11.4 Simple Harmonic Motion (SHM), Pendulum, Mass-Spring & Resonance

Simple Harmonic Motion (SHM), Pendulum, Mass-Spring & Resonance

Oscillatory and vibrational mechanics, detailed in of the Intermediate Physics Part I syllabus, represent one of the most conceptually rich and mathematically tested areas of the UET ECAT. Simple Harmonic Motion (SHM) describes periodic back-and-forth motion governed by a linear restoring force directed toward a stable equilibrium state. In engineering entrance exams, problems frequently test analytical phase differences, instantaneous kinetic and potential energy partitions, spring combinations, effective pendulum lengths across varying gravitational fields, and resonance phenomena. Developing rapid algebraic fluency with SHM formulas without relying on numerical calculators is essential for exam pacing.

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1. Kinematics and Fundamental Definition of SHM

The Fundamental Dynamical Definition

A mechanical system undergoes Simple Harmonic Motion (SHM) if and only if its acceleration $a$ is directly proportional to its displacement $x$ from the mean equilibrium position and is always directed toward that mean position:

$a \propto -x \implies a = -\omega^2 x$

Where:

• $x$ is the instantaneous displacement from equilibrium (meters, $\text{m}$).
• $\omega$ is the circular angular frequency of oscillation (radians per second, $\text{rad}\cdot\text{s}^{-1}$).
• The negative sign indicates that acceleration is invariably directed opposite to displacement—namely, pointing back toward $x = 0$.

By Newton's second law ($F = m a$):

$F = m(-\omega^2 x) = -m\omega^2 x = -k x$

Where $k = m\omega^2$ is the restoring spring stiffness constant ($\text{N}\cdot\text{m}^{-1}$), confirming Hooke's law as the dynamical origin of SHM.

Equations of Kinematics in SHM

Representing SHM as the projection of uniform circular motion of reference radius $x_0$ (amplitude) onto a diameter yields the standard kinematic wave functions:

1. Instantaneous Displacement ($x$):

$x(t) = x_0 \sin(\omega t + \phi)$

If timing starts at the mean position ($x=0$ at $t=0$), initial phase $\phi = 0 \implies x(t) = x_0 \sin(\omega t)$. If timing starts at maximum positive amplitude ($x = x_0$ at $t=0$), $\phi = \frac{\pi}{2} \implies x(t) = x_0 \cos(\omega t)$.

1. Instantaneous Velocity ($v$): Differentiating displacement with respect to time:

$v(t) = \frac{dx}{dt} = \omega x_0 \cos(\omega t + \phi)$

Using the identity $\cos\theta = \pm\sqrt{1 - \sin^2\theta} = \pm\sqrt{1 - (x/x_0)^2}$:

$v(x) = \pm \omega \sqrt{x_0^2 - x^2}$

• At the mean position ($x = 0$): Speed attains its maximum value: $v_{\text{max}} = \omega x_0$.
• At the extreme positions ($x = \pm x_0$): Speed drops to zero: $v = 0$.

1. Instantaneous Acceleration ($a$): Differentiating velocity with respect to time:

$a(t) = \frac{dv}{dt} = -\omega^2 x_0 \sin(\omega t + \phi) = -\omega^2 x$

• At the mean position ($x = 0$): Acceleration is zero: $a = 0$.
• At the extreme positions ($x = \pm x_0$): Acceleration attains maximum magnitude: $a_{\text{max}} = \omega^2 x_0$.

Phase Relationships Between Kinematic Variables

A central focus of ECAT multiple-choice questions is the phase angle difference between SHM kinematic variables:

• Displacement vs Velocity: Velocity leads displacement by a phase angle of $\frac{\pi}{2}\text{ rad}$ ($90^\circ$).
• Velocity vs Acceleration: Acceleration leads velocity by a phase angle of $\frac{\pi}{2}\text{ rad}$ ($90^\circ$).
• Displacement vs Acceleration: Acceleration and displacement are in exact phase opposition, differing by a phase angle of $\pi\text{ rad}$ ($180^\circ$).

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2. Mass-Spring Systems

Horizontal and Vertical Oscillators

Consider an ideal block of mass $m$ attached to a massless spring of stiffness constant $k$.

• Restoring Force: $F = -k x$.
• Equating to Newton's law: $m a = -k x \implies a = -\left(\frac{k}{m}\right)x$.

