Oscillatory Motion

Periodic Motion

Any regular and repeating motion is termed periodic motion. Vibrations of objects, atomic and molecular movements in solids, planetary orbits, heartbeats, and breathing are all examples. The key characteristics are:

• Period (T): The time required for one complete repetition of the motion.
• Frequency (f): The number of repetitions per unit time. Related to period by f = 1/T.
• In one dimension, position function x(t) satisfies x(t + T) = x(t).
• The path can be linear, circular, elliptical, or other curves (rectangular, square, saw-toothed, sinusoidal waves, etc.).

Simple Harmonic Motion (SHM)

A specific type of periodic motion where displacement, velocity, and acceleration are sinusoidal functions of time. SHM occurs under two conditions:

1. An object is displaced from its position of stable equilibrium and released.
2. The restoring force is directly proportional to the object's displacement from the equilibrium position and always directed towards it.

Parameters used to describe SHM are period (T), amplitude (A, maximum displacement), and phase (δ or φ, related to initial position).

Kinematics of Simple Harmonic Motion

The motion of an object in SHM can be described by sinusoidal functions:

• Angular frequency (ω): ω = 2π/T = 2πf
• Displacement: x(t) = A sin(ωt + δ) or x(t) = A cos(ωt + δ)
  • δ is the phase constant, determined by initial conditions.
  • A positive δ shifts the sine function to the left; a negative δ shifts it to the right.
• Velocity: v(t) = dx/dt = ωA cos(ωt + δ)
  • Maximum speed is v_max = ωA.
• Acceleration: a(t) = dv/dt = -ω²A sin(ωt + δ) = -ω²x(t)
  • Maximum acceleration is a_max = ω²A.

Dynamics of Simple Harmonic Motion

Applying Newton's second law (F=ma) to SHM:

• Restoring Force: F = ma(t) = m(-ω²x) = -mω²x.
• This force is a "linear restoring force":
  • It's restoring (minus sign) because it directs the object back to x=0.
  • It's linear because its magnitude is proportional to x.
• SHM occurs when the restoring force is proportional to displacement from a stable equilibrium point.
• Stable Equilibrium: Small displacement causes forces to restore the system.
• Unstable Equilibrium: Small displacement causes forces to move the system further away.

Simple Harmonic Oscillators (SHOs)

Systems executing SHM are called Simple Harmonic Oscillators.

1. The Simple Spring

• Hooke's Law: The restoring force of a spring is F_s = -kx, where k is the spring constant.
• Comparing with F = -mω²x: mω² = k
• Angular Frequency: ω = sqrt(k/m)
• Period: T = 2π sqrt(m/k)
• For vertical springs, the equilibrium position is shifted due to gravity, but oscillations around this new equilibrium still follow SHM with the same period.

2. The Simple Pendulum

• A point mass attached to a massless string, oscillating for small angles (θ < 0.1 rad).
• Restoring Force (tangential component of gravity): F_s = -mg sinθ ≈ -mg(s/L) for small θ, where s is arc length and L is string length.
• Comparing with F = -mω²s: mω² = mg/L
• Angular Frequency: ω = sqrt(g/L)
• Period: T = 2π sqrt(L/g)

3. The Torsional Pendulum

• A mass suspended by a wire, oscillating rotationally.
• Restoring Torque: τ = -Kθ, where K is the torsion constant and θ is the angular displacement.
• Comparing with rotational analog of F=-mω²x (τ=-Iω²θ): Iω² = K, where I is the moment of inertia.
• Angular Frequency: ω = sqrt(K/I)
• Period: T = 2π sqrt(I/K)

4. The Physical Pendulum

• Any rigid body oscillating about a pivot point not at its center of gravity.
• Restoring Torque: τ = -mgh sinθ ≈ -mghθ for small angles, where h is distance from pivot to center of mass.
• Comparing with τ = -Iω²θ: Iω² = mgh
• Angular Frequency: ω = sqrt(mgh/I)
• Period: T = 2π sqrt(I/(mgh))

5. Floating Object Oscillating in a Fluid

• A cylinder of mass m, cross-sectional area A, floating in a fluid of density ρ.
• Restoring Force (Buoyancy): When displaced by x, the additional displaced volume is Ax. Buoyant force is F_B = -(ρgA)x (using the mass of displaced fluid, m_fd = ρAx).
• Comparing with F = -mω²x: mω² = ρgA
• Angular Frequency: ω = sqrt(ρgA/m)
• Period: T = 2π sqrt(m/(ρgA))

6. Fluid Oscillating in a U-Tube

• Liquid of density ρ, total length L, and cross-sectional area A in a U-tube.
• When liquid is depressed by x in one arm, the height difference is 2x.
• Restoring Force (weight of liquid column of length 2x): F = -(2Aρg)x
• Total mass of liquid: m = ALρ.
• Comparing with F = -mω²x: (ALρ)ω² = 2Aρg
• Angular Frequency: ω = sqrt(2g/L)
• Period: T = 2π sqrt(L/(2g))

Summary Table of SHOs

The document provides a table summarizing the restoring force, constant of motion, angular frequency, and period for simple spring, simple pendulum, torsional pendulum, physical pendulum, floating object, and liquid in U-tube.

