The total mechanical energy of a spring-mass system in simple harmonic motion is \[E = \frac{1}{2}m \omega^2 A^2 .\] Suppose the oscillating particle is replaced by another particle of double the mass while the amplitude *A* remains the same. The new mechanical energy will

#### Options

become 2

*E*become

*E*/2become \[\sqrt{2}E\]

remain

*E*

#### Solution

remain *E*

Mechanical energy (E) of a spring-mass system in simple harmonic motion is given by, \[E_{} = \frac{1}{2}m \omega^2 A^2\]

where *m* is mass of body, and \[\omega\] is angular frequency.

Let *m*_{1} be the mass of the other particle and *ω*_{1} be its angular frequency.

New angular frequency ω_{1} is given by,\[\omega_1 = \sqrt{\frac{k}{m_1}} = \sqrt{\frac{k}{2m}} ( m_1 = 2m)\]

New energy* **E*_{1} is given as,

\[E_1 = \frac{1}{2} m_1 \omega_1^2 A^2 \]

\[ = \frac{1}{2}(2m)(\sqrt{\frac{k}{2m}} )^2 A^2 \]

\[ = \frac{1}{2}m \omega^2 A^2 = E\]