Comparing with $a = -\omega^2 x$ confirms that the angular frequency is:

$\omega = \sqrt{\frac{k}{m}}$

From $\omega = 2\pi f = \frac{2\pi}{T}$, the time period $T$ and frequency $f$ are:

$T = 2\pi \sqrt{\frac{m}{k}}, \qquad f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Crucial ECAT Observations:

• The time period depends solely on inertial mass $m$ and elastic stiffness $k$.
• $T$ is completely independent of amplitude $x_0$ (isochronous property).
• For a vertical mass-spring system, gravity stretches the spring to a new equilibrium extension $x_{\text{eq}} = \frac{mg}{k}$. The period about this new equilibrium remains completely unchanged: $T = 2\pi\sqrt{\frac{m}{k}} = 2\pi\sqrt{\frac{x_{\text{eq}}}{g}}$.

Combination of Springs

When multiple springs are coupled together, their effective spring constant $k_{\text{eq}}$ dictates the resulting time period:

1. Series Combination (End-to-End):
   • Each spring experiences the identical tension force $F$.
   • Total displacement is the sum of individual extensions: $x_{\text{total}} = x_1 + x_2$.
   • Since $x = F/k$, $\frac{F}{k_{\text{eq}}} = \frac{F}{k_1} + \frac{F}{k_2}$:

$\frac{1}{k_{\text{eq}}} = \frac{1}{k_1} + \frac{1}{k_2} \implies k_{\text{eq}} = \frac{k_1 k_2}{k_1 + k_2}$

For two identical springs ($k_1 = k_2 = k$):

$k_{\text{eq}} = \frac{k}{2} \implies T_{\text{series}} = 2\pi\sqrt{\frac{m}{k/2}} = \sqrt{2} T_0$

1. Parallel Combination (Side-by-Side):
   • Both springs undergo identical displacement $x$.
   • Total restoring force is the sum of individual spring forces: $F_{\text{total}} = F_1 + F_2 = k_1 x + k_2 x$:

$k_{\text{eq}} = k_1 + k_2$

For two identical springs ($k_1 = k_2 = k$):

$k_{\text{eq}} = 2k \implies T_{\text{parallel}} = 2\pi\sqrt{\frac{m}{2k}} = \frac{T_0}{\sqrt{2}}$

1. Cutting a Spring:
   • The spring constant of a uniform spring is inversely proportional to its resting length: $k \propto \frac{1}{L} \implies k L = \text{constant}$.
   • If a spring of stiffness $k$ and length $L$ is cut into $n$ equal segments, each segment has length $L' = L/n$, so its new spring constant is:

$k' = n k$

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3. The Simple Pendulum

Dynamical Derivation for Small Angular Displacements

A simple pendulum consists of a small heavy bob of mass $m$ suspended by a light, inextensible string of length $l$. Displacing the bob by a small angle $\theta$ from the vertical:

• Weight $mg$ resolves into two components: $mg\cos\theta$ along the string (balanced by string tension $T_{\text{string}}$) and $mg\sin\theta$ perpendicular to the string, acting as the restoring force toward equilibrium:

$F_{\text{restoring}} = -m g \sin\theta$

For small angular amplitudes (typically $\theta \le 10^\circ \approx 0.175\text{ rad}$), the small-angle approximation $\sin\theta \approx \theta$ applies. Expressing $\theta = \frac{x}{l}$ (where $x$ is linear arc displacement):

$F = -m g \left(\frac{x}{l}\right) = -\left(\frac{m g}{l}\right) x$

By Newton's second law ($F = m a$):

$m a = -\left(\frac{m g}{l}\right) x \implies a = -\left(\frac{g}{l}\right) x$

Comparing with the definition $a = -\omega^2 x$:

$\omega = \sqrt{\frac{g}{l}}$

The time period $T$ and frequency $f$ are therefore:

$T = 2\pi \sqrt{\frac{l}{g}}, \qquad f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}$

Key Laws of the Simple Pendulum for ECAT

1. Law of Length: $T \propto \sqrt{l}$. If pendulum length is quadrupled ($4l$), the time period doubles ($2T$).
2. Law of Gravity: $T \propto \frac{1}{\sqrt{g}}$. If gravitational acceleration decreases (e.g., atop a high mountain or on the Moon where $g_m = g/6$), the period increases ($T_{\text{moon}} = \sqrt{6} T \approx 2.45 T$), causing the clock to run slow.
3. Law of Mass Independence: $T$ is strictly independent of the mass $m$ and material of the bob. Replacing an aluminum bob with a lead bob of identical dimensions leaves $T$ unaltered.
4. Law of Isochronism: $T$ is independent of angular amplitude $\theta$, provided $\theta$ remains small ($\theta \le 10^\circ$).

