Simple Harmonic Motion Equations to Know for College Physics I – Introduction

Simple harmonic motion describes how objects oscillate back and forth around an equilibrium position. Key concepts include angular frequency, period, and energy dynamics, all crucial for understanding mechanics, sound, and waves in physics.

1. Angular frequency: ω = 2πf = √(k/m)

   • Represents how quickly an object oscillates in radians per second.
   • Directly related to the spring constant (k) and mass (m) of the system.
   • Higher angular frequency indicates faster oscillations.

2. Period: T = 2π√(m/k) = 1/f

   • The time it takes to complete one full cycle of motion.
   • Inversely related to frequency; as frequency increases, the period decreases.
   • Depends on mass and spring constant; heavier masses result in longer periods.

3. Frequency: f = 1/T = (1/2π)√(k/m)

   • The number of cycles per unit time, measured in Hertz (Hz).
   • Higher frequency means more oscillations in a given time frame.
   • Influenced by the mass and stiffness of the spring.

4. Displacement: x(t) = A cos(ωt + φ)

   • Describes the position of the oscillating object at any time (t).
   • A is the amplitude, representing the maximum displacement from equilibrium.
   • φ is the phase constant, determining the starting position of the motion.

5. Velocity: v(t) = -Aω sin(ωt + φ)

   • Indicates the speed and direction of the oscillating object at any time (t).
   • Maximum velocity occurs as the object passes through the equilibrium position.
   • Velocity is zero at maximum displacement (amplitude).

6. Acceleration: a(t) = -Aω² cos(ωt + φ)

   • Shows how quickly the velocity of the object is changing at any time (t).
   • Always directed towards the equilibrium position, indicating restoring force.
   • Maximum acceleration occurs at maximum displacement.

7. Maximum velocity: vmax = Aω

   • The highest speed reached by the oscillating object.
   • Directly proportional to both amplitude and angular frequency.
   • Important for understanding the energy dynamics of the system.

8. Maximum acceleration: amax = Aω²

   • The greatest acceleration experienced by the object during its motion.
   • Proportional to the square of the angular frequency and amplitude.
   • Indicates the strength of the restoring force at maximum displacement.

9. Total energy: E = (1/2)kA² = (1/2)mω²A²

   • Represents the total mechanical energy in the system, conserved throughout the motion.
   • Composed of potential and kinetic energy, varying as the object oscillates.
   • Maximum when the object is at maximum displacement.

10. Potential energy: PE = (1/2)kx²

    • Energy stored in the system due to its position (displacement from equilibrium).
    • Increases as the object moves away from the equilibrium position.
    • Maximum at maximum displacement (amplitude).

11. Kinetic energy: KE = (1/2)mv²

    • Energy of the object due to its motion.
    • Maximum when the object is at the equilibrium position.
    • Converts back and forth with potential energy during oscillation.

12. Spring force: F = -kx

    • The restoring force exerted by the spring, proportional to the displacement.
    • Negative sign indicates that the force acts in the opposite direction of displacement.
    • Fundamental to understanding the dynamics of simple harmonic motion.