Question 316: Oscillations

Question

A body is executing a linear S.H.M. Its potential energies at the displacement '$$\mathrm{x}$$' and '$$\mathrm{y}$$' are '$$\mathrm{E}_1$$' and '$$E_2$$' respectively. Its potential energy at displacement $$(\mathrm{x}+\mathrm{y})$$ will be

Accepted Answer

We know,
Potential Energy $$\mathrm{E}_{\mathrm{P}}=\frac{1}{2} \mathrm{Kx}^2$$
$$	herefore \quad \mathrm{E}_1=\frac{1}{2} \mathrm{Kx}^2 \Rightarrow \mathrm{x}=\sqrt{\frac{2 \mathrm{E}_1}{\mathrm{~K}}}$$ .... (i)
and
$$E_2=\frac{1}{2} K y^2 \Rightarrow y=\sqrt{\frac{2 E_2}{K}}$$ ..... (ii)
Given, total displacement $$=(x+y)$$
$$	herefore \quad$$ Potential energy at displacement $$(x+y)$$,
$$\mathrm{E}$$ is $$\frac{1}{2} \mathrm{~K}(\mathrm{x}+\mathrm{y})^2$$
$$\begin{aligned}
& =\frac{1}{2} K\left(\sqrt{\frac{2 E_1}{K}}+\sqrt{\frac{2 E_2}{K}}ight)^2 \
& =\frac{1}{2} K\left[\frac{2 E_1}{K}+\frac{2 E_2}{K}+2\left(\sqrt{\frac{E_1}{K}}ight)\left(\sqrt{\frac{2 E_2}{K}}ight)ight] \
& =\left(\sqrt{E_1}+\sqrt{E_2}ight)^2=\left(E_1+E_2+2 \sqrt{E_1 E_2}ight)
\end{aligned}$$

Answer

$$\mathrm{E}_1+\mathrm{E}_2$$

Answer

$$\left(\sqrt{\mathrm{E}_1}+\sqrt{\mathrm{E}_2}\right)^2$$

Answer

$$\quad \mathrm{E}_1-\mathrm{E}_2$$

Answer

$$\left(\sqrt{\mathrm{E}_2}-\sqrt{\mathrm{E}_1}\right)^2$$

Oscillations

A body is executing a linear S.H.M. Its potential energies at the displacement '$\mathrm{x}' and '\mathrm{y}' are '\mathrm{E}_1' and 'E_2' respectively. Its potential energy at displacement (\mathrm{x}+\mathrm{y})$ will be

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