MHT CET · Physics · Oscillations
A body is executing S.H.M. Its potential energy is ' \(\mathrm{P}_1\) ' and ' \(\mathrm{P}_2\) ' at displacements ' \(x\) ' and ' \(y\) ' respectively. The potential energy at displacement \((\mathrm{x}+\mathrm{y})\) is
- A \(\sqrt{\mathrm{P}_1}-\sqrt{\mathrm{P}_2}=\sqrt{\mathrm{p}}\)
- B \(\mathrm{p}_1+\mathrm{p}_2=\mathrm{p}\)
- C \(\mathrm{p}_1-\mathrm{p}_2=\mathrm{p}\)
- D \(\sqrt{\mathrm{p}_1}+\sqrt{\mathrm{p}_2}=\sqrt{\mathrm{p}}\)
Answer & Solution
Correct Answer
(D) \(\sqrt{\mathrm{p}_1}+\sqrt{\mathrm{p}_2}=\sqrt{\mathrm{p}}\)
Step-by-step Solution
Detailed explanation
At displacement \(\mathrm{x}\)
At displacement y
\(\mathrm{E}_1=\frac{1}{2} \mathrm{~m} \omega^2 \mathrm{x}^2\)
From equations
(1), Or (2) and (3) it follows that
\(\sqrt{\mathrm{E}}=\sqrt{\mathrm{E}_1}+\sqrt{\mathrm{E}_2}\)
Or \(\mathrm{E}=\mathrm{E}_1+\mathrm{E}_2+2 \sqrt{\mathrm{E}_1 \mathrm{E}_2}\)
At displacement y
\(\mathrm{E}_1=\frac{1}{2} \mathrm{~m} \omega^2 \mathrm{x}^2\)
From equations
(1), Or (2) and (3) it follows that
\(\sqrt{\mathrm{E}}=\sqrt{\mathrm{E}_1}+\sqrt{\mathrm{E}_2}\)
Or \(\mathrm{E}=\mathrm{E}_1+\mathrm{E}_2+2 \sqrt{\mathrm{E}_1 \mathrm{E}_2}\)
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