# If the equation of motion of a particle moving along the y-axis is y = P + Q sin ωt, the particle must be executing simple harmonic motion with amplit

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If the equation of motion of a particle moving along the y-axis is y = P + Q sin ωt, the particle must be executing simple harmonic motion with amplitude
1. P
2. Q
3. P + Q
4. $\sqrt {{P^2} + {Q^2}}$

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Correct Answer - Option 2 : Q

Concept:

Simple harmonic motion:

• It is a special type of periodic motion, in which a particle moves to and fro repeatedly about a mean position.
• In linear S.H.M. a restoring force which is always directed towards the mean position and whose magnitude at any instant is directly proportional to the displacement of the particle from the mean position at that instant

Restoring force ∝  Displacement of the particle from mean position.

F = -kx

Where k is known as force constant. It's S.I. unit is Newton/meter

The general expression for the wave is given by:
y = A Sin (ωt + θ)

y = 0 is the mean position of particle

Where A = amplitude of wave, ω = angular frequency, t is time and θ = phase angle,

Amplitude: It is the maximum displacement of the object from the equilibrium position.

Calculation:

Motion of particle is given by

y = P + Q sin ωt

⇒ y - P =  Q sin ωt

So, particles execute SHM with mean position y = P.

Let y -P = y'

⇒ y' =  Q sin ωt

Comparing this equation by equation of SHM,

We will get amplitude

A = Q

Hence, particles execute simple harmonic motion with amplitude Q.