Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Knowing initial position \(x_0\) and initial momentum \(p_0\) is enough to determine the position and momentum at any time \(t\) for a simple harmonic motion with a given angular frequency \(\omega\).
Reason (R): The amplitude and phase can be expressed in terms of \(x_0\) and \(\mathrm{p}_0\).
In the light of the above statements, choose the correct answer from the options given below :

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Correct Answer

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

Step-by-step Solution

Detailed explanation

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

Same subject

List\(-I\)	List\(-II\)
\(UV\) rays	\((i)\) Diagnostic tool in medicine
\(X-\) rays	\((ii)\) Water purification
Microwave	\((iii)\) Communication, Radar
Infrared wave	\((iv)\) Improving visibility in foggy days

Choose the correct answer from the options given below.

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