Comprehensive Study Guide on Oscillations and Simple Harmonic Motion and Damped Motion

Simple Harmonic Motion: Definitions and Fundamental Expressions

• Definition of Linear Simple Harmonic Motion (SHM):

  • Linear simple harmonic motion is defined as periodic motion in which the restoring force (or acceleration) is always directed towards the mean position and is directly proportional to the displacement from the mean position.

  • Mathematically: $F \propto -x$

  • $F = -kx$

  • Where $k$ is the force constant.

• Derivation for Angular Frequency ($\omega$):

  • We know that according to Newton's second law: $F = ma$

  • Therefore, acceleration $a = \frac{F}{m} = \frac{-kx}{m}$

  • This can be written as $a = -\omega^2 x$

  • Comparing the two expressions for acceleration: $\omega^2 = \frac{k}{m}$

  • $\omega = \sqrt{\frac{k}{m}}$

  • Where $\omega$ is the angular frequency of the body.

Differential Equation of Linear SHM

• Establishment of the Equation:

  • Let a particle of mass $m$ undergo SHM about its mean position $O$.

  • The restoring force is given by: $F = -kx$

  • Acceleration is the rate of change of velocity: $a = \frac{dv}{dt}$

  • Velocity is the rate of change of displacement: $v = \frac{dx}{dt}$

  • Thus, acceleration is the second derivative of displacement: $a = \frac{d}{dt} \left( \frac{dx}{dt} \right) = \frac{d^2x}{dt^2}$

  • Substituting this into the force equation $F = ma$:

    • $F = m \times \frac{d^2x}{dt^2}$

  • Relating the two force equations:

    • $-kx = m \frac{d^2x}{dt^2}$

    • $m \frac{d^2x}{dt^2} + kx = 0$

  • Dividing the entire equation by $m$:

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

  • Since $\omega^2 = \frac{k}{m}$, the differential equation becomes:

    • $\frac{d^2x}{dt^2} + \omega^2 x = 0$

Kinematic Expressions Derived from the Differential Equation

• Expression for Acceleration ($a$):

  • From the differential equation: $\frac{d^2x}{dt^2} + \omega^2 x = 0$

  • Since $a = \frac{d^2x}{dt^2}$, the equation can be written as:

    • $a + \omega^2 x = 0$

    • $a = -\omega^2 x$

• Expression for Velocity ($v$):

  • From the differential equation: $\frac{d^2x}{dt^2} + \omega^2 x = 0$

  • Let $a = \frac{d^2x}{dt^2}$, so $a + \omega^2 x = 0$

  • Acceleration can be rewritten using the chain rule: $a = \frac{dv}{dt} = \frac{dv}{dx} \times \frac{dx}{dt}$

  • Since $v = \frac{dx}{dt}$, then $a = v \frac{dv}{dx}$

  • The equation becomes: $v \frac{dv}{dx} + \omega^2 x = 0$

    • $v \frac{dv}{dx} = -\omega^2 x$

    • $v \, dv = -\omega^2 x \, dx$

  • Integrating both sides:

    • $\int v \, dv = -\omega^2 \int x \, dx$

    • $\frac{v^2}{2} = -\omega^2 \frac{x^2}{2} + C$

  • At the extreme position, displacement $x = \pm A$ and velocity $v = 0$. Substituting these values to find the constant $C$:

    • $0 = \frac{-\omega^2 A^2}{2} + C$

    • $C = \frac{\omega^2 A^2}{2}$

  • Substituting $C$ back into the equation:

    • $\frac{v^2}{2} = \frac{-\omega^2 x^2}{2} + \frac{\omega^2 A^2}{2}$

    • $v^2 = \omega^2 (A^2 - x^2)$

    • $v = \pm \omega \sqrt{A^2 - x^2}$

• Expression for Displacement ($x$):

  • Considering the magnitude of velocity: $v = \omega \sqrt{A^2 - x^2}$

  • Since $v = \frac{dx}{dt}$, then:

    • $\frac{dx}{dt} = \omega \sqrt{A^2 - x^2}$

    • $\frac{dx}{\sqrt{A^2 - x^2}} = \omega \, dt$

  • Integrating both sides:

    • $\int \frac{dx}{\sqrt{A^2 - x^2}} = \int \omega \, dt$

    • $\sin^{-1} \left( \frac{x}{A} \right) = \omega t + \alpha$

    • $\frac{x}{A} = \sin(\omega t + \alpha)$

    • $x = A \sin(\omega t + \alpha)$

Fundamental Definitions in SHM

• Periodic Motion: A motion which repeats itself after equal intervals of time is called periodic motion.

• Oscillation: In SHM, the particle performs the same set of movements again and again. One such set of movement is called an oscillation.

• Amplitude of SHM ($A$): The magnitude of maximum displacement of the particle from its mean position while performing SHM.

• Period of SHM ($T$): The time taken by the particle to complete one oscillation.

• Frequency of SHM ($n$): The number of oscillations performed by the particle in one second while performing SHM.

