Oscillations

JEE Main level, single correct, medium

107 of 194

Question 107 of 194

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Knowing initial position x0x_0 and initial momentum p0p_0 is enough to determine the position and momentum at any time tt for a simple harmonic motion with a given angular frequency ω\omega.
Reason (R): The amplitude and phase can be expressed in terms of x0x_0 and p0\mathrm{p}_0.
In the light of the above statements, choose the correct answer from the options given below :

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Answer: (D) Both (A)(\mathbf{A}) and (R)(\mathbf{R}) are true and (R)(\mathbf{R}) is the correct explanation of (A)(\mathbf{A})

 If we express position x(t)=Asin⁡(ωt+ϕ) then x0=Asin⁡ϕv0=Aωcos⁡ϕ⇒tan⁡ϕ=ωx0v0A=x02+v02ω2\begin{aligned} & \text { If we express position } x(t)=A \sin (\omega t+\phi) \\ & \text { then } x_0=A \sin \phi \\ & v_0=A \omega \cos \phi \\ & \Rightarrow \tan \phi=\frac{\omega x_0}{v_0} \\ & A=\sqrt{x_0^2+\frac{v_0^2}{\omega^2}}\end{aligned}
Hence both position and linear momentum of a particle can be expressed as a function of time if we know initial momentum and position

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