Comprehensive Study Notes on Simple Harmonic Motion

Overview of Oscillatory Motion

• Definition of Oscillatory Motion: Oscillatory motion is defined as a back-and-forth ("to and fro") motion around a mean position. It is a periodic motion, meaning it repeats itself after equal intervals of time.
• The Mean Position: All bodies undergoing vibrational or oscillatory motion have an equilibrium position or mean position.
• Restoring Force: When a body is displaced from its mean position, a restoring force acts upon it to bring it back toward the equilibrium position, which sustains the oscillation.
• Natural Examples of Periodic and Oscillatory Motion:
  • Celestial Motion: The motion of the Earth around the Sun and the Moon around the Earth.
  • Waves: Water waves and sound waves.
  • Everyday Objects: A rocking chair, a swing, and a tuning fork.
  • Microscopic Level: Atoms or molecules in solid substances oscillating about their mean positions.
  • Biology: The wings of birds during flight.
  • Musical Instruments: Strings on guitars or violins vibrating to produce music.
  • Sound Propagation: Air molecules oscillating as sound waves travel through the medium.

Terms Related to Oscillatory Motion

• Vibration: Defined as one complete round trip or cycle of a vibrating body about its mean position. Alternatively, it is the motion from one extreme position to the other and back to the starting extreme position, crossing the mean position twice.
  • Example: For a simple pendulum, the path from position A to B and back to A constitutes one vibration.
• Instantaneous Displacement ($x$): The distance of a vibrating body from its mean position at any specific instant.
• Amplitude ($x_o$): The magnitude of the maximum displacement of the vibrating body on either side of its mean position.
  • Energy Connection: The amplitude of a wave measures the energy it carries. The greater the amplitude, the greater the energy, and vice versa.
• Time Period ($T$): The time taken to complete one full vibration or cycle.
  • SI Unit: Second ($s$).
  • Mathematical Relations: $T = \frac{1}{f}$ and T = \frac{2\times\text{\pi}}{\omega}.
• Frequency ($f$): The number of vibrations or oscillations ($n$) completed by a body in one second.
  • Mathematical Formula: $f = \frac{n}{t}$.
  • SI Unit: Hertz ($Hz$), where $1\,Hz$ is one oscillation per second.
  • Dimension: $[T^{-1}]$.
• Angular Frequency ($\omega$): The angular displacement per unit time.
  • Mathematical Formula: $\omega = \frac{\theta}{t}$. For one full revolution, \omega = \frac{2\times\text{\pi}}{T}, which results in \omega = 2\times\text{\pi}\times f.
  • SI Unit: $rad\,s^{-1}$.
  • Dimension: $[T^{-1}]$.

Simple Harmonic Motion (S.H.M)

• Definition: A specific type of oscillation produced under the action of a restoring force.
• Necessary Conditions and Characteristics:
  1. The restoring force must be directly proportional to the displacement from the mean position ($F \boldsymbol{\propto} -x$).
  2. The restoring force must be proportional to the inertia of the body.
  3. The system must follow Hooke's Law: $F = -k\times x$, where $k$ is the spring constant.
  4. Acceleration ($a$) must be proportional to displacement ($a \boldsymbol{\propto} -x$).
  5. The negative sign indicates that acceleration is always directed towards the mean position.
• Note: Not all periodic vibrations are S.H.M. For example, an electrocardiogram (ECG) trace is periodic, but the recorder needle motion is not S.H.M because the restoring force is not proportional to displacement.

Practical S.H.M Systems: Mass Attached to a Spring

• System Setup: A body of mass $m$ is attached to a spring with spring constant $k$ on a smooth horizontal surface.
• Mechanism:
  • Displacing the mass by $x$ creates an applied force $F = k\times x$.
  • Simultaneously, the spring exerts an elastic restoring force equal and opposite to the applied force.
  • When released, inertia causes the body to cross the mean position and oscillate between extreme points A and B.
• Mathematical Analysis:
  • According to Newton's Second Law: $F = m\times a$.
  • Equating forces: $m\times a = -k\times x$.
  • Solving for acceleration: $a = -(\frac{k}{m})\times x$.
  • Since $\frac{k}{m}$ is constant, $a \boldsymbol{\propto} -x$, proving the motion is S.H.M.
• Derived Physics Parameters:
  • Angular Frequency: \omega = \n\sqrt{\frac{k}{m}}.
  • Frequency: f = \frac{1}{2\times\text{\pi}}\times\sqrt{\frac{k}{m}}.
  • Time Period: T = 2\times\text{\pi}\times\sqrt{\frac{m}{k}}.

