Revision: Oscillations Physics HSC Science (General) 12th Standard Board Exam Maharashtra State Board

When a body oscillates under the influence of an external periodic force, not with its own natural frequency but with the frequency of the external periodic force, its oscillations are called forced oscillations.

The particular case of forced oscillations in which the frequency of the driving force equals the natural frequency of the oscillator, and the amplitude of oscillations is very large — such oscillations are called resonant oscillations and the phenomenon is called resonance.

The motion which repeats itself after equal intervals of time is called periodic motion.

The periodic (to and fro) and bounded motion of a body about a fixed point is called oscillatory motion.

When a body, capable of oscillation, is slightly displaced from its position of equilibrium and left to itself, it starts oscillating with a frequency of its own — such oscillations are called free oscillations.

The oscillations in which the amplitude decreases gradually with the passage of time are called damped oscillations.

When energy is continuously supplied from outside to an oscillating system at the same rate at which energy is lost, so that the amplitude remains constant, such oscillations are called maintained oscillations.

If the restoring force/torque acting on a body in oscillatory motion is always directly proportional to its displacement from the equilibrium position and directed towards it, then the motion is called simple harmonic motion (SHM).

Time period: The time period is defined as the time taken by a particle to complete one oscillation. It is usually denoted by T. For one complete revolution, the time taken is t = T, therefore,

`ω"T"` = 2π ⇒ 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 (Hz).

Mathematically, frequency is related to the time period by f = `1/"T"`

Any function that repeats itself at regular intervals of its argument is called a periodic function.

The physical quantity which represents the position and direction of motion at any instant of the particle executing SHM is called phase.

The constant φ in the equation of SHM x = Acos⁡(ωt + ϕ) is called phase constant or initial phase.

Linear S.H.M. is defined as the linear periodic motion of a body in which the restoring force (or acceleration) is always directed towards its mean position and its magnitude is directly proportional to the displacement from the mean position.
Consider a particle ‘P’ moving along the circumference of a circle of radius 'a' and centre O, with uniform angular speed of 'ω' in anticlockwise direction as shown.
Particle P along the circumference of the circle has its projection particle on diameter AB at point M.

• When a particle is subjected simultaneously to two SHMs having the same period and along the same path, the resultant motion is also SHM along the same path.
• The resultant displacement of the particle at any instant is equal to the vector sum of its individual displacements due to both the SHMs at that instant.