Simple Harmonic Motion Notes

Introduction

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is a fundamental concept in physics, especially relevant for JEE Advanced, as it lays the groundwork for understanding various physical phenomena.

Characteristics of Simple Harmonic Motion

Definition and Basic Concepts

• Restoring Force: In SHM, the force that brings the system back to its equilibrium position is called the restoring force. Mathematically, it is given by: $$ F = -kx $$ where $k$ is the force constant (or spring constant) and $x$ is the displacement from the equilibrium position.
• Displacement: The distance of the particle from its equilibrium position.
• Amplitude ($A$): The maximum displacement from the equilibrium position.
• Time Period ($T$): The time taken to complete one full cycle of the motion. It is given by: $$ T = 2\pi \sqrt{\frac{m}{k}} $$ where $m$ is the mass of the particle and $k$ is the spring constant.
• Frequency ($f$): The number of cycles per unit time. It is the reciprocal of the time period: $$ f = \frac{1}{T} $$
• Angular Frequency ($\omega$): It is related to the frequency by: $$ \omega = 2\pi f = \sqrt{\frac{k}{m}} $$

Differential Equation of SHM

The motion of a particle undergoing SHM can be described by the second-order differential equation: $$ \frac{d^2x}{dt^2} + \omega^2 x = 0 $$

Solution of the Differential Equation

The general solution to this differential equation is: $$ x(t) = A \cos(\omega t + \phi) $$ where $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase constant, which depends on the initial conditions of the motion.

Energy in Simple Harmonic Motion

Potential Energy

The potential energy (PE) in SHM is given by: $$ PE = \frac{1}{2} k x^2 $$

Kinetic Energy

The kinetic energy (KE) in SHM is given by: $$ KE = \frac{1}{2} m v^2 $$ where $v$ is the velocity of the particle.

Total Energy

The total mechanical energy (E) in SHM is constant and is given by: $$ E = KE + PE = \frac{1}{2} k A^2 $$

Examples of Simple Harmonic Motion

Mass-Spring System

Consider a mass $m$ attached to a spring with a spring constant $k$. The system undergoes SHM with a time period: $$ T = 2\pi \sqrt{\frac{m}{k}} $$

Simple Pendulum

For small angular displacements, a simple pendulum of length $L$ and mass $m$ undergoes SHM with a time period: $$ T = 2\pi \sqrt{\frac{L}{g}} $$ where $g$ is the acceleration due to gravity.

Phase and Phase Difference

Phase

The phase of the SHM at any time $t$ is given by: $$ \theta = \omega t + \phi $$

Phase Difference

The phase difference between two SHM equations $x_1(t) = A \cos(\omega t + \phi_1)$ and $x_2(t) = A \cos(\omega t + \phi_2)$ is: $$ \Delta \phi = \phi_2 - \phi_1 $$

Damped Simple Harmonic Motion

Introduction to Damping

In real-world scenarios, SHM is often subject to damping forces, which gradually reduce the amplitude of oscillation.

Equation of Damped SHM

The differential equation for damped SHM is: $$ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 $$ where $b$ is the damping coefficient.

Types of Damping

1. Underdamped: The system oscillates with a gradually decreasing amplitude.
2. Critically Damped: The system returns to equilibrium as quickly as possible without oscillating.
3. Overdamped: The system returns to equilibrium without oscillating, but more slowly than in the critically damped case.

Forced Simple Harmonic Motion

Introduction to Forced Oscillations

When an external periodic force drives the system, it undergoes forced oscillations.

Equation of Forced SHM

The differential equation for forced SHM is: $$ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = F_0 \cos(\omega t) $$

Resonance

Resonance occurs when the frequency of the external force matches the natural frequency of the system, resulting in a maximum amplitude of oscillation.

Conclusion

Simple Harmonic Motion is a fundamental concept in physics that describes a wide range of physical phenomena. Understanding SHM, including its energy properties, phase relations, and the effects of damping and external forces, is crucial for mastering advanced topics in physics and excelling in exams like JEE Advanced.