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**1. 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?

**2. Homework Equations**

E = (1/2)mv^2 + (1/2)kx^2 = (1/2)kA^2

**3. 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

**1. Homework Statement**

**2. Homework Equations**

**3. The Attempt at a Solution**