Haven Barrows PhD
Nov 19, 2024
Harmonic Oscillator Equation question!?
At what time ( t_1 ) does the block return to its original equilibrium position ( ( x = 0 ) ) for the first time? Please express your answer in terms of variables, including ( A ), ( k ), and ( m ). I attempted to solve the problem myself, but I ended up with an incorrect answer: ( \frac{\arccos(0^2 \cdot m)}{k} ). I would appreciate some guidance in finding the correct solution.
2 Answers
The differential equation for a harmonic oscillator is
m d²x(t)/dt² = -kx
with the general solution
x(t) = A*cos(ωt +φ)
ω = √(k/m)
φ can be derived from initial condition. I think for this problem the oscillator started its motion at x=A, hence φ=0
x(t1) = A*cos(ωt1) = 0
or
cos(ωt1) = 0
The cosine is zero for
ωt1 = 1/2 π , 3/2 π, 5/2 π;7/2 π,…
hence
t1 = π / (2ω) = π/2 * √(m/k)
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