Effects of Doubling Mass and Amplitude

1. **Period of Motion**: The period of a mass-spring system in SHM is given by the formula:

T = 2π√(m/k)

When the mass is doubled (let's say from m to 2m), the new period becomes:

T' = 2π√(2m/k) = 2π√(2)√(m/k) = √2 * T

This means that the period increases by a factor of √2. Thus, the motion becomes slower because a larger mass takes more time to complete each oscillation.

2. **Energy of the System**: The total mechanical energy (E) in SHM is given by:

E = (1/2)kA²

When the amplitude is doubled (from A to 2A), the new energy becomes:

E' = (1/2)k(2A)² = (1/2)k(4A²) = 4 * (1/2)kA² = 4E

This indicates that the energy of the system increases by a factor of 4 when the amplitude is doubled. More energy means that the particle will oscillate with greater maximum speeds and displacements.

Summary of Changes

To summarize:

• The period of the motion increases by a factor of √2 due to the doubling of mass.
• The total mechanical energy increases by a factor of 4 due to the doubling of amplitude.

These changes illustrate how mass and amplitude influence the dynamics of simple harmonic motion. By understanding these relationships, we can predict how a system will behave under different conditions, which is crucial in fields ranging from engineering to physics.