• Phase of SHM: The physical quantity which describes the state of oscillation. In the equation $x = A \sin(\omega t + \alpha)$, the term $(\omega t + \alpha)$ is the phase.

• Epoch of SHM ($\alpha$): The physical quantity which describes the state of oscillation of the particle at the start of motion ($t = 0$). It is also called the phase constant.

Graphical Representation of SHM

• Particle Starting from Mean Position ($x = 0$ at $t = 0$):

  • Displacement: Starts at zero, follows a sine curve $x = A \sin(\omega t)$.

  • Velocity: Starts at maximum value +$\omega A$, follows a cosine curve $v = \omega A \cos(\omega t)$.

  • Acceleration: Starts at zero, but follows a negative sine curve $a = -\omega^2 A \sin(\omega t)$.

• Particle Starting from Extreme Position ($x = A$ at $t = 0$):

  • Displacement: Starts at $+A$, follows a cosine curve $x = A \cos(\omega t)$.

  • Velocity: Starts at 0, follows a negative sine curve $v = -\omega A \sin(\omega t)$.

  • Acceleration: Starts at negative maximum $-A \omega^2$, following a negative cosine curve $a = -\omega^2 A \cos(\omega t)$.

Energetics of Simple Harmonic Motion

• Kinetic Energy (K.E.):

  • Instantaneous velocity: $v = \pm \omega \sqrt{A^2 - x^2}$

  • $v^2 = \omega^2(A^2 - x^2)$

  • $K.E. = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2)$

  • Since $k = m \omega^2$, then:

    • $K.E. = \frac{1}{2} k (A^2 - x^2)$

• Potential Energy (P.E.):

  • Consider a particle at distance $x$ from the mean position. The restoring force is $F = -kx$.

  • Work done ($dW$) against the restoring force for a small displacement $dx$:

    • $dW = -F \, dx = -(-kx) \, dx = kx \, dx$

  • Total work done to displace the particle from $0$ to $x$:

    • $W = \int_{0}^{x} kx \, dx = k \left[ \frac{x^2}{2} \right]_0^x = \frac{1}{2} k x^2$

  • This work is stored as Potential Energy: $P.E. = \frac{1}{2} k x^2$

  • P.E. at Mean Position ($x = 0$): $P.E. = 0$

  • P.E. at Extreme Position ($x = \pm A$): $P.E. = \frac{1}{2} k A^2$

• Total Energy (T.E.):

  • $T.E. = K.E. + P.E.$

  • $T.E. = \frac{1}{2} k (A^2 - x^2) + \frac{1}{2} k x^2$

  • $T.E. = \frac{1}{2} k A^2 - \frac{1}{2} k x^2 + \frac{1}{2} k x^2$

  • $T.E. = \frac{1}{2} k A^2$

  • Substituting $k = m \omega^2$ and $\omega = 2 \pi n$:

    • $T.E. = \frac{1}{2} m (2 \pi n)^2 A^2 = \frac{1}{2} m (4 \pi^2 n^2) A^2 = 2 \pi^2 m n^2 A^2$

• Laws of Total Energy:

  • Total energy is directly proportional to the square of the amplitude: $T.E. \propto A^2$

  • Total energy is directly proportional to the square of the frequency: $T.E. \propto n^2$

  • Total energy is conserved (constant at all points in the motion).

Analytical Composition of Two SHMs

• Setup:

  • Consider two SHMs with the same period, parallel to each other, but having different amplitudes ($A_1, A_2$) and initial phases ($\alpha_1, \alpha_2$):

    • $x_1 = A_1 \sin(\omega t + \alpha_1)$

    • $x_2 = A_2 \sin(\omega t + \alpha_2)$

• Resultant Displacement:

  • $x = x_1 + x_2 = A_1 \sin(\omega t + \alpha_1) + A_2 \sin(\omega t + \alpha_2)$

  • Using trigonometric expansion and collecting terms:

    • $x = \sin(\omega t) [A_1 \cos \alpha_1 + A_2 \cos \alpha_2] + \cos(\omega t) [A_1 \sin \alpha_1 + A_2 \sin \alpha_2]$

  • Let $R \cos \delta = A_1 \cos \alpha_1 + A_2 \cos \alpha_2$

  • Let $R \sin \delta = A_1 \sin \alpha_1 + A_2 \sin \alpha_2$

  • Then $x = R \sin(\omega t + \delta)$, which describes a new SHM.

• Resultant Amplitude ($R$):

  • $R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\alpha_1 - \alpha_2)}$

• Specific Cases for Phase Difference ($\alpha_1 - \alpha_2$):

  • Case (i): Phase difference is 0:

    • $R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(0)} = \sqrt{(A_1 + A_2)^2} = A_1 + A_2$

  • Case (ii): Phase difference is $\pi$:

    • $R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\pi)} = \sqrt{(A_1 - A_2)^2} = A_1 - A_2$

  • Case (iii): Phase difference is $\frac{\pi}{2}$:

    • $R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos \left( \frac{\pi}{2} \right)} = \sqrt{A_1^2 + A_2^2}$

  • Case (iv): Phase difference is $\frac{\pi}{3}$:

    • $R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos \left( \frac{\pi}{3} \right)} = \sqrt{A_1^2 + A_2^2 + A_1 A_2}$

The Simple Pendulum

• Proving SHM for a Simple Pendulum:

  • Consider a pendulum of length $L$ with mass $m$, displaced by a small angle $\theta$ from the vertical.

