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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)

Equating these:

(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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