Exhaustive Physics Notes on Oscillations and Simple Harmonic Motion

Fundamental Concepts of Periodic and Oscillatory Motion

• Periodic Motion: A motion that repeats itself after a definite, regular, or fixed interval of time. It can occur along any path.

  • Examples of Periodic Motion:
    • Uniform Circular Motion or orbital motion.
    • The motion of a bouncing ball.
    • The motion of the hands of a clock.
    • Motion of a pendulum.
    • Motion of the moon revolving around the Earth.
    • Motion of planets in the solar system (orbital).

• Oscillatory Motion: A motion in which a body moves back and forth (also described as to-and-fro or up-and-down) repeatedly about a fixed point (mean position) in a definite time interval.

  • Examples of Oscillatory Motion:
    • Motion of a pendulum.
    • Spring-mass system.
    • Swinging a pendulum in a clock.
    • To-and-fro motion of atoms in a substance.
    • Vibration of strings in a guitar.

• Relationship Between Periodic and Oscillatory Motion:

  • All oscillatory motions (with no loss of energy) are periodic. Examples include the motion of a pendulum or a spring-mass system.
  • All periodic motions are NOT necessarily oscillatory. For example, uniform circular motion is periodic but not oscillatory because it does not move back and forth about a fixed point.
  • Rule: All oscillatory motions are periodic, but not all periodic motions are oscillatory.

Characteristics of Oscillatory Motion

• Restoring Force: There is a tendency to return to the equilibrium or mean position after being disturbed.
• Force Proportionality: The restoring force is governed by the equation $F = -K x^n$. The restoring force is always proportional to the displacement and directed towards the mean position.
• Energy Conservation: In an ideal oscillatory motion, total energy is conserved.
• Examples of Periodic but Non-Oscillatory Motion:
  • Motion of the seconds hand of a watch.
  • Motion of fan blades rotating with constant angular velocity $\omega$.
  • Uniform circular motion.
  • Orbital motion of planets.

Fundamental Parameters of Oscillation

• Time Period ($T$): The time taken for one complete oscillation or rotation.
  • In one complete rotation, the angular displacement $\theta = 2̀̀\pi$.
• Frequency ($f$): The number of oscillations completed in one second.
  • SI Unit: Hertz ($Hz$) or per second ($s^{-1}$).
• Angular Velocity / Angular Frequency ($\omega$):
  • Defined by the relationship: $\omega = \frac{\theta}{t} = \frac{2\pi}{T}$.
  • Relationship with frequency: $\omega = 2\pi f$.
  • Inversely, $f = \frac{1}{T}$.

Simple Harmonic Motion (S.H.M)

• Definition: When a particle moves to and fro in a straight line about its equilibrium position such that the force acting upon it is always directly proportional to its displacement and directed towards the mean position.
• Condition for S.H.M:
  • $F = -K x$
  • Where $F$ is the restoring force, $K$ is the spring constant, and $x$ is the displacement from the mean position.
• Key Rules of Restoring Force:
  • It is always directed towards the Mean Position.
  • It is maximum when displacement is maximum (at Amplitude).
  • If force is on the x-axis, $F = -Kx$. If on the y-axis, $F = -Ky$.
• SHM as Projection of Uniform Circular Motion: SHM can be defined as the motion of the projection of a particle on any diameter of a circle of reference.

Kinematics of SHM

• Displacement ($y$):
  • Represents the displacement at any instant of time $t$.
  • $y = A \sin(\theta) = A \sin(\omega t)$.
  • Amplitude ($A$): The maximum displacement from the mean or equilibrium position. When a particle reaches the extreme position, it undergoes maximum displacement ($\theta = 90^{\circ}$, so $y = A$).
• Velocity ($v$):
  • Derived by differentiating displacement: $v = \frac{dy}{dt} = \frac{d}{dt} (A \sin(\omega t)) = A \omega \cos(\omega t)$.
  • In terms of displacement: $v = \omega \sqrt{A^2 - y^2}$.
  • Maximum Velocity ($v_{max}$): Occurs at the mean position where $y = 0$. $v_{max} = \omega A$.
• Acceleration ($a$):
  • Derived by differentiating velocity: $a = \frac{dv}{dt} = \frac{d}{dt} (A \omega \cos(\omega t)) = -A \omega^2 \sin(\omega t)$.
  • In terms of displacement: $a = -\omega^2 y$.
  • Maximum Acceleration ($a_{max}$): Occurs at extreme positions where displacement is maximum ($A$). $a_{max} = \omega^2 A$.

Graphical and Tabular Representation of SHM

• Sine Function (Mean position start):
  • At Mean Position ($t = 0, \omega t = 0$): $y = 0$ (min), $v = A \omega$ (max), $a = 0$ (min).
  • At Extreme Position ($\omega t = \pi/2$ or $t = T/4$): $y = A$ (max), $v = 0$ (min), $a = -A \omega^2$ (max).
• Cosine Function (Extreme position start):
  • At start ($t = 0$): $x = A$, $v = 0$, $a = -\omega^2 A$.
  • At $t = T/4$: $x = 0$, $v = -\omega A$, $a = 0$.
  • At $t = T/2$: $x = -A$, $v = 0$, $a = \omega^2 A$.

