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• If a cylinder of mass m, length L, density of material p and uniform area of cross section A,
oscillates vertically in a liquid of density o, then the time period of oscillation is given by

• Undamped and Damped Simple Harmonic Oscillations
Undamped Simple Harmonic oscillations: When a simple harmonic system oscillates with a
constant amplitude which does not change with time, its oscillations are called undamped simple
harmonic oscillations.
Damped Simple Harmonic oscillations: When a simple harmonic system oscillates with a
decreasing amplitude with time, its oscillations are called damped simple harmonic oscillations.
The angular frequency of the damped oscillator is given by

• A system is said to execute free oscillations, if on being displaced or disturbed from its position of
equilibrium, it oscillates itself without outside interference.
When a system is compelled to oscillate with a frequency other than its natural frequency, it is said
to execute forced oscillations.
The external force which causes forced oscillation, is of sinusoidal nature. It is given as

• Resonance is the phenomenon of setting a body into oscillations with large amplitude under the
influence of some external periodic force whose frequency is exactly equal to the natural
frequency of the given body. Such oscillations are called the “resonant oscillations”.
• The two or more oscillations linked together in such a way that the exchange of energy takes
place between them are called coupled oscillators. The oscillations produced by coupled
oscillators are known as coupled oscillations.
• The speed of a mechanic wave depends upon the properties of the medium in which it is
travelling. If E is the elastic constant and ρ is the density of the medium then the speed of the
wave is given by