Chapter 11: Periodic Motion and Simple Harmonic Motion (SHM)

Periodic Motion

Definition and Characteristics

Periodic motion refers to any motion that repeats itself in a regular cycle over a set period of time. This type of motion is fundamental in physics and is observed in many natural and engineered systems.

• Periodic Motion: Motion that repeats in a set time period.

• Examples: Pendulums, springs, planetary orbits.

Simple Harmonic Motion (SHM)

Restoring Force and Equation of Motion

Simple harmonic motion is a specific type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.

• Restoring Force: $F_x = -kx$, where x is the displacement from equilibrium and k is the force constant.

• Equation of Motion: $a = -\frac{k}{m}x$, where m is the mass of the object.

• Sinusoidal Nature: The motion is described by sine and cosine functions.

Key Terms in SHM

• Amplitude (A): The maximum value of displacement from equilibrium.

• Period (T): The time required to complete one full cycle of motion (measured in seconds).

• Frequency (f): The number of cycles per second (measured in Hertz, Hz). $1\ \text{Hz} = 1\ \text{cycle/s} = 1/s = s^{-1}$

• Relationship: $f = \frac{1}{T}$, $T = \frac{1}{f}$

• Angular Frequency (\omega): $\omega = 2\pi f = \frac{2\pi}{T}$

Mathematical Description of SHM

Equations of Motion

The position and velocity of an object in SHM as a function of time can be described using trigonometric functions:

• Position: $x(t) = A \cos(\omega t)$

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

Summary Table: SHM Parameters for a Spring-Mass System

Parameter	Spring Mass Formula
Angular Frequency	$\omega = \sqrt{\frac{k}{m}}$
Frequency	$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$
Period	$T = 2\pi \sqrt{\frac{m}{k}}$

Physical Interpretation

• Equilibrium Position: $x = 0$ is the point where the net force is zero.

• Maximum Displacement: At $x = \pm A$, the object is at its furthest from equilibrium.

• Sinusoidal Motion: The motion can be visualized as the projection of uniform circular motion onto one axis.

• Example: The shadow of a ball moving in a circle creates a sinusoidal pattern, analogous to SHM.

Energy in Simple Harmonic Motion

Conservation of Mechanical Energy

In the absence of friction, the total mechanical energy of an object in SHM remains constant. Energy oscillates between kinetic and potential forms.

• Total Energy: $E = \frac{1}{2}mv_x^2 + \frac{1}{2}kx^2 = \text{constant}$

• At Maximum Displacement ($x = \pm A$): All energy is potential ($E = \frac{1}{2}kA^2$).

• At Equilibrium ($x = 0$): All energy is kinetic ($E = \frac{1}{2}mv_{max}^2$).

Velocity as a Function of Position

• Velocity: $v(x) = \sqrt{\frac{k}{m}(A^2 - x^2)}$

• Maximum Speed: Occurs at $x = 0$, $v_{max} = \omega A$

Example Problems

Spring-Mass System Calculations

• Example 1: A block is hung from an ideal spring with force constant $k$ and oscillates with amplitude $A$. Given the period $T$ and amplitude, calculate the mass, period, and frequency.

• Example 2: A block with mass $4.0\ \text{kg}$ attached to a horizontal spring with force constant $k$ oscillates with amplitude $0.50\ \text{m}$ and maximum acceleration $20\ \text{m/s}^2$. Find the spring constant $k$.

• Example 3: Two springs with spring constant $7580\ \text{N/m}$ support a mass of $0.245\ \text{kg}$. Find the frequency of oscillation.

Note: For all calculations, use the formulas provided above for period, frequency, and angular frequency.

Additional info:

• Simple harmonic motion is a foundational concept in physics, underlying phenomena such as sound waves, electromagnetic waves, and mechanical vibrations.

• Real-world systems may include damping (friction), which causes energy loss and modifies the equations above.