I think it's more interesting if we try to work it from the paradigm SHM for a particle of mass mm, ie without digging out equations that from memory might lead us straight there.
So, for displacement, we have the generalised idea of SHM:
x(t) = A cos (omega t + psi)x(t)=Acos(ωt+ψ)
Kinetic Energy: T = 1/2 m (dot x)^2 T=12m(.x)2
Velocity: dot x = - omega A sin (omega t + psi) .x=−ωAsin(ωt+ψ)
implies T(t) = 1/2 m omega^2 A^2 sin^2 (omega t + psi) ⇒T(t)=12mω2A2sin2(ωt+ψ)
And:
sin^2 theta in [0,1] implies T_(max) = 1/2 m omega^2 A^2
SHM is conservative and is in essence a system where energy oscillates between potential and kinetic energy. So at certain points in time, the energy is all either Kinetic or Potential Energy.
If follows for Potential Energy U(t) that:
U_(max) = 1/2 m omega^2 A^2
In terms of Total Energy E_T(t), we also can see that:
E_T(t) = T(t) + U(t) = const implies E_T = 1/2 m omega^2 A^2
Now at x = A/2, and if at that time, t = tau , then:
A/2 = A cos (omega tau + psi) implies cos (omega tau + psi) = 1/2
T(tau) = 1/2 m omega^2 A^2 sin^2 (omega tau + psi)
= 1/2 m omega^2 A^2 (1 - cos^2 (omega tau + psi) )
= 3/8 m omega^2 A^2
So
(T(tau))/(E_T(tau)) = ( 3/8 m omega^2 A^2)/(1/2 m omega^2 A^2)
= color(blue)(3/4)
And:
(U(tau))/(E_T(tau)) = 1 - 3/4 = color(blue)(1/4)