The Second's Pendulum

A second's pendulum is defined as a pendulum whose half-period (time for one single tick from one extreme to the other) is exactly $1\text{ second}$, corresponding to a full vibration time period of:

$T = 2.0\text{ s}$

To find its length on Earth's surface where $g = 9.8\text{ m}\cdot\text{s}^{-2}$:

$T = 2\pi\sqrt{\frac{l}{g}} \implies 2.0 = 2\pi\sqrt{\frac{l}{9.8}} \implies 1 = \pi\sqrt{\frac{l}{9.8}}$

$l = \frac{9.8}{\pi^2} \approx \frac{9.80}{9.87} \approx 0.992\text{ m} = 99.2\text{ cm} \approx 1.0\text{ m}$

Pendulum in Non-Inertial Frames (Accelerated Systems)

If the suspension pivot of a simple pendulum undergoes linear acceleration $\mathbf{a}_0$, an effective gravitational acceleration $\mathbf{g}_{\text{eff}} = \mathbf{g} - \mathbf{a}_0$ must be used:

• Elevator accelerating upward with $+a$: $g_{\text{eff}} = g + a \implies T = 2\pi\sqrt{\frac{l}{g+a}}$ (Period decreases, clock runs fast).
• Elevator accelerating downward with $-a$: $g_{\text{eff}} = g - a \implies T = 2\pi\sqrt{\frac{l}{g-a}}$ (Period increases, clock runs slow).
• Freely Falling Elevator ($a = g$): $g_{\text{eff}} = g - g = 0 \implies T = 2\pi\sqrt{\frac{l}{0}} \to \infty$ (Pendulum ceases oscillating).
• Geostationary / Orbiting Satellite: Apparent weightlessness ($g_{\text{eff}} = 0$) means $T \to \infty$; a pendulum cannot tick inside an orbiting spacecraft.

Comparison of Classical Oscillating Systems

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4. Energy Conservation and Partition in SHM

In an ideal undamped simple harmonic oscillator, total mechanical energy continuously transfers between kinetic energy ($K$) and potential energy ($U$), while their sum remains strictly constant.

Instantaneous Potential Energy ($U$)

The work done against the restoring force $F = kx$ to displace the mass from 0 to $x$ is stored as elastic potential energy:

$U = \int_0^x kx' dx' = \frac{1}{2} k x^2$

Substituting $x = x_0 \sin(\omega t)$:

$U(t) = \frac{1}{2} k x_0^2 \sin^2(\omega t)$

• At the mean position ($x = 0$): $U_{\text{min}} = 0$.
• At the extreme positions ($x = \pm x_0$): $U_{\text{max}} = \frac{1}{2} k x_0^2$.

Instantaneous Kinetic Energy ($K$)

The kinetic energy of mass $m$ moving with speed $v = \omega\sqrt{x_0^2 - x^2}$ is:

$K = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (x_0^2 - x^2)$

Recalling that $k = m\omega^2$:

$K = \frac{1}{2} k (x_0^2 - x^2)$

Substituting $v = \omega x_0 \cos(\omega t)$:

$K(t) = \frac{1}{2} k x_0^2 \cos^2(\omega t)$

• At the mean position ($x = 0$): $K_{\text{max}} = \frac{1}{2} k x_0^2$.
• At the extreme positions ($x = \pm x_0$): $K_{\text{min}} = 0$.

Total Mechanical Energy ($E_{\text{total}}$)

Summing kinetic and potential energies at any arbitrary displacement $x$:

$E_{\text{total}} = K + U = \frac{1}{2} k (x_0^2 - x^2) + \frac{1}{2} k x^2 = \frac{1}{2} k x_0^2 = \frac{1}{2} m \omega^2 x_0^2 = \text{constant}$

Total mechanical energy is proportional to the square of the amplitude ($E_{\text{total}} \propto x_0^2$) and the square of the frequency ($E_{\text{total}} \propto f^2$).

Specific Energy Partition Locations

1. Equipartition Location ($K = U$):

$\frac{1}{2} k (x_0^2 - x^2) = \frac{1}{2} k x^2 \implies x_0^2 - x^2 = x^2 \implies 2x^2 = x_0^2$

$x = \pm \frac{x_0}{\sqrt{2}} \approx \pm 0.707 x_0$

Kinetic and potential energies are equal at approximately $70.7\%$ of maximum displacement.

