Question

A damped oscillation is subjected to a damping force of $F_d = -b V$, where 'V' is the velocity of the oscillator and 'b' is the damping constant. The angular velocity of the damped oscillator is ($k$ - force constant and $m$ - mass of the oscillator):

• A. $\omega = \sqrt{\frac{k}{m} - \frac{4m^2}{b^2}}$
• B. $\omega = \sqrt{\frac{b^2k}{4m^2} - b}$
• C. $\omega = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$
• D. $\omega = \sqrt{4m^2kb}$

Hint:

Critical damping occurs when the term inside the square root becomes zero, i.e., $b^2 = 4mk$.
Remembering this boundary condition helps easily verify the correct denominator term ($4m^2$) in the options.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to identify the mathematical equation for the angular frequency (angular velocity) of a damped harmonic oscillator.

Step 2: Key Formula and Approach:
The motion of a damped harmonic oscillator is governed by the differential equation:
\[ m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0 \] The solution for underdamped oscillations yields an angular frequency $\omega$ related to the natural frequency $\omega_0 = \sqrt{\frac{k}{m}}$ and the damping factor.

Step 3: Detailed Explanation:

• Derive the frequency equation:
Let us define the damping coefficient $\gamma = \frac{b}{2m}$.
The general equation for the damped angular frequency is:
\[ \omega = \sqrt{\omega_0^2 - \gamma^2} \]

• Substitute physical parameters:
\[ \omega = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2} \] \[ \omega = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}} \]

Step 4: Final Answer:
The angular velocity is $\omega = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}}$, which corresponds to Option (C).