Calculate the ratio (pure number) of the kinetic energy to the potential energy of a

simple harmonic oscillator when its displacement is half its amplitude

well, if amplitude is 1/2, and PE is related to the square of amplitude, then PE must be 1/4, so KE is 3/4 max, so ratio is 3/1 check my thinking.

To calculate the ratio of the kinetic energy to the potential energy of a simple harmonic oscillator when its displacement is half its amplitude, we first need to understand the formulas for kinetic energy and potential energy in such a system.

In a simple harmonic oscillator, the kinetic energy (K) and potential energy (U) can be determined using the following equations:

Kinetic energy (K) = (1/2) * m * v^2

Potential energy (U) = (1/2) * k * x^2

where:
m = mass of the object oscillating
v = velocity of the object
k = spring constant
x = displacement from the equilibrium position

To calculate the ratio, we need to find the values of K and U when the displacement (x) is half the amplitude. Let's assume the amplitude A, then the displacement will be x = A/2.

Also, it is important to note that the total mechanical energy (E) of the system remains constant in a simple harmonic oscillator, and it can be given by:

Total mechanical energy (E) = K + U

Since the total mechanical energy is constant, we can write:

E = K + U

The ratio of K to U can be obtained by dividing the kinetic energy by the potential energy:

K/U = (K / (K + U))

Now, let's substitute the values:

Kinetic energy (K) = (1/2) * m * v^2
Potential energy (U) = (1/2) * k * x^2

Substituting x = A/2 into the potential energy equation:

U = (1/2) * k * (A/2)^2
= (1/2) * k * A^2 / 4
= (k * A^2) / 8

Considering that the total mechanical energy (E) remains constant, we can write:

E = K + U
= (1/2) * m * v^2 + (k * A^2) / 8

Now, we can express the kinetic energy in terms of the total mechanical energy:

K = E - U
= (1/2) * m * v^2 + (k * A^2) / 8 - (k * A^2) / 8
= (1/2) * m * v^2

Finally, substitute the expressions for K and U into the ratio equation:

K/U = [(1/2) * m * v^2] / [(k * A^2) / 8]

By simplifying this expression, you can calculate the ratio of the kinetic energy to the potential energy of the simple harmonic oscillator when its displacement is half its amplitude.