I have been given the potential of a simple harmonic oscillator: V=1/2kx^2 I want to calculate the value x(t) of a particle moving in this potential, with initial conditions x(t=0)=0 and v(t=0)=v_0. How would I go about doing this in simple classical mechanics?

misyjny76 2022-09-23 Answered
I have been given the potential of a simple harmonic oscillator:
V = 1 2 k x 2
I want to calculate the value x ( t ) of a particle moving in this potential, with initial conditions x ( t = 0 ) = 0 and v ( t = 0 ) = v 0 . How would I go about doing this in simple classical mechanics?
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Answers (2)

LilsGroolonip86
Answered 2022-09-24 Author has 9 answers
Solving the equation:
You can guess the solution x = A cos k / m t + B sin k / m t or you can use substitution x = e λ t and find λ
Anyway:
x ( t = 0 ) = A cos 0 + B sin 0 = A = 0
v ( t = 0 ) = A k / m sin 0 + B k / m cos 0 = B k / m = v 0
therefore:
x ( t ) = m k v 0 sin ( k m t )
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batejavizb
Answered 2022-09-25 Author has 4 answers
With given potential you can define force acting on your oscillator as
F = d V d x = k x
This gives you Hooks law (no surprise). Than just use Newton's second law:
m a = k x
Than create second order differential equation replacing a with x ¨ and solve it with your initial conditions.
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