MHT CET · Physics · Oscillations
A body performing simple harmonic motion has potential energy '\(\mathrm{P}_{1}\)' at displacement ' \(x_{1}\) '. Its potential energy is ' \(\mathrm{P}_{2}\) ' at displacement ' \(x_{2}\) '. The potential energy 'P' at displacement \(\left(x_{1}+x_{2}\right)\) is
- A \(\mathrm{P}_{1}+\mathrm{P}_{2}\)
- B \(\sqrt{\mathrm{P}_{1} \mathrm{P}_{2}}\)
- C \(\sqrt{\mathrm{P}_{1}^{2}+\mathrm{P}_{2}^{2}}\)
- D \(\mathrm{P}_{1}+\mathrm{P}_{2}+2 \sqrt{\mathrm{P}_{1} \mathrm{P}_{2}}\)
Answer & Solution
Correct Answer
(D) \(\mathrm{P}_{1}+\mathrm{P}_{2}+2 \sqrt{\mathrm{P}_{1} \mathrm{P}_{2}}\)
Step-by-step Solution
Detailed explanation
(B)
\(\begin{aligned} P &=\frac{1}{2} k\left(x_{1}+x_{2}\right)^{2} \\ &=\frac{1}{2} k\left(x_{1}^{2}+x_{2}^{2}+2 x_{1} x_{2}\right) \\ &=\frac{1}{2} k x_{1}^{2}+\frac{1}{2} k x_{2}^{2}+2\left(\sqrt{\frac{k}{2}} x\right)\left(\sqrt{\frac{k}{2}} x_{2}\right) \\ &=P_{1}+P_{2}+2 \sqrt{P_{1} P_{2}} \end{aligned}\)
\(\begin{aligned} P &=\frac{1}{2} k\left(x_{1}+x_{2}\right)^{2} \\ &=\frac{1}{2} k\left(x_{1}^{2}+x_{2}^{2}+2 x_{1} x_{2}\right) \\ &=\frac{1}{2} k x_{1}^{2}+\frac{1}{2} k x_{2}^{2}+2\left(\sqrt{\frac{k}{2}} x\right)\left(\sqrt{\frac{k}{2}} x_{2}\right) \\ &=P_{1}+P_{2}+2 \sqrt{P_{1} P_{2}} \end{aligned}\)
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