Phase and Resultant SHM

• Phase: Expresses the position and direction of motion at a specific instant. It is expressed as an angle ($\theta$) or in terms of periodic time ($T$). If a particle is perpendicular to its initial position, its phase angle is $\pi/2$.
• Linear Combination of Sine and Cosine: When waves are superimposed, the resultant displacement is $z(t) = A \sin(\omega t) + B \cos(\omega t)$.
  • Resultant Amplitude ($D$): $D = \sqrt{A^2 + B^2}$.
  • Phase Angle ($\phi$): $\tan(\phi) = \frac{B}{A}$.
  • Resultant Equation: $z(t) = D \sin(\omega t + \phi)$.

Energy in Simple Harmonic Motion

• Potential Energy ($PE$):
  • Based on Hooke's Law and work done: $PE = \frac{1}{2} k x^2$.
  • In terms of time: $PE = \frac{1}{2} k A^2 \cos^2(\omega t)$.
• Kinetic Energy ($KE$):
  • Based on velocity: $KE = \frac{1}{2} m v^2$.
  • In terms of displacement: $KE = \frac{1}{2} k (A^2 - x^2)$.
  • In terms of time: $KE = \frac{1}{2} k A^2 \sin^2(\omega t)$.
• Total Energy ($TE$):
  • $TE = PE + KE = \frac{1}{2} k A^2$.
  • At the mean position, $KE$ is maximum and $PE$ is minimum ($0$).
  • At extreme positions, $PE$ is maximum and $KE$ is minimum ($0$).
  • $PE = KE$ at a distance of $x = \pm \frac{A}{\sqrt{2}}$ from the mean position.

Time Period Dynamics: Pendulums and Springs

• Simple Pendulum:
  • Time Period formula: $T = 2\pi \sqrt{\frac{L}{g}}$.
  • Seconds Pendulum: A pendulum with a time period specifically of $2\,s$.
  • Independence: The time period depends on length ($L$) and gravity ($g$), but not on the mass ($m$) of the bob.
• Mass-Spring System (Horizontal):
  • Time Period formula: $T = 2\pi \sqrt{\frac{m}{k}}$.
• Specific Scenarios:
  • Altitude: If a pendulum is taken to a higher altitude (e.g., a mountain or the moon), $g$ decreases, so $T$ increases and the pendulum slows down.
  • Center of Gravity Changes: If a person standing on a swing sits down, their center of gravity (C.G.) lowers, effective length ($L$) increases, and $T$ increases. Conversely, standing up reduces $L$ and decreases $T$.
  • Hollow Sphere with Draining Liquid: As water/mercury drains from the bottom, the C.G. first moves downward (increasing $L$ and $T$). As the liquid drains further and becomes nearly empty, the C.G. moves back upward to the center (decreasing $L$ and $T$).
  • Car Stability: For greater stability, a car should have a stiffer spring (higher spring constant $k$) to resist body roll and weight transfer during cornering.

Free Oscillations, Forced Oscillations, and Resonance

• Free Oscillation: A body vibrates with its own natural frequency without help from any external periodic force (e.g., a swing moving after a single push).
• Forced (Driven) Oscillation: A body vibrates under the influence of an external periodic force with a frequency ($\omega_d$) different from its natural frequency ($\omega$).
• Natural Frequency: The characteristic frequency at which a system vibrates when disturbed. It depends on size, shape, and material properties.
• Resonance:
  • The phenomenon where the amplitude of oscillation increases significantly because the frequency of the external driving force ($\omega_d$) is very close to or equal to the natural frequency of the body ($\omega$).
  • Conditions for Maximum Amplitude:
    1. $\omega = \omega_d$.
    2. Damping constant ($b$) should be minimum.
  • Real-world Implications:
    • Aircraft Design: Engine frequency must not match wing natural frequency to prevent violent flapping.
    • Soldiers on Bridges: Soldiers are ordered to "route step" (walk out of step) because marching in rhythm can match the bridge's natural frequency, causing resonance and possible collapse.
    • Earthquakes: Tectonic plate movement/vibrations are often explained through concepts like resonance and oscillatory motion.

Questions & Discussion

• Heartbeat Calculation: A patient's heartbeat period is $0.8\,s$. Express beats per minute.
  • $f = 1/T = 1 / 0.8 = 1.25\,s^{-1}$.
  • Beats per minute = $1.25 \times 60 = 75\,bpm$.
• Insect Wings: Moves wings up and down $144$ times in $3\,s$. Period ($T$) = ??
  • $f = 144 / 3 = 48\,Hz$.
  • $T = 1 / 48 \approx 0.0208\,s$.
• Resultant Amplitude Calculation: Given $x = 6 \cos(\omega t) + 8 \sin(\omega t)$.
  • $A = 8$; $B = 6$.
  • Resultant amplitude $D = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = 10\,m$.
• Acceleration-Displacement Relationship: For a particle in SHM, which relationship is true ($\beta$ is acceleration, $y$ is displacement)?
  • Relationship is $\beta = -\omega^2 y$. Therefore, $\beta = -5y$ is possible, while $\beta = 3x$ (positive) or $\beta = -50y^2$ (quadratic) are incorrect.
• Lab Practices: Recording timing for $20$ oscillations instead of one is done to minimize measurement errors and calculate a more accurate average time period, though the question notes a common misconception about the period varying.