In Simple Harmonic Motion (SHM), when the displacement is equal to the amplitude, the fraction of the total energy that is kinetic is 50%.
In SHM, the total mechanical energy (E) of the system is the sum of its kinetic energy (KE) and potential energy (PE):
Total Energy (E) = Kinetic Energy (KE) + Potential Energy (PE)
At the amplitude of the oscillation, the displacement from the equilibrium position is equal to the amplitude (A). At this point, the entire energy of the system is in the form of kinetic energy, and the potential energy is zero. This is because when the displacement is maximum (equal to the amplitude), the potential energy is at its minimum, and all the energy is in the form of kinetic energy.
Therefore, at the amplitude of the SHM, the kinetic energy (KE) is equal to the total energy (E), and the fraction of total energy that is kinetic is:
Fraction of Total Energy that is Kinetic = KE / E = KE / (KE + PE) = KE / (KE + 0) = KE / KE = 1
So, at the amplitude of SHM, 100% of the total energy is kinetic.
On the other hand, at the equilibrium position (where the displacement is zero), the kinetic energy is zero, and all the energy is in the form of potential energy. Thus, the fraction of total energy that is kinetic at the equilibrium position is 0%.
As the particle oscillates between the amplitude and the equilibrium position, the fraction of total energy that is kinetic will vary sinusoidally, with 100% at the amplitude and 0% at the equilibrium position. At any other point in the oscillation, the fraction of total energy that is kinetic can be calculated using the respective values of kinetic and potential energy at that point.