Question

Two particles vibrating simple harmonically with amplitude A and frequency co along the same straight line. They pass one another when going in opposite direction. Each time when they cross are at a distance 0.707A. What is the phase difference between them?

Solution

Let the equation of motion of two particles be,

$straight y equals straight A space space sin space ωt space and space space space straight y space equals space straight A space sin open parentheses ωt plus straight ϕ close parentheses$
where,
ϕ is the phase difference between two particles.
Each time the particles cross at a distance 0.707A.
Therefore,
0.707A=A sin $omega t$

and 0.707A=A sin(wt+ $straight ϕ$ )

$rightwards double arrow$ 0.707=sin $omega t$ and,
0.707= sn $space space space open parentheses ωt plus straight ϕ close parentheses$
Now,

sin( $left parenthesis ωt plus straight ϕ right parenthesis$ =sin( $left parenthesis wt plus straight ϕ right parenthesis$ =sin
$left parenthesis wt right parenthesis cos left parenthesis straight ϕ right parenthesis plus cos left parenthesis wt right parenthesis sin left parenthesis straight ϕ right parenthesis$
Since,
$s i n space omega t equals 0.707 space rightwards double arrow space c o s space omega t space equals 0.707 space therefore space 0.707 equals 0.707 space c o s left parenthesis straight ϕ right parenthesis plus 0.707 space space s i n left parenthesis straight ϕ right parenthesis space rightwards double arrow space space space space c o s left parenthesis straight ϕ right parenthesis plus s i n left parenthesis straight ϕ right parenthesis equals 1 space rightwards double arrow space space space space c o s open parentheses straight ϕ close parentheses equals 1 minus s i n left parenthesis straight ϕ right parenthesis space rightwards double arrow space space space space c o s squared left parenthesis straight ϕ right parenthesis equals 1 plus s i n squared left parenthesis straight ϕ right parenthesis minus 2 s i n left parenthesis straight ϕ right parenthesis space rightwards double arrow space space space 1 minus s i n squared left parenthesis straight ϕ right parenthesis equals 1 plus s i n squared left parenthesis straight ϕ right parenthesis minus 2 s i n left parenthesis straight ϕ right parenthesis space rightwards double arrow space space space space space space 2 space s i n squared left parenthesis straight ϕ right parenthesis equals 2 space s i n left parenthesis straight ϕ right parenthesis space rightwards double arrow space space space space s i n left parenthesis straight ϕ right parenthesis space equals space 1 space o r space 0 space rightwards double arrow space space space space space space straight ϕ space equals space 0 to the power of degree space space o r space 90 degree$
We can see that ϕ = 0 is not possible.
Therefore the phase difference between two particles is 90°.

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Mechanical Properties Of Fluids

Two particles vibrating simple harmonically with amplitude A and frequency co along the same straight line. They pass one another when going in opposite direction. Each time when they cross are at a distance 0.707A. What is the phase difference between them?

Some More Questions From Mechanical Properties of Fluids Chapter

Define frequency.

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.

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