When an object or a particle moves back and forth repeatedly about a reference point for some duration of time it is said to have Oscillatory (or vibratory) motion.
For a SHM, the acceleration or force on the particle is directly proportional to its displacement from a fixed point and always directed towards that fixed point. The force is
Fx = − k x
where k is a constant whose dimension is force per unit length, called as force constant.
In Simple harmonic motion, the displacement, y = A sin ωt.
In Simple harmonic motion, the velocity, v = A ω cos ωt = ω√[A2-y2]
In Simple harmonic motion, the acceleration, a =
The time period is defined as the time taken by a particle to complete one oscillation. It is usually denoted by T. Time period T = 2π/ω
The number of oscillations produced by the particle per second is called frequency. It is denoted by f. SI unit for frequency is S−1 or hertz (In symbol, Hz). Mathematically, frequency is related to time period by f =1/T.
The frequency of the angular harmonic motion is
For n springs connected in series, the effective spring constant in series is
For n springs connected in parallel, the effective spring constant is
The time period for U-tube oscillation is T
For a conservative system in one dimension, the force field can be derived from a scalar potential energy:
In a simple harmonic motion, potential energy is U ( x).
In a simple harmonic motion, kinetic energy is KE.
Total energy for a simple harmonic motion is .
Types of oscillations – Free oscillations, Damped oscillations, Maintained oscillations and Forced oscillations.
Resonance is a special case of forced oscillations.