Practical S.H.M Systems: Simple Pendulum

• Structure: Consists of a small heavy solid bob of mass $m$ suspended by a light, inextensible string of length $l$ from a rigid support.
• Forces at Extreme Position:
  • The weight $m\times g$ is resolved into components: $m\times g\times\cos(\theta)$ and $m\times g\times\sin(\theta)$.
  • The component $m\times g\times\cos(\theta)$ is balanced by the tension $T$ in the string.
  • The component $m\times g\times\sin(\theta)$ provides the restoring force: $F = -m\times g\times\sin(\theta)$.
• Derivation for Small Angles:
  • If the angle $\theta$ is small (less than $10^{\circ}$), $\sin(\theta) \boldsymbol{\approx} \theta$ (in radians).
  • From geometry, $\theta = \frac{x}{l}$, where $x$ is the displacement (arc length).
  • $m\times a = -m\times g\times(\frac{x}{l}) \Rightarrow a = -(\frac{g}{l})\times x$.
  • Since $a \boldsymbol{\propto} -x$, the motion is S.H.M.
• Derived Physics Parameters:
  • Angular Frequency: \omega = \n\sqrt{\frac{g}{l}}.
  • Time Period: T = 2\times\text{\pi}\times\sqrt{\frac{l}{g}}.
• Key Dependencies: The time period of a simple pendulum is independent of the mass of the bob and the amplitude, provided the angle is small.
• Second Pendulum: A pendulum with a time period of exactly $2\,s$.
  • Calculation for length at $g = 9.8\,m/s^2$: l = \frac{g\times T^2}{4\times\text{\pi}^2} \boldsymbol{\approx} 0.994\,m (or $99.4\,cm$).

S.H.M and Uniform Circular Motion

• The Projection Principle: When a particle moves in a circular path with uniform angular speed, its projection on the diameter of the circle executes S.H.M.
• Quantitative Analysis (Radius $x_o$):
  • Instantaneous Displacement: $x = x_{o}\times\cos(\theta) = x_{o}\times\cos(\omega\times t)$.
  • Instantaneous Velocity ($v$): The velocity of the projection is the horizontal component of the tangential velocity $v_p = r\times \omega$.
    • v = \n\omega\times\sqrt{x_{o}^2 - x^2}.
    • At mean position ($x = 0$), velocity is maximum: $v_{max} = x_{o}\times\omega$.
    • At extreme position ($x = x_o$), velocity is zero.
  • Instantaneous Acceleration ($a$): The acceleration of the projection is the horizontal component of centripetal acceleration $a_c = -x_{o}\times\omega^2$.
    • $a = -\omega^2\times x$.
    • At mean position ($x=0$), acceleration is zero.
    • At extreme positions ($x = x_o$), acceleration is maximum: $a_{max} = -x_{o}\times\omega^2$.

Phase in S.H.M

• Definition: Phase is the angle \theta = \n\omega\times t + \n\phi that specifies the displacement and the direction of motion of an oscillator.
• General Equation: x = x_{o}\times\cos(\omega\times t + \n\phi), where $\phi$ is the phase constant or initial phase angle.
• Phase Differences:
  • In Phase: Phase difference of $0^{\circ}$ or $360^{\circ}$.
  • Out of Phase: Phase difference of $180^{\circ}$.
• Relationships between Parameters:
  • The phase difference between velocity and displacement is \frac{\text{\pi}}{2}.
  • The phase difference between acceleration and displacement is \text{\pi}.
  • The phase difference between acceleration and velocity is \frac{\text{\pi}}{2}.

