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 $p_{0}$
In the light of the above statements, choose the correct answer from the options given below:
Select the correct option:
A
(A) is false but (R) is true
B
Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
C
(A) is true but (R) is false
D
Both (A) and (R) are true and (R) is the correct explanation of (A)
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
$\mathrm{x}=\mathrm{A} \sin (\omega \mathrm{t}+\phi) ; \mathrm{x}_{0}=\mathrm{A} \sin \phi$
$\mathrm{p}=\mathrm{mA} \omega \cos (\omega \mathrm{t}+\phi) \mathrm{p} ; p_{0}=m A \omega \cos \phi$
(2) / (1) $\Rightarrow \tan \phi=\left(\frac{x_{0}}{p_{0}}\right) m \omega$
$\sin \phi=\frac{\mathrm{x}_{0} \mathrm{~m} \omega}{\sqrt{\left(\mathrm{~m} \omega \mathrm{x}_{0}\right)^{2}+\mathrm{p}_{0}^{2}}}$
From (1), $A=\frac{x_{0}}{\sin \phi}=\frac{\sqrt{\left(m \omega x_{0}\right)^{2}+p_{0}^{2}}}{m \omega}$
This means we can explain assertion with the given reason.
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