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  • The motion of a mass on a spring, with spring constant K is as shown in figure. The equation of motion is given by <img alt="x

The motion of a mass on a spring, with spring constant K is as shown in figure. The equation of motion is given by x(t)=A \sin \omega t+$ $B \cos \omega \mathrm{t}$ with $\omega=\sqrt{\frac{\mathrm{K}}{\mathrm{m}}} Suppose that at time t=0, the position of mass is x(0) and velocity v(0), then its displacement can also be represented as x(t)=\operatorname{Cos}(\omega t-\phi), where C$ and $\phi are :
Option: 1 \mathrm{C}=\sqrt{\frac{2 \mathrm{v}(0)^{2}}{\omega^{2}}+\mathrm{x}(0)^{2}}, \phi=\tan ^{-1}\left(\frac{\mathrm{v}(0)}{\mathrm{x}(0) \omega}\right)
Option: 2 \mathrm{C}=\sqrt{\frac{2 \mathrm{v}(0)^{2}}{\omega^{2}}+\mathrm{x}(0)^{2}}, \phi=\tan ^{-1}\left(\frac{\mathrm{x}(0) \omega}{2 \mathrm{v}(0)}\right)
Option: 3 C=\sqrt{\frac{v(0)^{2}}{\omega^{2}}+x(0)^{2}}, \phi=\tan ^{-1}\left(\frac{x(0) \omega}{v(0)}\right)
Option: 4 C=\sqrt{\frac{v(0)^{2}}{\omega^{2}}+x(0)^{2}}, \phi=\tan ^{-1}\left(\frac{v(0)}{x(0) \omega}\right)

Answers (1)

best_answer

\begin{aligned} X &=A \sin \omega t+B \cos \omega t \\ V &=A \omega \cos \omega t+B \omega(-\sin \omega t) \\ \end{aligned}

\begin{aligned} X(0) &= B \\ V(0) &=A \omega \\ \end{aligned}

\begin{aligned} \text { Aso, } X(t) &=C(\cos \omega t \cos \phi+\sin \omega t \sin \phi] \\ &=(c \sin \phi) \cos \omega t+(\cos \phi) \sin \omega t \end{aligned}

            \begin{aligned} c\cos \Phi =B \end{aligned}

            \begin{aligned} c \sin \Phi =A \end{aligned}

\begin{aligned} c^{2}\left(\sin ^{2} \phi+\cos ^{2} \phi\right) &=A^{2}+B^{2} \\ \end{aligned}

                            \begin{aligned} C^{2} &=\frac{V^{2}(0)}{\omega^{2}} +X^{2}_{\left ( 0 \right )}\\ \end{aligned}

\begin{aligned} c=\sqrt{\frac{V^{2}(0)}{\omega^{2}}+X^{2}(0)} \end{aligned}

\tan \Phi =\frac{A}{B}=\left [ \frac{V_{\left ( 0 \right )}}{x_{\left ( 0 \right )}} \right ]\frac{1}{\omega }

\Phi =\tan ^{-1}\left [ \frac{V\left ( 0 \right )}{X\left ( 0 \right )\omega } \right ]

The correct option is (4)

Posted by

vishal kumar

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