Chapter 11: Periodic Motion – Simple Harmonic Motion (SHM) Study Notes

Periodic Motion

Introduction to Simple Harmonic Motion (SHM)

Periodic motion refers to any motion that repeats itself at regular intervals. Simple Harmonic Motion (SHM) is a specific type of periodic motion where the restoring force acting on an object is directly proportional to its displacement from the equilibrium position and is directed towards that position.

• Definition: SHM is a back-and-forth motion of an object in which the restoring force is proportional to its displacement from equilibrium.

• Hooke's Law: The restoring force in SHM obeys Hooke's law: $F = -kx$, where $k$ is the spring constant and $x$ is the displacement.

• Example: A mass attached to a spring exhibits SHM when displaced from its equilibrium position.

Forces in SHM

Understanding the forces acting on an object in SHM is crucial for analyzing its motion.

• Restoring Force: Always acts towards the equilibrium position and changes with time as the displacement changes.

• Other Forces: Depending on the system, gravity and friction may also act, but in ideal SHM, only the restoring force is considered.

Key Terminology in SHM

• Oscillation: A repetitive back-and-forth motion.

• Equilibrium Position: The position where the net force on the object is zero.

• Restoring Force: The force that brings the object back to equilibrium.

• Time Period (T): The time required for one complete cycle of motion. Units: seconds [s].

• Frequency (f): The number of cycles per second. Units: hertz [Hz].

• Angular Frequency ($\omega$): The rate of change of the phase of the oscillation, measured in radians per second. $\omega = 2\pi f$.

Position, Velocity, and Acceleration in SHM

The motion of an object in SHM can be described mathematically using sinusoidal functions.

• Position: $x(t) = A \cos(\omega t + \phi)$, where $A$ is amplitude, $\omega$ is angular frequency, $\phi$ is phase constant.

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

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

• Maximum velocity: $v_{max} = A\omega$ (occurs at equilibrium position, $x=0$)

• Maximum acceleration: $a_{max} = A\omega^2$ (occurs at maximum displacement, $x=\pm A$)

Direction: Velocity is directed towards equilibrium when moving from maximum displacement, and acceleration is always directed towards equilibrium.

Energy in SHM

Energy in a simple harmonic oscillator is continuously exchanged between kinetic and potential forms, but the total mechanical energy remains constant (conserved).

• Kinetic Energy (K): $K = \frac{1}{2}mv^2$

• Potential Energy (U): $U = \frac{1}{2}kx^2$

• Total Mechanical Energy (E): $E = K + U = \frac{1}{2}kA^2$ (constant for a given amplitude)

• At maximum displacement ($x = \pm A$), all energy is potential; at equilibrium ($x = 0$), all energy is kinetic.

Relationship Between SHM and Uniform Circular Motion

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

• The x-component of a particle moving in a circle at constant angular speed follows SHM.

• The frequency of the circular motion matches the frequency of the SHM.

• Mathematical Relation: If a particle moves in a circle of radius $A$ at angular speed $\omega$, its x-position is $x(t) = A \cos(\omega t)$.

Factors Affecting SHM

• Spring Constant (k): Higher $k$ leads to higher frequency.

• Mass (m): Greater mass leads to lower frequency.

• Amplitude (A): Does not affect frequency, but affects energy.

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

Worked Example: Air Track Problem

A spring oscillates with amplitude 0.04 m when a 0.5 kg object is attached. The restoring force of the spring is 6 N when the displacement is 0.03 m.

• Force constant of the spring: $k = \frac{F}{x} = \frac{6}{0.03} = 200 \text{ N/m}$

• Maximum velocity: $v_{max} = A\omega = A \sqrt{\frac{k}{m}}$

• Maximum acceleration: $a_{max} = A\omega^2 = A \frac{k}{m}$

• Velocity and acceleration at $x = 0.02$ m: Use $v = \omega \sqrt{A^2 - x^2}$ and $a = -\omega^2 x$

Summary Table: SHM Quantities

Quantity	Expression	Maximum Value
Position	$x(t) = A \cos(\omega t + \phi)$	$A$
Velocity	$v(t) = -A\omega \sin(\omega t + \phi)$	$A\omega$
Acceleration	$a(t) = -A\omega^2 \cos(\omega t + \phi)$	$A\omega^2$
Kinetic Energy	$K = \frac{1}{2}mv^2$	$\frac{1}{2}kA^2$
Potential Energy	$U = \frac{1}{2}kx^2$	$\frac{1}{2}kA^2$
Total Energy	$E = \frac{1}{2}kA^2$	Constant

Additional info:

• Angular frequency $\omega$ is related to the spring constant and mass by $\omega = \sqrt{\frac{k}{m}}$.

• Energy conservation in SHM assumes no energy loss due to friction or air resistance.

• Simulation tools (such as PhET) can help visualize how changing $k$, $m$, or $A$ affects SHM.