Unit VII – Oscillations

Defining Periodic Motion and Oscillations

Motions that repeat themselves over and over again are classified as periodic motion or oscillations. These repetitive movements are common in various physical systems. Examples provided by Dr. Sanjukta Lahiri include the swinging motion of a pendulum in a grandfather clock, the vibrations of a quartz crystal within a watch, and the back-and-forth reciprocating motions of pistons in a car engine.

Mechanics of a Simple Spring-Block Oscillator

A foundational example of a simple oscillator is a block connected to a spring on a horizontal surface where friction is negligible. When the block is pulled and then released, the spring applies a restorative force that attempts to return the block to its equilibrium position. In a frictionless environment, the block's momentum carries it past the equilibrium position, which then compresses the spring. The restorative force then acts to pull the block back toward equilibrium again. Without friction, the block oscillates indefinitely between point A and point B, passing through the equilibrium in a uniform periodic motion. In this ideal scenario, there is no loss in total mechanical energy. Conversely, if the surface possessed friction, some energy would be lost, causing the oscillation distance to decrease over time until points A and B eventually collapse at the origin, stopping the motion.

Fundamental Properties: Amplitude, Period, and Frequency

The amplitude, denoted as $A$, of periodic motion is defined as the maximum magnitude of displacement from the equilibrium point. This value is always positive. In an ideal spring system, the maximum displacement during the stretching phase is identical to the maximum displacement during the compression phase. At the point of amplitude, the displacement $\Delta x$ is at its maximum, and the block reverses direction toward the equilibrium position.

The time period, denoted as $T$, describes the total time in seconds required for the block to complete one full cycle. Quantitatively, this is often referred to as "seconds per cycle." For the spring-block system starting at equilibrium point 0, the period covers the duration the block takes to move from 0 to point A, then back from A to point B, and finally from B back to 0. This is conceptually analogous to a particle in circular motion, where $T$ represents the time for one full revolution around a center.

Frequency, represented as $f$, is the number of cycles the block completes in exactly one second. This unit is denoted as Hertz ($Hz$), named in honor of Heinrich Hertz, a 19th-century German physicist. There is also the angular frequency, $\omega$, which is mathematically defined as $\omega = 2\pi f$. Physically, angular frequency indicates the number of radians the oscillation completes per second and is expressed in $\text{rad/sec}$. The relationship between these values dictates that the period of an oscillation is the reciprocal of its frequency: $T = \frac{1}{f}$ and $f = \frac{1}{T}$. In terms of angular frequency, the relationships are $\omega = 2\pi f = \frac{2\pi}{T}$.

Simple Harmonic Motion (SHM) and Hooke’s Law

Simple Harmonic Motion occurs when the restoring force acting on an object is directly proportional to its displacement from the equilibrium position. According to Hooke’s Law, this force is expressed as $F = -kx$. When this proportional restorative force is present, the resulting oscillation is termed Simple Harmonic Motion. Using Newton’s 2nd Law, where acceleration is $a = \frac{F}{m}$, we can determine the acceleration for a simple spring system as $a = -\left(\frac{k}{m}\right)x$. The negative sign in this equation implies that in SHM, the acceleration and displacement always possess opposite signs. Any object undergoing this specific type of motion is known as a harmonic oscillator.

The Relationship Between SHM and Uniform Circular Motion

Experiments and mathematical proofs show that Simple Harmonic Motion is effectively the projection of uniform circular motion onto a diameter. Consider an object X oscillating in SHM between $+A$ and $-A$, and an object Y rotating around a fixed point at a radius $A$. Their motions are identical if the amplitude of X equals the radius of rotation for Y, and the angular frequency of X equals the angular speed of Y. For a point Q moving around a circle of radius $R = A$ with an angular speed $\omega$, its projection point P on the diameter oscillates with the same angular frequency. The time taken for point Q to complete a revolution is identical to the time taken for point P to complete one oscillation cycle.

Mathematical Derivations of SHM Parameters

As point Q moves in a circle, the $x$-component of the radius vector $OQ$ at any time $t$ is the $x$-coordinate of point Q: $x = R\cos(\phi)$. The centripetal acceleration felt by point Q is $a_Q = \omega^2 R$. The acceleration of the projection point P ($a_P$) is the $x$-component of this centripetal acceleration, defined as $a_P = -a_Q\cos(\phi)$. Through substitution, we find $a_P = -\omega^2 R\cos(\phi)$. Since $x = R\cos(\phi)$, the acceleration is $a_P = -\omega^2 x$. By comparing this to the derivation from Hooke’s Law ($a_P = -\left(\frac{k}{m}\right)x$), it is established that $\omega^2 = \frac{k}{m}$. Consequently, the angular frequency is $\omega = \sqrt{\frac{k}{m}}$ and the time period is $T = 2\pi\sqrt{\frac{m}{k}}$. Linear frequency can be found using $f = \frac{\omega}{2\pi}$ and the time period as $T = \frac{1}{f} = \frac{2\pi}{\omega}$.

