Question

When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $\frac{x}{ 8}$ , where x = _________.

Answer 1

Step 1: Define Total Energy - The total energy E of a simple harmonic oscillator is given by:

E = $\frac{1}{2}KA^2$

- Where K is the spring constant and A is the amplitude.

Step 2: Calculate Potential Energy U at Displacement $\frac{A}{3}$ - When the displacement is $\frac{A}{3}$:

U = $\frac{1}{2}K \left(\frac{A}{3}\right)^2 = \frac{KA^2}{18} = \frac{E}{9}$

Step 3: Calculate Kinetic Energy KE - Kinetic energy is the difference between total energy and potential energy:

KE = E - U = E - $\frac{E}{9} = \frac{8E}{9}$

Step 4: Calculate the Ratio of Total Energy to Kinetic Energy :

$\frac{\text{Total Energy}}{\text{KE}} = \frac{E}{\frac{8E}{9}} = \frac{9}{8}$

Step 5: Determine x - Since the ratio is $\frac{x}{8}$, we have x = 9.

So, the correct answer is: x = 9