Question

Consider the following differential equation that describes the oscillations of a physical system:
\[ \alpha \frac{d^2y}{dt^2} + \beta \frac{dy}{dt} + \gamma y = 0 \] If \( \alpha \) and \( \beta \) are held fixed, and \( \gamma \) is increased, then:

• A. The frequency of oscillations increases
• B. The oscillations decay faster
• C. The frequency of oscillations decreases
• D. The oscillations decay slower

Hint:

In damped oscillations, increasing the stiffness constant (\( \gamma \)) increases the restoring force, thereby increasing the oscillation frequency.

Answer 1

The Correct Option is A

Step 1: Identify the type of motion.
The given differential equation represents a damped harmonic oscillator, where \( \alpha \) is the inertia coefficient, \( \beta \) is the damping constant, and \( \gamma \) is the restoring coefficient.
Step 2: Write the angular frequency of damped oscillations.
The angular frequency is given by: \[ \omega = \sqrt{\frac{\gamma}{\alpha} - \left(\frac{\beta}{2\alpha}\right)^2}. \] Step 3: Analyze the effect of increasing \( \gamma \).
If \( \gamma \) increases while \( \alpha \) and \( \beta \) are fixed, the term \( \frac{\gamma}{\alpha} \) increases, so the value under the square root increases. Thus, \( \omega \) increases, implying higher oscillation frequency.
Step 4: Final Answer.
The frequency of oscillations increases with \( \gamma \).