In simple harmonic motion (SHM), the ratio of potential energy (P.E) to kinetic energy (K.E) depends on the displacement of the oscillating object. The total mechanical energy of the system remains constant throughout the motion.
When the displacement is equal to half of the amplitude (x₀/2), the ratio of P.E to K.E can be calculated as follows:
At maximum displacement, the total mechanical energy of the system is given by:
E = P.E + K.E
Since the total mechanical energy is conserved, it remains constant at all points in the motion. At maximum displacement, all the energy is in the form of potential energy (P.E), and the kinetic energy (K.E) is zero.
When the displacement is equal to half of the amplitude (x₀/2), the potential energy is at its maximum, and the kinetic energy is at its minimum. At this point, the potential energy is equal to the total mechanical energy (E), and the kinetic energy is zero:
P.E = E K.E = 0
Therefore, the ratio of P.E to K.E at this specific displacement is:
P.E/K.E = E/0 = ∞
At this point, the ratio of P.E to K.E is infinite, indicating that all the energy is stored as potential energy and there is no kinetic energy.