Question

A particle is executing SHM along a straight line. Its velocities at distance x₁ and x₂ from the mean position are v₁ and v₂, respectively. Its time period is

$2 pi square root of fraction numerator straight x subscript 1 superscript 2 plus straight x subscript 2 superscript 2 over denominator straight v subscript 1 superscript 2 plus straight v subscript 2 superscript 2 end fraction end root$
$2 pi square root of fraction numerator straight x subscript 2 superscript 2 minus straight x subscript 1 superscript 2 over denominator straight v subscript 1 superscript 2 minus straight v subscript 2 superscript 2 end fraction end root$
$2 pi square root of fraction numerator straight v subscript 1 superscript 2 plus straight v subscript 2 superscript 2 over denominator straight x subscript 1 superscript 2 plus straight x subscript 2 superscript 2 end fraction end root$
$2 pi square root of fraction numerator straight v subscript 1 superscript 2 minus straight v subscript 2 superscript 2 over denominator straight x subscript 1 superscript 2 minus straight x subscript 2 superscript 2 end fraction end root$

Solution

2 pi square root of fraction numerator straight x subscript 2 superscript 2 minus straight x subscript 1 superscript 2 over denominator straight v subscript 1 superscript 2 minus straight v subscript 2 superscript 2 end fraction end root

Let A be the amplitude of oscillation then,
$straight v subscript 1 superscript 2 space equals space straight omega squared space left parenthesis straight A squared minus straight x subscript 1 superscript 2 right parenthesis space space space space... space left parenthesis straight i right parenthesis straight v subscript 1 superscript 2 space equals space straight omega squared left parenthesis straight A squared minus straight x squared right parenthesis space space space space.... space left parenthesis ii right parenthesis subtracting space Eq. space left parenthesis ii right parenthesis thin space from space Eq space left parenthesis straight i right parenthesis comma space we space get straight v subscript 1 superscript 2 minus straight v subscript 2 superscript 2 space equals space straight omega squared space left parenthesis straight x subscript 2 superscript 2 minus straight x subscript 1 superscript 2 right parenthesis rightwards double arrow space straight omega space equals space square root of fraction numerator straight v subscript 1 superscript 2 minus straight v subscript 2 superscript 2 over denominator straight x subscript 2 superscript 2 minus straight x subscript 1 superscript 2 end fraction end root rightwards double arrow space fraction numerator 2 straight pi over denominator straight T end fraction equals space square root of fraction numerator straight v subscript 1 superscript 2 minus straight v subscript 2 superscript 2 over denominator straight x subscript 2 superscript 2 minus straight x subscript 1 superscript 2 end fraction end root straight T equals space 2 straight pi square root of fraction numerator straight x subscript 2 superscript 2 minus straight x subscript 1 superscript 2 over denominator straight v subscript 1 superscript 2 minus straight v subscript 2 superscript 2 end fraction end root$

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Mechanical Properties Of Fluids

Some More Questions From Mechanical Properties of Fluids Chapter

What is the condition for motion to be simple harmonic motion?

In the displacement equation y = r sin(ωt/ + ϕ) what does ϕ called and what information does it give?

What determines the natural frequency of an oscillator?

Is there any relation between uniform circular motion and simple harmonic motion?

What is periodic motion?

Write down the expression for the instantaneous velocity and acceleration of particle vibrating in simple harmonic motion.

How do you define the mean position of particle executing simple harmonic motion?

Define amplitude.

What is the net displacement of the particle executing simple harmonic motion in one complete oscillation?

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