pi \sqrt{\frac{x}{gx/l}}$.
• Equation: $T = 2\pi \sqrt{\frac{l}{g}}$.

Factors Affecting Time Period

1. Length: $T \propto \sqrt{l}$. (If a girl standing on a swing sits down, the center of mass lowers, length increases, and $T$ increases).
2. Gravity: $T \propto \frac{1}{\sqrt{g}}$. (At higher altitudes, $g$ decreases, so $T$ increases).
3. Independence: Time period does not depend on the mass of the bob or the amplitude of oscillation.
4. Temperature: If the suspension is a wire, increasing temperature increases length, thus increasing $T$.

Pendulum in a Lift

• At rest/constant velocity: $T = 2\pi \sqrt{\frac{l}{g}}$.
• Accelerating upward ($a$): $T = 2\pi \sqrt{\frac{l}{g+a}}$ (period decreases).
• Accelerating downward ($a$): $T = 2\pi \sqrt{\frac{l}{g-a}}$ (period increases).
• Free fall ($a=g$): $T = \infty$ (no oscillation).

Horizontal Oscillation of a Mass-Spring System

System Setup

• A spring with force constant $k$ is attached to a mass $m$ on a frictionless horizontal surface.

Dynamics and Period

• Restoring force (Hooke\'s Law): $F = -kx$.
• Newton's Second Law: $F = ma \implies ma = -kx \implies a = -\frac{k}{m}x$.
• Acceleration is proportional to displacement, therefore it is SHM.
• Time period: $T = 2\pi \sqrt{\frac{m}{k}}$.

Points to Remember

• The period does not depend on gravity ($g$), the length of the spring, or amplitude.
• Harder springs (higher $k$) result in smaller time periods.
• Heavier masses result in larger time periods.
• Spring constant formula: $k = m \omega^2 = \frac{4\pi^2 m}{T^2}$.

Damped and Forced Oscillations

Un-damped Oscillation

• Oscillation where amplitude remains constant over time due to the absence of resistive forces. Total mechanical energy is constant.

Damped Oscillation

• Oscillation where amplitude and energy decrease exponentially over time due to resistive forces like friction or viscosity.
• Damping Force ($F_d$): $F_d = -bv$, where $b$ is the damping constant and $v$ is velocity.
• Differential Equation: $m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0$.
• Displacement: x = A e^{-bt/2m} \cos(\omega\' t).
• Amplitude: $R = A e^{-bt/2m}$. Amplitude becomes zero as $t \rightarrow \infty$.
• Energy: $E = \frac{1}{2} k A^2 e^{-bt/m}$.

Forced Oscillation

• Oscillation maintained by an external periodic driving force $F = F_o \cos(\omega_d t)$.
• Differential Equation: $m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = F_o \cos(\omega_d t)$.
• The system eventually vibrates with the driving frequency $\omega_d$ instead of its natural frequency $\omega$.
• Amplitude of forced oscillation: $A = \frac{F_o}{[m^2(\omega^2 - \omega_d^2)^2 + \omega_d^2 b^2]^{1/2}}$.

Resonance

Definition

• Resonance is a special type of forced oscillation that occurs when the frequency of the driving force is equal or very close to the natural frequency of the oscillator ($f_d = f$).
• This results in a sharp increase in the amplitude and velocity of the oscillation.

Condition for Resonance

• $\omega = \omega_d$ or $\frac{\omega}{\omega_d} = 1$.

Sharpness of Resonance

• The sharpness or flatness of the resonance peak depends on damping.
• Smaller damping results in a taller and narrower (sharper) resonance peak.

Examples of Resonance

• Pendulums on a String: If several pendulums of different lengths are hung from a rubber band and one is set in motion, others of the same length will oscillate with large amplitudes due to resonance.
• Tuning Fork: A vibrating fork near an air column of an appropriate length produces a loud sound at resonance.
• Suspension Bridges: Armies break step when crossing a bridge to prevent the frequency of their steps from matching the natural frequency of the bridge, which could cause collapse.

Questions & Discussion

Check Your Understanding Questions

1. Motion of Halley’s Comet around the sun: Periodic.
2. Motion of the pendulum of a wall clock: Oscillatory.
3. Motion of liquid in a U-tube when compressed and left: Oscillatory.
4. A human heart beats 75 times/min. Calculate frequency and period:
   • $f = \frac{75}{60} = 1.25\,\text{Hz}$
   • $T = \frac{1}{f} = 0.8\,\text{s}$
5. Particle takes 32s for 20 oscillations:
   • $T = \frac{32}{20} = 1.6\,\text{s}$
   • $f = \frac{20}{32} = 0.625\,\text{Hz}$
6. Net external force at equilibrium position: Zero.
7. Why must pendulum amplitude be small?: If angular displacement is large, the restoring force is no longer proportional to displacement, and the motion is not SHM.

Revision Questions

1. If $g$ decreases by $10\%$ on a hill, how should the pendulum length change to maintain accuracy?
2. A pendulum ($T = 4\,\text{s}$, $l = 4\,\text{m}$). What length is needed for 30 oscillations in 1 minute?
3. Will a pendulum clock and a spring clock taken to the moon show accurate time? (Reason: Pendulum depends on $g$, spring does not).
4. A $10\,\text{g}$ body has velocity $6\,\text{cm/s}$ after 1s starting from mean. If $T = 6\,\text{s}$, find KE, PE, and Total Energy.
5. Two identical pendulums with amplitudes $2\,\text{cm}$ and $6\,\text{cm}$. Calculate the ratio of their energies ($E \propto A^2$).
6. PE is $2.5\,\text{J}$ when displacement is half the amplitude. What is Total Energy?
7. Displacement $x = 3 \sin(314t) + 4 \cos(314t)$. Find amplitude, phase angle, and frequency.
8. Pendulum ($T = 2\,\text{s}$, $A = 5\,\text{cm}$). Write equation and find displacement/velocity after 1.5s starting from mean.
9. A spring balance reads $0-50\,\text{kg}$ over $20\,\text{cm}$. A body oscillates with $T = 0.60\,\text{s}$. Find the weight of the body.
10. SHM with $T = 3\,\text{s}$. After what interval from $t = 0$ is displacement half of the amplitude?