Kinematics of Simple Harmonic Motion

If at time $t = 0$ the phasor $OQ$ makes an angle $\phi$ with the $x$-axis, and at time $t$ it makes an angle $\Theta$, the relationship from rotational motion is $\Theta = \omega t + \phi$. The linear displacement of point P over time is $x = A\cos(\Theta)$, or $x = A\cos(\omega t + \phi)$. Because the cosine of an angle always ranges between $-1$ and $+1$, the displacement $x$ swings between $+A$ and $-A$. The function repeats whenever $t$ increases by one time period $T$, specifically when it completes a displacement of $2\pi$ radians, such that $\omega T = \sqrt{\frac{k}{m}} T = 2\pi$.

If the initial phase $\phi = 0$ at $t = 0$, the equation simplifies to $x = A\cos(2\pi t/T)$. Mapping this over one period yields specific displacements: at $t = 0$, $x = +A$; at $t = \frac{1}{4}T$, $x = 0$; at $t = \frac{1}{2}T$, $x = -A$; at $t = \frac{3}{4}T$, $x = 0$; and at $t = T$, $x = +A$. The tangential velocity of the circular projection point Q is $v_Q = R\omega = A\omega$. The linear velocity $v$ of the projection point P is the $x$-component of $v_Q$, given by $v = -v_Q\sin(\Theta)$. Thus, the velocity equation is $v = -A\omega\sin(\omega t)$.

Energy Conservation in Simple Harmonic Motion

In SHM, mechanical energy is conserved. For a block of mass $m$ on a frictionless surface with a spring constant $k$, the mass of the spring is considered negligible. Vertical forces (weight and normal force) balance and do no work. The total mechanical energy $E$ is the sum of spring potential energy $U_s = \frac{1}{2}kx^2$ and kinetic energy $K = \frac{1}{2}mv^2$. At maximum displacement ($x = A$ or $x = -A$), the velocity is zero, meaning kinetic energy is zero and total energy equals potential energy: $E = \frac{1}{2}kA^2$. At the equilibrium position ($x = 0$), potential energy is zero, and the energy is entirely kinetic: $E = \frac{1}{2}mv_{max}^2$. As the block oscillates, potential energy converts to kinetic energy and back, maintaining a constant total mechanical energy $E = K + U_s = \frac{1}{2}kA^2$.

Applications: Vertical Simple Harmonic Motion

When a block of mass $m$ is attached to a spring with constant $k$ and hung vertically, the oscillations are vertical. At equilibrium, the spring is stretched by a length $l$ where the weight of the mass $mg$ is balanced by the vertical force $kl$ exerted by the spring ($kl = mg$). This implies $\frac{m}{k} = \frac{l}{g}$. The time period for this vertical oscillation is $T = 2\pi\sqrt{\frac{m}{k}}$ or $T = 2\pi\sqrt{\frac{l}{g}}$. Correspondingly, the frequency is $f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}$.

The Simple Pendulum: Mechanics and Approximations

A simple pendulum consists of a bob of mass $m$ suspended by a massless, unstretchable string of fixed length $L$. When released from an angle, it oscillates about equilibrium. The arc length traversed is $x = \Theta L$. The restorative force is the tangential component of the weight: $F = -mg\sin(\Theta)$. For very small angles (less than $0.1\,\text{rad}$ or $60$, as noted in the transcript), trigonometry allows the approximation $\sin(\Theta) = \Theta$. Thus, $F = -mg\Theta$. Substituting $\Theta = \frac{x}{L}$, the force becomes $F = -\left(\frac{mg}{L}\right)x$. This shows the restorative force is linearly proportional to displacement, with a force constant $k = \frac{mg}{L}$. The time period is $T = 2\pi\sqrt{\frac{m}{k}} = 2\pi\sqrt{\frac{L}{g}}$, and the frequency is $f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$.

Energy Dynamics in a Simple Pendulum

The pendulum bob oscillates in SHM between maximum rotational amplitudes $+Q$ and $-Q$. At these amplitudes, the instantaneous velocity is zero, resulting in zero kinetic energy and maximum gravitational potential energy. As the bob swings toward the equilibrium position, it gains tangential velocity and kinetic energy while losing potential energy. At the equilibrium point, kinetic energy is at its maximum and potential energy is zero. Throughout the swing, the total energy of the pendulum-earth system remains conserved.

Comprehensive Summary of SHM Equations

For a horizontal spring-mass system: Displacement is $x = A\cos(\omega t)$, velocity is $v = -A\omega\sin(\omega t)$, and acceleration is $a = -\omega^2 A\cos(\omega t)$. The time period is $T = 2\pi\sqrt{\frac{m}{k}}$ and frequency is $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$. Energy components are $U_s = \frac{1}{2}kx^2$ and $K = \frac{1}{2}mv^2$.

For vertical SHM: The period is $T = 2\pi\sqrt{\frac{l}{g}}$ and frequency is $f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}$.

For a simple pendulum: Using the small angle approximation $\sin(\Theta) = \Theta$, the period is $T = 2\pi\sqrt{\frac{L}{g}}$ and frequency is $f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$.