Understanding Simple Harmonic Motion
In simple harmonic motion (SHM), a particle oscillates about a mean position, and its energy is composed of kinetic energy (KE) and potential energy (PE). The relationship between these energies can reveal important information about the particle's displacement.
Energy Ratio in SHM
- The given ratio of kinetic energy to potential energy is 1:3.
- This implies that at this point in its motion, the potential energy is three times the kinetic energy.
Energy Formulae
- The total mechanical energy (E) in SHM is constant and can be expressed as:
- E = KE + PE.
- When KE = 1x, then PE = 3x, thus:
- Total Energy, E = 1x + 3x = 4x.
Displacement Calculation
- In SHM, the kinetic energy is given by the formula:
- KE = (1/2)mv^2.
- The potential energy is given by:
- PE = (1/2)kx^2.
- The displacement (x) can be related to the energy states. Given the ratio of energies, we can express this situation in terms of the maximum potential energy (when displacement is maximum, A):
- PE = (1/2)kA^2.
Finding Displacement
- Using the ratio of energies:
- Since PE = 3x and KE = 1x, at this point, we can write:
- 3(1/2)kx^2 = (1/2)kA^2.
- The displacement (x) is determined by:
- x = A/sqrt(3).
- Therefore, the displacement is one-third of the amplitude (A) in this energy state.
Conclusion
- The ratio of kinetic to potential energy helps us find the specific displacement where the particle is located in its oscillatory path. In this case, the displacement is A/sqrt(3), indicating a position where potential energy dominates in the oscillation cycle.