Simple Harmonic Motion (SHM) Basics

Simple Harmonic Motion (SHM)

Definition

• Simple harmonic motion is a type of periodic motion where the restoring force is proportional to the displacement from the equilibrium position.
• The motion is sinusoidal and repeats at regular intervals.

Characteristics

• Displacement: The distance from the equilibrium position.
• Amplitude: The maximum displacement from the equilibrium position.
• Period: The time taken to complete one cycle of motion.
• Frequency: The number of cycles per second, measured in Hz (1/s).
• Phase: The initial angle of the motion.

Equations of Motion

• Displacement equation: x(t) = A cos(ωt + φ)
  • x(t) = displacement at time t
  • A = amplitude
  • ω = angular frequency (2πf)
  • φ = phase angle
• Velocity equation: v(t) = -Aω sin(ωt + φ)
• Acceleration equation: a(t) = -Aω^2 cos(ωt + φ)

Energy in SHM

• Kinetic energy: maximum at the equilibrium position, minimum at the amplitude
• Potential energy: maximum at the amplitude, minimum at the equilibrium position
• Total energy: constant, converted between kinetic and potential energy

Examples

• Mass on a spring
• Pendulum
• Vibrating string
• Electric circuits with capacitors and inductors

Simple Harmonic Motion (SHM)

• SHM is a type of periodic motion where the restoring force is proportional to the displacement from the equilibrium position and the motion is sinusoidal.

Characteristics of SHM

• Displacement is the distance from the equilibrium position.
• Amplitude is the maximum displacement from the equilibrium position.
• Period is the time taken to complete one cycle of motion.
• Frequency is the number of cycles per second, measured in Hz (1/s).
• Phase is the initial angle of the motion.

Equations of SHM

• The displacement equation is x(t) = A cos(ωt + φ), where x(t) is the displacement at time t, A is the amplitude, ω is the angular frequency, and φ is the phase angle.
• The velocity equation is v(t) = -Aω sin(ωt + φ).
• The acceleration equation is a(t) = -Aω^2 cos(ωt + φ).

Energy in SHM

• Kinetic energy is maximum at the equilibrium position and minimum at the amplitude.
• Potential energy is maximum at the amplitude and minimum at the equilibrium position.
• Total energy is constant and is converted between kinetic and potential energy.

Examples of SHM

• A mass on a spring exhibits SHM.
• A pendulum shows SHM.
• A vibrating string demonstrates SHM.
• Electric circuits with capacitors and inductors can also exhibit SHM.

Simple Harmonic Motion (SHM)

Definition

• SHM is a type of periodic motion where the restoring force is proportional to the displacement from the equilibrium position.
• The motion is sinusoidal and repeats at regular intervals.

Characteristics

• Displacement is the distance from the equilibrium position.
• Amplitude is the maximum displacement from the equilibrium position.
• Period is the time taken to complete one cycle of motion.
• Frequency is the number of cycles per second, measured in Hz (1/s).
• Phase is the initial angle of the motion.

Equations of Motion

• Displacement equation: x(t) = A cos(ωt + φ), where x(t) is displacement at time t, A is amplitude, ω is angular frequency (2πf), and φ is phase angle.
• Velocity equation: v(t) = -Aω sin(ωt + φ).
• Acceleration equation: a(t) = -Aω^2 cos(ωt + φ).

Energy in SHM

• Kinetic energy is maximum at the equilibrium position and minimum at the amplitude.
• Potential energy is maximum at the amplitude and minimum at the equilibrium position.
• Total energy is constant and converted between kinetic and potential energy.

Examples of SHM

• Mass on a spring.
• Pendulum.
• Vibrating string.
• Electric circuits with capacitors and inductors.

Important Concepts

• Resonance and damping are important concepts in SHM.