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
$E_1=\frac{1}{2}m{w}_1^2{a}_1^2$
$=\frac{1}{2}m\times 10\times 10\times 4\times 4$
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