The equation of a damped simple harmonic motion is `m(d^2x)/(dt^2)+b(dx)/(dt)+kx=0`. Then the angular frequency of oscillation is

Options

A. `omega=(k/m-(b^2)/(4m^2))^(1//2)`
B. `omega=(k/m-(b)/(4m))^(1//2)`
C. `omega=(k/m-(b^2)/(4m))^(1//2)`
D. `omega=(k/m-(b^2)/(4m^2))^(1//2)`

Explanation

To find the angular frequency of oscillation for the given damped simple harmonic motion, we start with the equation provided:

1. **Write the equation**: The equation of damped simple harmonic motion is given as:
\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \]

2. **Rearrange the equation**: We can divide the entire equation by \(m\) to simplify it:
\[ \frac{d^2x}{dt^2} + \frac{b}{m} \frac{dx}{dt} + \frac{k}{m} x = 0 \]

3. **Identify the standard form**: The standard form of the damped simple harmonic motion equation is:
\[ \frac{d^2x}{dt^2} + 2\zeta \omega_n \frac{dx}{dt} + \omega_n^2 x = 0 \]
where \(\zeta\) is the damping ratio and \(\omega_n\) is the natural frequency.

4. **Compare coefficients**: By comparing the coefficients from both equations, we can identify:
- \(2\zeta \omega_n = \frac{b}{m}\)
- \(\omega_n^2 = \frac{k}{m}\)

5. **Solve for \(\omega_n\)**: From the second equation, we can express \(\omega_n\):
\[ \omega_n = \sqrt{\frac{k}{m}} \]

6. **Conclusion**: The angular frequency of oscillation in damped simple harmonic motion is:
\[ \omega_n = \sqrt{\frac{k}{m}} \]