Question:
The displacement of two identical particles performing S.H.M are represented by equation \(x_{1}=6\,\sin (5t+\pi /4)\) and \(x_{2}=4\,\cos \omega t\). The energies of both the particles will be same, if the value of \(\omega\) is
The displacement of two identical particles performing S.H.M are represented by equation \(x_{1}=6\,\sin (5t+\pi /4)\) and \(x_{2}=4\,\cos \omega t\). The energies of both the particles will be same, if the value of \(\omega\) is
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The energy of a particle in S.H.M. is proportional to the square of amplitude times the square of angular frequency.
Updated On: Nov 12, 2026
- \(9.5\) unit
- \(6.5\) unit
- \(8.5\) unit
- \(7.5\) unit
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
Energy in S.H.M. is \(E=\frac{1}{2}m\omega^{2}A^{2}\). The particles are identical, so \(A^{2}\omega^{2}\) must be equal.
Step 2: Read the amplitudes
For \(x_{1}=6\sin(5t+\frac{\pi}{4})\): \(A_{1}=6\) and \(\omega_{1}=5\). For \(x_{2}=4\cos\omega t\): \(A_{2}=4\) and angular frequency \(\omega\).
Step 3: Calculation
\(6^{2}\times5^{2}=4^{2}\times\omega^{2}\), so \(\omega^{2}=\frac{900}{16}=56.25\) and \(\omega=7.5\).
Final Answer:
The value of omega is 7.5 units. \[ \boxed{\text{(D) }7.5\ \text{unit}} \]
Energy in S.H.M. is \(E=\frac{1}{2}m\omega^{2}A^{2}\). The particles are identical, so \(A^{2}\omega^{2}\) must be equal.
Step 2: Read the amplitudes
For \(x_{1}=6\sin(5t+\frac{\pi}{4})\): \(A_{1}=6\) and \(\omega_{1}=5\). For \(x_{2}=4\cos\omega t\): \(A_{2}=4\) and angular frequency \(\omega\).
Step 3: Calculation
\(6^{2}\times5^{2}=4^{2}\times\omega^{2}\), so \(\omega^{2}=\frac{900}{16}=56.25\) and \(\omega=7.5\).
Final Answer:
The value of omega is 7.5 units. \[ \boxed{\text{(D) }7.5\ \text{unit}} \]
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