Mechanical Properties of Fluids

The velocity of a particle executing simple harmonic motion is v when displacement is x 1 and v 2 when displacement is x 2 . What is the amplitude of vibration of the particle? - Wired Faculty

Question

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The velocity of a particle executing simple harmonic motion is v when displacement is x₁ and v₂ when displacement is x₂. What is the amplitude of vibration of the particle?

Solution

The velocity of the particle executing simple harmonic motion when at a distance x from mean position is given by,
$straight v space equals space straight omega space square root of straight r squared minus straight y squared end root Here comma space straight v space equals space straight v subscript 1 space at space space straight x space equals space straight x subscript 1 space comma space and space straight v space equals space straight v subscript 2 space end subscript at space straight x space equals space straight x subscript 2 space Therefore comma space straight v subscript 1 space end subscript equals space straight omega space square root of straight r squared minus straight x subscript 1 squared end root space space space space space space space... space left parenthesis 1 right parenthesis space and straight v subscript 2 space equals space straight omega space square root of straight r squared minus straight x subscript 2 squared end root space Squaring space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis space and space dividing comma space straight v subscript 1 squared over straight v subscript 2 squared space equals space fraction numerator straight omega squared left parenthesis straight r squared minus straight x subscript 1 squared right parenthesis over denominator straight omega squared left parenthesis straight r squared minus straight x subscript 2 squared right parenthesis end fraction space space space space space space space space space space space space equals space fraction numerator left parenthesis straight r squared minus straight x subscript 1 squared right parenthesis over denominator left parenthesis straight r squared minus straight x subscript 2 squared right parenthesis end fraction space space space space space space space space space straight v subscript 1 squared straight r squared space minus space straight v subscript 1 squared straight x subscript 2 squared space equals space straight v subscript 2 squared straight r squared space minus space straight v subscript 2 squared straight x subscript 1 squared space space rightwards double arrow space space straight r squared space left parenthesis straight v subscript 2 squared space minus space straight v subscript 1 squared right parenthesis thin space equals space straight v subscript 2 squared straight x subscript 1 squared space minus space straight v subscript 1 squared straight x subscript 2 squared space straight i. straight e. comma space space space space space space space space space space space space space space space space space space space space space straight r space equals space square root of fraction numerator straight v subscript 2 squared straight x subscript 1 squared space minus space straight v subscript 1 squared straight x subscript 2 squared over denominator straight v subscript 2 squared minus straight v subscript 1 squared end fraction end root$

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