Question

The displacements of two particles of same mass executing SHM are represented by the equations ${{x}_{1}}=4\sin \left( 10t+\frac{\pi }{6} \right)$ and ${{x}_{2}}=5\cos (\omega t)$ The value of co for which the energies of both the particles remain same is

• A. 16 unit
• B. 6 unit
• C. 4 unit
• D. 8 unit

Answer 1

Answer: (D)

8 unit

Answer 2

The equation of displacement ${{x}_{1}}=4\sin \left( 10t+\frac{\pi }{6} \right)$ The energy of this equation ${{E}_{1}}=\frac{1}{2}mw_{1}^{2}a_{1}^{2}$ $=\frac{1}{2}m\times 10\times 10\times 4\times 4$ The second equation of displacement ${{x}_{2}}=5\cos (\omega t)$ The energy of this equation ${{E}_{2}}=\frac{1}{2}m\omega _{2}^{2}a_{2}^{2}$ $=\frac{1}{2}m\omega _{2}^{2}\times 5\times 5$ According to question $\because$ ${{E}_{1}}={{E}_{2}}$ $\therefore$ $\frac{1}{2}m\omega _{2}^{2}\times 5\times 5=\frac{1}{2}m\times 10\times 10\times 4\times 4$ $\omega _{2}^{2}=\frac{10\times 10\times 4\times 4}{5\times 5}$ $\omega _{2}^{2}=2\times 2\times 4\times 4$ Or ${{\omega }_{2}}=\sqrt{2\times 2\times 4\times 4}$ $=2\times 4=8$ unit