Given below are two statements. One is labell | vedclass

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(D) For a simple harmonic motion,the position is given by $x(t) = A \sin(\omega t + \phi)$.At $t = 0$,$x_0 = A \sin \phi$  $..........(1)$The momentum

Vedclass · Accepted Answer

Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A)$.

Vedclass · Answer

(D) For a simple harmonic motion,the position is given by $x(t) = A \sin(\omega t + \phi)$.At $t = 0$,$x_0 = A \sin \phi$  $..........(1)$The momentum is $p(t) = m \frac{dx}{dt} = m A \omega \cos(\omega t + \phi)$.At $t = 0$,$p_0 = m A \omega \cos \phi$  $..........(2)$From equations $(1)$ and $(2)$,we can solve for the two unknowns,amplitude $A$ and phase constant $\phi$,using the initial conditions $x_0$ and $p_0$.Specifically,$	an \phi = \frac{m \omega x_0}{p_0}$ and $A = \sqrt{x_0^2 + (p_0 / m \omega)^2}$.Since $A$ and $\phi$ are uniquely determined by $x_0$ and $p_0$,the state of the system at any time $t$ is fully determined.Thus,both Assertion $(A)$ and Reason $(R)$ are true,and $(R)$ is the correct explanation of $(A)$.

Similar Questions

Define simple harmonic motion $(SHM)$.

What is the value of frequency in the differential equation of $SHM$ $\frac{d^{2}x}{dt^{2}} + 100x = 0$?

Which one of the following is a simple harmonic motion?

For particle $P$ revolving around the centre $O$ with radius of circular path $r$ and angular velocity $\omega$,as shown in the figure,the projection of $OP$ on the $x$-axis at time $t$ is .................

$A$ body oscillates with $SHM$ according to the equation (in $SI$ units):
$x = 5 \cos (2 \pi t + \pi / 4)$
At $t = 1.5 \, s$,calculate the:
$(a)$ displacement
$(b)$ speed
$(c)$ acceleration of the body.

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