Chapter 14: Periodic Motion – Study Notes

Periodic Motion

Introduction to Periodic Motion

• Periodic motion refers to any motion that repeats itself at regular intervals, such as the swinging of a pendulum, vibrations in musical instruments, or the movement of pistons in engines.

• Oscillation is a type of periodic motion where an object moves back and forth about an equilibrium position.

• Periodic motion is fundamental in physics and underlies many natural and engineered systems.

What Causes Periodic Motion?

Restoring Force and Equilibrium

• When a body attached to a spring is displaced from its equilibrium position, the spring exerts a restoring force that acts to return the object to equilibrium.

• This restoring force is responsible for the oscillatory (back-and-forth) motion.

• At equilibrium (x = 0), the net force and acceleration are zero.

• Displacement to the right (x > 0): Restoring force is negative, pulling the object back.

• Displacement to the left (x < 0): Restoring force is positive, pushing the object back.

Characteristics of Periodic Motion

• Amplitude (A): Maximum displacement from equilibrium.

• Period (T): Time for one complete cycle of motion.

• Frequency (f): Number of cycles per unit time.

• Angular frequency ($\omega$): Related to frequency by$\omega$ = $2\pi\cdot f\left(frequency\right)$.

• Frequency and period are reciprocals: \(f = 1/T\) and \(T = 1/f\).

Simple Harmonic Motion (SHM)

Definition and Hooke's Law

• SHM occurs when the restoring force is directly proportional to displacement and directed toward equilibrium: \(F_x = -kx\) (Hooke's Law).

• Graphically, the force vs. displacement is a straight line with negative slope.

• In real systems, this proportionality holds for small displacements (small amplitude).

Projection Interpretation

• SHM can be visualized as the projection of uniform circular motion onto one axis (reference circle).

• The rotating vector (phasor) represents the state of the oscillator.

Mathematical Description of SHM

Equations and Parameters

• For a mass \(m\) on a spring with force constant \(k\):

  • Angular frequency: $\omega = \sqrt{\frac{k}{m}}$

  • Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

  • Period: $T = 2\pi \sqrt{\frac{m}{k}}$

• Displacement as a function of time: $x(t) = A \cos(\omega t + \phi)$

• Amplitude \(A\): Maximum displacement.

• Phase angle \(\phi\): Determines initial position at \(t = 0\).

Effect of Parameters on SHM

• Increasing mass \(m\) increases the period (slower oscillation).

• Increasing force constant \(k\) decreases the period (faster oscillation).

• Changing amplitude \(A\) does not affect the period.

• Changing phase \(\phi\) shifts the motion in time but does not affect period or amplitude.

Graphs in SHM

Displacement, Velocity, and Acceleration

• Displacement: $x(t) = A \cos(\omega t + \phi)$

• Velocity: $v_x(t) = -\omega A \sin(\omega t + \phi)$

• Acceleration: $a_x(t) = -\omega^2 A \cos(\omega t + \phi)$

• The velocity graph is shifted by one-quarter cycle relative to displacement; acceleration is shifted by half a cycle.

Energy in SHM

Conservation of Mechanical Energy

• Total mechanical energy is conserved:

• $E = \frac{1}{2} m v_x^2 + \frac{1}{2} k x^2 = \frac{1}{2} k A^2 = \text{constant}$

• At maximum displacement (\(x = \pm A\)), all energy is potential (kinetic energy is zero).

• At equilibrium (\(x = 0\)), all energy is kinetic (potential energy is zero).

• At intermediate points, energy is shared between kinetic and potential forms.

Vertical SHM

• When a mass hangs from a vertical spring, the restoring force is still proportional to displacement from equilibrium: $F = -k x$.

• Equilibrium is reached when the upward spring force balances the weight: $k \Delta l = mg$.

• Oscillations about this equilibrium are SHM.

Angular SHM

• For rotational systems, a coil spring exerts a restoring torque: $\tau = -\kappa \theta$.

• \(\kappa\) is the torsion constant; \(\theta\) is angular displacement.

• This leads to angular simple harmonic motion, as seen in balance wheels of watches.

SHM in Molecules

Potential Energy of a Two-Atom System

• Potential energy near equilibrium can be approximated by a parabola.

• At equilibrium separation \(r = R_0\), the potential energy is minimum.

Vibrations of Molecules

• For small displacements \(x\) from equilibrium, the restoring force is:

• $F_r = -\left(72 U_0 / R_0^2\right)x$

• The motion is SHM with force constant $k = 72 U_0 / R_0^2$.

Pendulums

The Simple Pendulum

• A simple pendulum consists of a point mass (bob) on a massless, unstretchable string.

• For small amplitudes (small \(\theta\)), the motion is approximately SHM.

The Physical Pendulum

• A physical pendulum is any real pendulum with an extended body.

• For small amplitudes, the motion is also approximately SHM.

• Example: The leg of a Tyrannosaurus rex can be modeled as a physical pendulum.

Damped Oscillations

• Real systems experience dissipative forces (like friction or air resistance) that reduce amplitude over time.

• This reduction is called damping, and the motion is damped oscillation.

• Stronger damping (larger damping constant \(b\)) causes amplitude to decrease more rapidly and increases the period.

Forced Oscillations and Resonance

• A damped oscillator will eventually stop unless a periodic driving force is applied.

• When a periodic driving force with angular frequency \(\omega_d\) is applied, the system undergoes forced oscillation.

• The amplitude of the forced oscillator is given by:

$A = \frac{F_{\text{max}}}{\sqrt{(k - m \omega_d^2)^2 + b^2 \omega_d^2}}$

• If the driving frequency matches the system's natural frequency, resonance occurs, resulting in a large amplitude response.

• Example: The wings of a lady beetle are driven by muscles applying a periodic force, causing forced oscillation at the driving frequency.