Question

If the total energy of the particle executing SHM is $e$ and amplitude $a$ if the mass is $m$. Find the new total energy if a new mass $m$ is attached to the particle at its extreme?

Answer 1

Every object has energy, whether it is moving or stationary. The object moves back and forth along the same route in simple harmonic motion. An item possesses energy in simple harmonic motion while traversing the same path again and over. The energy in a simple harmonic oscillator is split between kinetic and potential energy. The system's overall energy remains constant. A continual exchange of kinetic and potential energy occurs in simple harmonic motion.

Complete answer:
We know that the total energy of a SHM motion at its most extreme point is given by $\dfrac{1}{2}m{\omega ^2}{A^2}$ (where $A$ is the amplitude of the motion)
Now, if we gently apply a mass m to this particle, its kinetic energy will be $0$ , and because the velocity of a particle in SHM at its extreme point is also zero, the system's new total energy will be,
$\dfrac{1}{2}m{\omega ^2}{A^2} + 0 = e$
As a result, the overall energy will remain unchanged.

Additional Information:
Simple harmonic motion can be used to mimic a number of motions, but it is best exemplified by the oscillation of a mass on a spring when it is subjected to Hooke's law's linear elastic restoring force. The motion has a single resonance frequency and is sinusoidal in time.

Note: It should be remembered that the mass continues to fluctuate as long as there is no energy loss in the system. As a result, simple harmonic motion is a periodic motion. When energy is lost in a system, the mass oscillates in a damped manner.