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A harmonic oscillator has angular frequency ω and amplitude A.?
For a mass undergoing simple harmonic oscillator, the displacement of the mass as a function of time is given as:
x(t) = A*sin(ω*t) [to within a phase factor]
where ω = sqrt(k/m), k being the force constant for the restoring force, and m being the mass.
The velocity of oscillating mass is given by the time derivative of the position:
v(t) = A*ω*cos(ω*t)
The potential energy of the oscillator is given by:
U(t) = (k/2)*(x(t)²) = (k/2)*A²*sin²(ω*t)
while the kinetic energy is given by:
K(t) = (1/2)*m*v² = (m/2)*A²*ω²*cos²(ω*t)
(k/2)*A²*sin²(ω*t) = (m/2)*A²*ω²*cos²(ω*t)
k*sin²(ω*t) = m*(k/m)²*cos²(ω*t)
sin²(ω*t) = cos²(ω*t)
tan²(ω*t) = 1
t = arctan(±1)/ω, for example, t = π/(4ω)
So when U(t) = K(t):
x(t) = A*sin(ω*π/(4ω)) = A*sin(π/4) = A/sqrt(2)
v(t) = A*ω*cos(ω*π/(4ω)) = A*ω*cos(π/4) = A*ω/sqrt(2)
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