Conical Pendulum

This is a pendulum that spins in a complete circle, maintaining a constant angle with the vertical. It performs periodic motion but is not SHM because the motion is circular, not oscillatory about an equilibrium point. The analysis involves balancing vertical forces and providing a horizontal centripetal force.

Simple Harmonic Motion and Uniform Circular Motion

SHM can be viewed as the projection of uniform circular motion onto one dimension (e.g., x-axis or y-axis). An object moving in a circle with constant angular speed ω has its x and y coordinates oscillating sinusoidally, 90° out of phase. This analogy helps visualize SHM and understand the relationship of ω.

Energy in Simple Harmonic Motion

The total mechanical energy (E_T) in an ideal SHM system is conserved and constant.

• Kinetic Energy (K.E.): 1/2 mv²
• Potential Energy (P.E.): 1/2 kx² (for a spring) or 1/2 mgh (for a pendulum)
• Total Mechanical Energy: E_T = K.E. + P.E. = 1/2 kx² + 1/2 mv² = 1/2 kA² = 1/2 mω²A².
• Energy conversion: When K.E. is max, P.E. is zero (at equilibrium, x=0, v=v_max). When P.E. is max, K.E. is zero (at extreme displacement, x=A, v=0).

Damping of Simple Harmonic Motion

In real systems, energy is dissipated (e.g., by friction, viscous forces), causing the amplitude of oscillation to decrease over time. This process is called damping.

• Damping Force: Often proportional to velocity, F_d = -bv, where b is the damping constant.
• Equation of Motion: m(d²x/dt²) + b(dx/dt) + kx = 0.
• Damped Displacement: x(t) = A e^(-bt/2m) sin(ω't + φ), where ω' is the damped angular frequency.
• Damped Angular Frequency: ω' = sqrt(k/m - (b/2m)²) = ω₀ sqrt(1 - (b/(2mω₀))²).

Types of Damping

1. Underdamped (0 < b < 2mω₀): Oscillations with gradually decreasing amplitude. Energy decays exponentially.
2. Critically Damped (b = 2mω₀): System returns to equilibrium as quickly as possible without oscillating. (Desired for shock absorbers).
3. Overdamped (b > 2mω₀): System returns to equilibrium without oscillating, but more slowly than critically damped.

Damping Parameter and Time Constant

• Damping Parameter (γ): γ = b/m.
• Time Constant (τ): The time after which the amplitude decreases by a factor of 1/e, and energy decreases by a factor of 1/e². τ = m/b = 1/γ.
• Amplitude decay: A(t) = A₀ e^(-t/2τ)
• Energy decay: E(t) = E₀ e^(-t/τ)

Quality Factor (Q)

A dimensionless measure of damping; high Q means low damping and slow decay.

• Definition: Q = (Energy stored in the oscillator) / (Energy dissipated per cycle/radian).
• Relation to damping: Q = ωτ = (mω)/b.
• Fractional energy loss per cycle: ΔE/E = 2π/Q.

Forced Oscillations and Resonance

When an external, periodic driving force (e.g., F_ext = F₀ sin(ω_d t)) is applied to a damped oscillator.

• Initially, there's a "transient effect" where the system tries to oscillate at its natural frequency (ω₀) and the driving frequency (ω_d).
• Eventually, the system settles into "forced oscillations" at the driving frequency (ω_d) with a constant amplitude and phase.
• Energy lost to damping is compensated by the driving force.
• Resonance: Occurs when the driving frequency (ω_d) is close to the natural frequency (ω₀), leading to a dramatic increase in the amplitude of oscillation. The frequency at which amplitude is maximum is the resonance frequency.

Resonance and Power Transfer

• The average power input by the driving force (P_av) varies with driving frequency.
• Maximum Power Transfer: Occurs at resonance, when ω_d = ω₀ (for light damping) and the phase difference between force and velocity is δ = ±π/2.
• The maximum power input is inversely proportional to the strength of damping (b).

Width of Power Resonance Curve

• The "sharpness" of the resonance peak is related to the damping and the Q-factor.
• Bandwidth (BW): The width of the power resonance curve at half its maximum value is BW = Δω = γ = ω₀/Q.
• High Q (low damping) means a sharp, narrow resonance peak. Low Q (high damping) means a broad, flat peak.

The document concludes with a series of worked examples applying these concepts.