Question

The velocity of a particle executing simple harmonic motion is V when displacement is x₁ and v₂ when displacement is x₂. What is the frequency of oscillation?

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Solution

The velocity of the particle executing simple harmonic motion when at a distance x from mean position is given by,

space space space space space space space space space straight v equals straight omega square root of straight r squared minus straight y squared end root space Here space we space have comma space space space space straight v equals space straight v subscript 1 space space at space straight x equals straight x subscript 1 space comma space and space space space straight v equals straight v subscript 2 space space at space straight x equals straight x subscript 2 space therefore space space space space space space space space space space space space straight v subscript 1 equals straight omega square root of straight r squared minus straight x subscript 1 squared end root space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space and space space space space space space space space space space straight v subscript 2 equals straight omega square root of straight r squared minus straight x subscript 2 squared end root space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis

Squaring (1) and (2) and subtracting,

straight v subscript 1 squared minus straight v subscript 2 squared equals straight omega squared left parenthesis straight x subscript 2 squared minus straight x subscript 1 squared right parenthesis rightwards double arrow space space space space space straight omega squared equals fraction numerator straight v subscript 1 squared minus straight v subscript 2 squared over denominator straight x subscript 2 squared minus straight x subscript 1 squared end fraction rightwards double arrow space space space space straight omega equals square root of fraction numerator straight v subscript 1 squared minus straight v subscript 2 squared over denominator straight x subscript 2 squared minus straight x subscript 1 squared end fraction end root

straight omega

is the frequency of the oscillation.

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