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Mention the expression for velocity and acceleration of a particle executing SHM.

Answer
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Hint:To find the expressions of velocity and acceleration of a particle executing simple harmonic motion, we need to use the equation of displacement of the particle from its equilibrium position in simple harmonic motion. The first derivative of the displacement with respect to time will give us the expression of velocity and by further differentiation, the second derivative of the displacement with respect to time will give us the expression of acceleration.

Formulas used:
x=Asin(ωt+ϕ),
Where, x is the displacement from equilibrium position, A is amplitude, ω is angular frequency, t is time and ϕ is initial face angle. Here, (ωt+ϕ)is called phase.
v=dxdt ,
Where, v is the velocity of particle, xis the displacement from equilibrium position and tis time
a=dvdt,
Where, a is the acceleration of the particle, v is the velocity of particle and t is time

Complete step by step answer:
We will first derive the expression for the velocity of the particle which executes simple harmonic motion by differentiating its displacement from the equilibrium position with respect to time. We know that the displacement from equilibrium position is given by:
x=Asin(ωt+ϕ)
v=dxdt
v=ddt(Asin(ωt+ϕ))=ωAcos(ωt+ϕ)
Now, at the equilibrium, initial phase angle is zero which means ϕ=0
v=ωAcosωt=ωA1sin2ωt

Also at equilibrium, x=Asinωtsin2ωt=x2A2
Putting this value in the equation of velocity, we get
v=Aω1x2A2=ωA2x2
Thus, the expression of the velocity of the particle executes simple harmonic motion is v=ωA2x2
Now, let us find the expression of acceleration of the particle executing simple harmonic motion by differentiating its velocity with respect to time.
a=dvdt
a=ddt(ωAcos(ωt+ϕ))=ω2Asin(ωt+ϕ)
But, we know that x=Asin(ωt+ϕ)
a=ω2x

Thus, the expressions for velocity and acceleration of a particle executing simple harmonic motion are v=ωA2x2and a=ω2x respectively.

Additional information:
v=ωA2x2
Squaring both sides, we get
v2=ω2(A2x2)v2ω2=A2x2v2ω2A2=1x2A2x2A2+v2ω2A2=1
This is the equation of an ellipse.
Therefore, we can say that the curve between displacement and velocity of a particle executing the simple harmonic motion is an ellipse.
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Also, the acceleration of the particle in simple harmonic motion is given by a=ω2x. And if we plot the graph of acceleration and displacement, it will be a straight line.
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Note:Here, we have seen that the displacement of the particle executing SHM is dependent on amplitude, angular frequency, time and initial face angle. We have derived the formula for velocity and acceleration of this particle using its displacement. That is why they both are also dependent on the same parameters.
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