  • The weight $mg$ is resolved into two components:

    1. Radial component: $mg \cos \theta$

    2. Tangential component: $mg \sin \theta$

  • The tangential component is the restoring force: $F = -mg \sin \theta$

  • For a small angle $\theta$ expressed in radians, $\sin \theta \approx \theta$.

  • Thus, $F = -mg \theta$.

  • Since $\theta = \frac{x}{L}$ (where $x$ is arc length/displacement), then $F = -mg \left( \frac{x}{L} \right)$.

  • As $m, g, L$ are constants, $F \propto -x$, proving the motion is linear SHM for small displacements.

• Expression for the Period ($T$):

  • Starting from $F = ma$ and $F = -mg \frac{x}{L}$:

    • $ma = -mg \frac{x}{L} \implies a = -\frac{g}{L} x$

  • We also know $a = -\omega^2 x$, so:

    • $\omega^2 = \frac{g}{L} \implies \omega = \sqrt{\frac{g}{L}}$

  • The period is defined as $T = \frac{2 \pi}{\omega}$:

    • $T = 2 \pi \sqrt{\frac{L}{g}}$

• Laws of Simple Pendulum:

  1. Law of Length: The period is directly proportional to the square root of its length ($T \propto \sqrt{L}$).

  2. Law of Acceleration Due to Gravity: The period is inversely proportional to the square root of acceleration due to gravity ($T \propto \frac{1}{\sqrt{g}}$).

  3. Law of Mass: The period does not depend on the mass of the bob.

  4. Law of Amplitude: The period does not depend on the amplitude (for small amplitudes).

• Seconds Pendulum:

  • A simple pendulum whose period is exactly two seconds is called a seconds pendulum.

  • For this pendulum, $T = 2 \, s$. Substituting into the period formula:

    • $2 = 2 \pi \sqrt{\frac{L}{g}} \implies 1 = \pi^2 \frac{L}{g}$

    • $L = \frac{g}{\pi^2}$

Linear SHM as a Projection of Uniform Circular Motion (UCM)

• A linear SHM is essentially the projection of a uniform circular motion along any of its diameters.

• Displacement Projection:

  • Particle $P$ moves in UCM along a circle of radius $r$ with angular velocity $\omega$.

  • At time $t$, the reference angle is $\theta = \omega t + \phi$.

  • Projection on the Y-axis: $y = r \sin \theta = r \sin(\omega t + \phi)$. This is the standard equation of SHM with amplitude $r$.

• Velocity Projection:

  • The velocity in UCM is $v_{UCM} = r \omega$. Its projection on the Y-axis is $v_y = r \omega \cos(\omega t + \phi)$.

• Acceleration Projection:

  • The centripetal acceleration in UCM is $a_{UCM} = r \omega^2$. Its projection on the Y-axis is $a_y = -r \omega^2 \sin(\omega t + \phi) = -\omega^2 y$.

Magnet Vibrating in a Uniform Magnetic Field (Angular SHM)

• If a bar magnet is given a small angular displacement $\theta$ in a magnetic field $B$ and released, it performs angular SHM.

• Restoring Torque (\tau):

  • $\tau = -\mu B \sin \theta$

  • For small $\theta$, $\sin \theta \approx \theta$, so $\tau = -\mu B \theta$

  • We also know $\tau = I \alpha$, where $I$ is the moment of inertia and $\alpha$ is angular acceleration.

  • $I \alpha = -\mu B \theta \implies \alpha = - \left( \frac{\mu B}{I} \right) \theta$

  • Since $\mu, B, I$ are constant, $\alpha \propto -\theta$, confirming angular SHM.

• Period of Vibration:

  • $T = 2 \pi \sqrt{\frac{\text{Angular Displacement}}{\text{Angular Acceleration}}}$

  • $T = 2 \pi \sqrt{\frac{I}{\mu B}}$

Damped Oscillations

• Definition: Periodic oscillations of gradually decreasing amplitude are called damped harmonic oscillations.

• Forces Involved:

  • In a damped system (e.g., a block on a spring in a liquid), a damping force ($F_d$) acts opposite to velocity: $F_d = -bv$.

  • The spring force is $F_s = -kx$.

• Differential Equation:

  • Total force $F_{total} = F_d + F_s$

  • $ma = -bv - kx \implies ma + bv + kx = 0$

  • Substituting $a = \frac{d^2x}{dt^2}$ and $v = \frac{dx}{dt}$, we get the differential equation:

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

• Angular Frequency and Period:

  • The angular frequency for damped oscillations is given by: $\omega = \sqrt{\frac{k}{m} - \left( \frac{b}{2m} \right)^2}$

  • The period of oscillation is: $T = \frac{2 \pi}{\omega}$