SHM (Simple Harmonic motion)

1. Displacement (x)

• Displacement is how far a particle has moved from its mean (center) position at any time.

• It is given by: x(t)=Acos⁡(ωt+ϕ)

  • A: Amplitude (maximum displacement).

  • ω: Angular frequency (how quickly it oscillates).

  • φ: Phase constant (where motion starts).

2. Amplitude (A)

• Amplitude is the maximum displacement from the mean position.

• The particle moves between +A and -A.

3. Phase (φ)

• Phase tells you the state of motion (position and velocity) at any time.

• The term (ωt + φ) determines where the particle is in its cycle.

4. Angular Frequency (ω)

• It describes how quickly the particle oscillates and is related to time period (T) by: ω=2π/T or ω=2πv

  • T: Time for one full cycle.

  • v: Frequency (oscillations per second).

5. Velocity (v)

• Velocity is how fast the particle is moving.

• Formula: v=ω *root of (A²+x²)

• At mean position (x = 0): Velocity is maximum v=ωA

• At extreme position (x = A): Velocity is zero.

6. Acceleration (a)

• Acceleration measures how quickly velocity changes.

• Formula: a=−ω²xa

• At mean position (x = 0): Acceleration is zero.

• At extreme position (x = A): Acceleration is maximum a=−ω²A

7. Force Law for SHM

• A particle in SHM experiences a force that brings it back to the mean position:

  F=−kx

  • k: Spring constant (stiffness of the system).

  • Force is proportional to displacement but in the opposite direction.

8. Energy in SHM

A particle in SHM has both kinetic energy (KE) and potential energy (PE).

Kinetic Energy (KE):

• Energy due to motion.

• Formula: KE=1/2k(A²−x²)

• At mean position: KE is maximum.

• At extreme positions: KE is zero.

Potential Energy (PE):

• Energy stored due to position.

• Formula: U=1/2kx²

• At mean position: PE is zero.

• At extreme positions: PE is maximum.

Total Energy (E):

• Total energy is constant and is the sum of KE and PE

  E=1/2kA²

9. Graphical Representation

• Displacement, velocity, and acceleration vary sinusoidally (like a wave).

• Energy graphs show:

  • KE is maximum at the mean position.

  • PE is maximum at extreme positions.

  • Total energy (KE + PE) remains constant.