Simple Harmonic Motion (SHM)

Introduction

• Definition: Simple Harmonic Motion (SHM) is a type of oscillatory motion where a body moves back and forth along the same path, and each cycle is identical.
• Subset: SHM is a subset of periodic motion characterized by constant amplitude and period.
• Ideal Motion: In ideal SHM, every cycle is exactly the same.

Key Properties

Constant Quantities

1. Amplitude (A)
   • Maximum displacement from the mean position.
2. Angular Velocity (ω)
   • Unit: radians per second.
   • Related formulas:
     • ω = 2πf
     • ω = 2π/T
3. Total Energy (E)
   • Sum of kinetic and potential energy, remains constant in SHM.

Non-Constant Quantities

1. Displacement (x)
   • Distance from the mean position at any point in time.
2. Velocity (v)
   • Speed of the moving particle at any point in time.
3. Acceleration (a)
   • Acceleration is proportional to displacement but in the opposite direction.

Equations of SHM

Displacement

• x = A cos(ωt)
• x = A sin(ωt + φ) (depending on the initial conditions)

Velocity

• v = -ωA sin(ωt)
• v = ±ω√(A² - x²)

Acceleration

• a = -ω²x

Graphical Representation

• Graphs: Use sinusoidal graphs to represent displacement, velocity, and acceleration over time, each phase-shifted by 90 degrees.

Energy in SHM

• Kinetic Energy: K = 1/2 m v²
• Potential Energy: U = 1/2 k x²
• Total Energy: E = K + U = Constant

Applications of SHM

Mass-Spring System

• Formula: ω = √(k/m)
• Period (T): T = 2π √(m/k)
• Where:
  • k = spring constant
  • m = mass

Pendulum

• Formula: ω = √(g/l)
• Period (T): T = 2π √(l/g)
• Where:
  • g = acceleration due to gravity
  • l = length of the pendulum
• SHM approximation valid for small angles of displacement.

Problem-Solving Tips

1. Identify the Type: Check if it is a simple or applied SHM problem.
2. Use Appropriate Equations: Based on whether the given problem involves displacement, velocity, or acceleration, select from the basic SHM equations.
3. Constants Identification: Identify and use given constants such as mass (m), spring constant (k), or length (l) appropriately.
4. Graph Analysis: Understand that displacement, velocity, and acceleration are phase-shifted sinusoidal functions.
5. Energy Considerations: Remember that total energy remains constant; use kinetic and potential energy formulas as needed.

Summary

• SHM is cyclic and idealized motion with constant amplitude and period.
• Core concepts include understanding displacement, velocity, and acceleration in the context of sinusoidal functions.
• Important formulas relate to the angular velocity, energy, and period of the motion.
• Familiarize with graphical representations and energy transformations in SHM for deeper understanding.