Chapter 14: Periodic Motion – Study Notes

Periodic Motion

Introduction to Periodic Motion

Periodic motion refers to any motion that repeats itself at regular intervals. Examples include the swinging of a pendulum, vibrations in musical instruments, and the movement of pistons in engines. Such motions are fundamental in physics and engineering.

• Periodic motion is also called oscillation.

• Many biological and mechanical systems exhibit periodic motion.

• Example: The stride of a dog versus a human is influenced by the periodic motion of their legs.

What Causes Periodic Motion?

Periodic motion is often caused by a restoring force that acts to return a system to its equilibrium position. A classic example is a mass attached to a spring.

• Restoring force: The force exerted by the spring when displaced from equilibrium.

• When the mass is displaced to the right (x > 0), the restoring force acts to the left (F_x < 0).

• When the mass is at equilibrium (x = 0), no restoring force acts (F_x = 0).

• When the mass is displaced to the left (x < 0), the restoring force acts to the right (F_x > 0).

Characteristics of Periodic Motion

Periodic motion is described by several key quantities:

• Amplitude (A): Maximum displacement from equilibrium.

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

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

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

• Frequency and period are reciprocals: $f = \frac{1}{T}$ and $T = \frac{1}{f}$.

Simple Harmonic Motion (SHM)

Definition and Conditions

Simple harmonic motion occurs when the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.

• SHM: $F_x = -kx$ (Hooke's Law)

• Occurs in ideal springs and other systems for small displacements.

• For larger displacements, real systems may deviate from perfect SHM.

SHM as a Projection of Circular Motion

SHM can be visualized as the projection of uniform circular motion onto one axis.

• The reference circle is the path of a point moving in uniform circular motion.

• The projection of this motion onto a diameter gives SHM.

• A rotating vector in this context is called a phasor.

Mathematical Characteristics of SHM

For a mass m attached to a spring with force constant k:

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

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

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

Increasing mass m decreases frequency and increases period; increasing k increases frequency and decreases period.

Displacement as a Function of Time in SHM

The displacement of an object in SHM as a function of time is given by:

• $x(t) = A \cos(\omega t + \phi)$

• A: Amplitude

• ω: Angular frequency

• φ: Phase angle (determines initial position)

Key effects:

• Increasing m (with constant A and k) increases the period.

• Increasing k (with constant A and m) decreases the period.

• Changing A does not affect the period.

• Changing φ shifts the graph horizontally.

Graphs of Displacement, Velocity, and Acceleration

Displacement, velocity, and acceleration in SHM are sinusoidal and related by phase shifts:

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

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

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

• Velocity is shifted by $\frac{1}{4}$ cycle from displacement; acceleration is shifted by $\frac{1}{2}$ cycle.

Energy in SHM

The total mechanical energy in SHM is conserved and is the sum of kinetic and potential energies:

• $E = \frac{1}{2} m v^2 + \frac{1}{2} k x^2 = \frac{1}{2} k A^2$ (constant)

• At maximum displacement, all energy is potential; at equilibrium, all energy is kinetic.

Energy Diagrams for SHM

Energy diagrams show how potential energy U, kinetic energy K, and total energy E vary with displacement:

• Potential energy: $U = \frac{1}{2} k x^2$

• Total energy: $E = \frac{1}{2} k A^2$ (constant)

• At $x = \pm A$, $K = 0$, $U = E$; at $x = 0$, $U = 0$, $K = E$

Applications of SHM

Vertical SHM

When an object oscillates vertically on a spring, the restoring force is still proportional to displacement, resulting in SHM.

• Restoring force: $F = kx$

• If a weight mg compresses the spring by $\Delta l$, then $k = \frac{mg}{\Delta l}$

Angular SHM

Angular SHM occurs when a coil spring exerts a restoring torque proportional to angular displacement.

• Restoring torque: $\tau_z = -k\theta$

• k is the torsion constant.

Potential Energy in Two-Atom Systems

In molecular systems, the potential energy near equilibrium can be approximated by a parabola, leading to SHM for small displacements.

• Equilibrium at $r = R_0$

• Restoring force: $F_r = -\left( \frac{72U_0}{R_0^2} \right)x$

• Spring constant: $k = \frac{72U_0}{R_0^2}$

The Simple Pendulum

A simple pendulum consists of a point mass suspended by a massless, unstretchable string. For small amplitudes, its motion is SHM.

• Restoring force is proportional to $\sin \theta$; for small $\theta$, $\sin \theta \approx \theta$

• Period: $T = 2\pi \sqrt{\frac{l}{g}}$ (for small angles)

The Physical Pendulum

A physical pendulum uses an extended object instead of a point mass. For small amplitudes, its motion is also SHM.

• Period depends on the object's moment of inertia and the distance from the pivot to the center of mass.

Biological Example: Tyrannosaurus Rex

The leg of a Tyrannosaurus rex can be modeled as a physical pendulum, illustrating the application of SHM to biomechanics.

Damped and Forced Oscillations

Damped Oscillations

Real systems experience dissipative forces (like friction) that reduce the amplitude of oscillation over time. This is called damping.

• Damping: The gradual decrease in amplitude due to energy loss.

• Damped oscillation: Oscillation with decreasing amplitude.

Forced Oscillations and Resonance

Oscillators can be driven by an external periodic force, maintaining constant amplitude. If the driving frequency matches the natural frequency, resonance occurs, resulting in large amplitude oscillations.

• Driving force: External force applied periodically.

• Resonance: Large response when driving frequency equals natural frequency.

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

Summary Table: Key Equations in SHM

Quantity	Equation	Description
Displacement	$x(t) = A \cos(\omega t + \phi)$	Position as a function of time
Velocity	$v_x = -\omega A \sin(\omega t + \phi)$	Rate of change of displacement
Acceleration	$a_x = -\omega^2 A \cos(\omega t + \phi)$	Rate of change of velocity
Angular frequency	$\omega = \sqrt{\frac{k}{m}}$	Oscillation rate in radians/sec
Period	$T = 2\pi \sqrt{\frac{m}{k}}$	Time for one cycle
Frequency	$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$	Cycles per second
Total energy	$E = \frac{1}{2} k A^2$	Conserved mechanical energy

Additional info: The notes include applications to molecular vibrations, pendulums, and biological systems, providing a broad context for SHM in physics and related fields.