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

Correct Answer: 9

Step 1: Total Energy Definition - The total energy E for a simple harmonic oscillator is computed using the formula:

E = \(\frac{1}{2}KA^2\)

- Here, K represents the spring constant and A denotes the amplitude.

Step 2: Potential Energy Calculation at Displacement \(\frac{A}{3}\) - When the displacement is \(\frac{A}{3}\), the potential energy U is calculated as:

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

Step 3: Kinetic Energy Calculation - Kinetic energy is determined by subtracting the potential energy from the total energy:

KE = E - U = E - \(\frac{E}{9} = \frac{8E}{9}\)

Step 4: Ratio of Total Energy to Kinetic Energy Calculation:

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

Step 5: Determine x - Given that the computed ratio is \(\frac{x}{8}\), it follows that x = 9.

Therefore, the final answer is: x = 9