Conservation of Energy in S.H.M

• Total Energy ($TE$): The sum of Kinetic Energy ($KE$) and Potential Energy ($PE$). In the absence of friction, $TE$ remains constant.
• Potential Energy ($PE$):
  • Formula: $PE = \frac{1}{2}\times k\times x^2$.
  • Maximum at extreme positions ($x = \boldsymbol{\pm} x_o$) and zero at the mean position ($x = 0$).
• Kinetic Energy ($KE$):
  • Formula: $KE = \frac{1}{2}\times k\times (x_{o}^2 - x^2)$.
  • Maximum at the mean position ($x = 0$) and zero at extreme positions ($x = \boldsymbol{\pm} x_o$).
• Total Energy Expression: $TE = \frac{1}{2}\times k\times x_{o}^2$ or $TE = \frac{1}{2}\times m\times a_{max}\times x_{o}$.
• Energy Oscillation: Energy continuously converts between $KE$ and $PE$ during oscillation, but the total sum is directly proportional to the square of the amplitude ($TE \boldsymbol{\propto} x_{o}^2$).

Free, Forced, and Damped Oscillations

• Free Oscillations: A body oscillates with its natural frequency without external interference (e.g., a simple pendulum slightly displaced). Total energy remains constant in ideal conditions.
• Forced Oscillations: A system subjected to an external periodic force (e.g., repeatedly striking a swing). The resulting frequency is called the driving frequency.
• Damped Oscillations: Oscillations where the amplitude decreases over time due to resistive forces (friction, air resistance).
  • Light Damping: Amplitude reduces gradually (e.g., a swing in a playground).
  • Heavy Damping: The body takes a long time to return to rest (e.g., a pendulum in thick oil).
  • Critical Damping: The object returns to equilibrium in the shortest possible time (e.g., car shock absorbers).

Resonance

• Definition: A phenomenon where the driving frequency of an external force matches the natural frequency of an oscillating body, resulting in a large increase in amplitude.
• Mathematical Condition: Resonance occurs at \text{f}_r = \n\text{n}\times \text{f}_o, where $\text{f}_o$ is the natural frequency and $n = 1, 2, 3...$.
• Sharpness of Resonance: Depends on damping. Smaller damping leads to sharper resonance and higher amplitude. Heavy damping results in a flatter resonance curve.
• Useful Applications:
  • Microwave Oven: Uses electromagnetic waves ($3\,GHz - 30\,GHz$) to excite water molecules in food. Friction from resulting oscillations produces heat.
  • Radio Tuning: Turning a knob changes the natural frequency of the electrical circuit to match a station's transmission frequency.
  • Magnetic Resonance Imaging (MRI): Strong radio frequencies cause nuclei to oscillate; energy absorption patterns are used for medical imaging.
  • Musical Instruments: Soundboards in pianos, violins, and guitars resonate with strings to amplify sound.
• Dangers/Circumstances to Avoid:
  • Bridges: Soldiers break step to avoid matching the bridge’s natural frequency. The Tacoma Narrows Bridge (1940) collapsed due to wind-induced resonant vibrations.
  • Aeroplane Wings: Must be designed so engine vibration or turbulence frequencies do not match the wings' natural frequency.

Resonance and Standing Waves

• Rubens Tube: A tube with holes and flammable gas that demonstrates acoustic standing waves. Sound waves create pressure nodes and antinodes, visualized by varying flame heights.
• Chladni Plate: A metal plate vibrated to show normal modes. Sand sprinkled on the plate moves from antinodes to stationary nodes, forming intricate geometric patterns.
• Acoustic Levitation: High-frequency ultrasonic waves create a standing wave between a transducer and reflector. Small particles (like water droplets or styrofoam) are trapped in the low-pressure nodes against gravity.

Questions & Discussion

• Why does a vibrating simple pendulum not produce sound? All vibrating bodies produce sound, but a standard pendulum's frequency is too low (below $20\,Hz$) for the human ear to perceive.
• Will the time period change if shift from Lahore to Karachi? It may change slightly if there is a variation in the local value of $g$.
• Would the time period be the same on Earth and moon? No, because the acceleration due to gravity ($g$) is different on the Moon compared to Earth, and $T \boldsymbol{\propto} \frac{1}{\sqrt{g}}$.
• Why use small amplitude for measuring time period? Using small amplitudes ensures the small-angle approximation ($\sin(\theta) \boldsymbol{\approx} \theta$) holds true, keeping the motion purely simple harmonic.
• Do marching troops break steps on bridges? Yes, to prevent the frequency of their steps from reaching the resonant frequency of the bridge structure, which could cause collapse.