Question

Which of the following equation can represent the equation of the velocity of a particle, if the particle is executing SHM?
1. v = aω cosωt
2. \(v= \omega \sqrt(a^2-x^2) \)
3. v_max = aω 
4. All the above equations represent velocity of particle executing SHM.

Answer 1

Correct Answer - Option 4 : All the above equations represent velocity of particle executing SHM.

CONCEPT:

• Simple harmonic motion occurs when the restoring force is directly proportional to the displacement from equilibrium.

F α -x

Where F = force and x = the displacement from equilibrium.

The equation of displacement in SHM is given by:

x = A sin(ωt + ϕ) .........(i)

where x is the distance from the mean position at any time t, A is amplitude, t is time, and ω is the angular frequency.​

The equation of velocity in SHM is given by:

v = Aω cos(ωt + ϕ)

or \(v = ω √{A^2-x^2}\)

where v is the velocity at any time t or displacement x, A is amplitude, t is time, and ω is the angular frequency.​

EXPLANATION:

The equation of displacement in SHM is given by:

x = A sinωt .........(i)

x/A = sinωt

x/A = √(1 - cos²ωt)

\(cosωt = { \sqrt{x^2-A^2} \over A}\) ...........(ii)

differentiate eq (i) with respect to time t

v = Aω cosωt

put the value of cosωt From equation (ii)

\(v= \omega \sqrt(A^2-x^2) \)

The maximum value of velocity can be obtained by eq(i) or (ii) both

v = Aω cosωt

The max value of cosθ is 1. so the max value of cosωt will be 1.

v_max = Aω (1)

vmax = Aω

So all the equations are for velocity.

Hence the correct answer is option 4.