Final answer:
The total energy of a particle in simple harmonic motion will become 4 times its initial energy when its amplitude is doubled, regardless of the time period.
Step-by-step explanation:
Total Energy of a Vibrating Particle in S.H.M.
The question asks about the total energy of a particle undergoing simple harmonic motion (S.H.M.) when its amplitude and time period are doubled. The total energy in S.H.M. is given by the sum of its kinetic energy and potential energy, which remains constant for a given amplitude and mass.
According to the formula for energy in S.H.M., E is proportional to the square of the amplitude (A). When the amplitude is doubled, the energy becomes (2A)2 = 4A2, hence the energy becomes four times the initial energy (4E).
The time period does not affect the total energy of the system.
Therefore, if the amplitude is doubled, the total energy of the particle in S.H.M. will become 4 times its initial energy (E).