1. Midway Position ($x = x_0 / 2$):

   • Potential energy: $U = \frac{1}{2} k (x_0/2)^2 = \frac{1}{4} \left(\frac{1}{2} k x_0^2\right) = \frac{1}{4} E_{\text{total}} = 25\%$.
   • Kinetic energy: $K = E_{\text{total}} - U = \frac{3}{4} E_{\text{total}} = 75\%$.
   • Ratio: $\frac{K}{U} = 3$.

2. Frequency of Energy Oscillations:

   • Because $\sin^2(\omega t) = \frac{1 - \cos(2\omega t)}{2}$ and $\cos^2(\omega t) = \frac{1 + \cos(2\omega t)}{2}$, both potential and kinetic energies oscillate with angular frequency $\omega_E = 2\omega$ and frequency $f_E = 2f$.
   • In one complete cycle of motion, kinetic energy drops to zero twice and peaks twice.

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5. Free, Damped, Forced Oscillations and Resonance

Free vs Damped Oscillations

1. Free Oscillations: An isolated system oscillates exclusively under internal restoring forces without external excitation or dissipative resistance. It vibrates at its natural frequency $f_0$ with permanent constant amplitude.
2. Damped Oscillations: Real oscillators encounter dissipative viscous drag or mechanical friction, transferring mechanical energy into thermal energy. The equation of motion includes a damping term $b\frac{dx}{dt}$:

$m\frac{d^2 x}{dt^2} + b\frac{dx}{dt} + kx = 0$

Three Damping Regimes:

• Underdamping ($b < 2\sqrt{km}$): The system oscillates with an amplitude decaying exponentially over time: $x(t) = x_0 e^{-\gamma t} \cos(\omega' t)$.
• Critical Damping ($b = 2\sqrt{km}$): The system returns to equilibrium in the minimum possible time without completing a single oscillation. Practical applications: shock absorbers in automobiles, dead-beat pointer galvanometers.
• Overdamping ($b > 2\sqrt{km}$): The system slowly creeps back to equilibrium without oscillating, hindered by heavy resistance (e.g., deep oil dashpots, automatic door closers).

Forced Oscillations and Resonance

When an oscillating system is driven by a periodic external force $F(t) = F_0 \cos(\omega_{\text{drive}} t)$, it initially exhibits transient behavior and subsequently settles into steady-state motion vibrating at the driving frequency $\omega_{\text{drive}}$.

Resonance Phenomenon

Resonance occurs when the driving frequency precisely matches the natural undamped frequency of the oscillating system:

$f_{\text{drive}} = f_0$

Under this resonance condition:

• The rate of energy absorption from the external driver reaches its absolute maximum.
• The oscillation amplitude reaches a pronounced peak.
• The phase difference between the driving force and displacement is exactly $\frac{\pi}{2}\text{ rad}$ ($90^\circ$).

Sharpness of Resonance

The profile of the amplitude vs driving frequency response curve depends on the degree of damping:

• Small Damping (Low Friction): Yields a high, narrow, sharp resonance peak. The system is highly selective (e.g., LC radio tuning circuits picking up a distinct transmission frequency).
• Large Damping (High Friction): Yields a low, broad, flat resonance peak. The amplitude remains modest over a wide frequency band.

Practical Engineering and Physical Manifestations of Resonance

1. Radio and Television Tuning: Variable air capacitors or varactor diodes alter capacitance $C$, shifting the electrical natural frequency $f_0 = \frac{1}{2\pi\sqrt{LC}}$ until it resonates with a targeted broadcast station's carrier wave.
2. Microwave Cooking: Microwave ovens radiate electromagnetic waves at $2.45\text{ GHz}$. Textbooks often cite this as resonance with water molecules; strictly, food is heated by dielectric loss as polar water molecules repeatedly try to realign with the oscillating field.
3. Magnetic Resonance Imaging (MRI): Strong magnetic fields align hydrogen nuclear spins, which are excited by resonant radiofrequency pulses to produce cross-sectional clinical imaging.
4. Destructive Mechanical Resonance: Marching soldiers break step across suspension bridges to prevent periodic footfalls from matching the bridge's natural fundamental vibration modes, which could induce destructive high-amplitude structural oscillations.