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)

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