Simple harmonic motion ?

Simple harmonic motion is a type of periodic motion in which an object oscillates back and forth around a central equilibrium position. The motion is characterized by a constant frequency and amplitude, and the restoring force acting on the object is directly proportional to its displacement from the equilibrium position. Examples of simple harmonic motion include a swinging pendulum, a vibrating guitar string, and a mass-spring system.

Simple Harmonic Motion (SHM)

Definition:
Simple harmonic motion is a periodic motion in which the restoring force acting on the system is directly proportional to the displacement from equilibrium and acts in the opposite direction.

Equation of SHM:
$$x(t) = A \cos(\omega t + \phi)$$

where:
- x(t) is the displacement at time t
- A is the amplitude (maximum displacement)
- ω is the angular frequency
- t is the time
- φ is the phase angle

Characteristics:

- Periodicity: SHM repeats itself at regular intervals called the period (T), where T = 2π/ω.
- Constant acceleration: The acceleration of the object in SHM is proportional to the displacement and is always directed towards equilibrium.
- Restoring force: The force acting on the object is proportional to the displacement and always acts to restore the object to equilibrium.

Examples:

- Pendulum
- Mass on a spring
- Oscillating LC circuit

Properties:

- Maximum velocity: Occurs at equilibrium (x = 0) and is proportional to the amplitude and angular frequency: v_max = Aω.
- Maximum acceleration: Occurs at maximum displacement (x = ±A) and is proportional to the square of the angular frequency and amplitude: a_max = Aω².
- Energy conservation: The total energy of the system in SHM is constant and consists of kinetic and potential energy.

Applications:

- Clocks
- Mechanical oscillators
- Musical instruments
- Vibrating systems

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