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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.

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.

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