- #1
kjartan
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Homework Statement
This is my first post here.
I'm particularly unsure about b(ii)
Thanks for any and all replies!
I apologize in advance if I haven't used the correct conventions, but I hope that this is legible. I will learn the correct conventions for future posts but was pressed for time here.
Question:
A harmonic oscillator has angular frequency w and amplitude A.
(a) What are the magnitudes of the displacement and velocity when the elastic potential energy is equal to the kinetic energy? (assume that U = 0 at equilibrium)
(b) (i) How often does this occur in each cycle?
(ii) What is the time between occurrences?
(c) At an instant when the displacement is equal to A/2, what fraction of the total energy of the system is kinetic and what fraction is potential?
Homework Equations
E = (1/2)mv^2 + (1/2)kx^2 = (1/2)kA^2
The Attempt at a Solution
(a) (1/2)mv^2 = (1/2)kx^2 --> x = +/- A/rad(2)
v = rad(k/m)*x = rad(k/m)*[A/rad(2)]
(b) (i) 4 times
(ii) x = Acos(wt + phi), choose phi = pi/2
so, x = Asin(wt)
x/A= sin(wt)
t = (1/w)arcsin(x/A)
subst. from (a) gives us t = (1/w)arcsin(1/rad(2)) = (1/w)*(pi/4)
So, change in t = pi/(2w)
(c) E = K + U
U = (1/2)kx^2 = (!/2)k(A/2)^2 = k(A^2/8)
K = k(A^2/2) - k(A^2/8) = k(3A^2/8)
so, K/E = 3/4 --> U/E = 1/4