Question

The speed of sound in an ideal gas at a given temperature T is v. The rms speed of gas molecules at that temperature is v_rms. The ratio of the velocities v and v_rms for helium and oxygen gases are X and X' respectively. Then \(\frac{X}{X'}\) is equal to

• A. \(\frac{5}{\sqrt{21}}\)
• B. \(\sqrt{\frac{5}{21}}\)
• C. \(\frac{21}{5}\)
• D. \(\frac{21}{\sqrt{5}}\)

Answer 1

The Correct Option is A

Given: 

• A block of mass \( m \) is connected to a light spring of force constant \( k \).
• The system is placed in a damping medium with damping constant \( b \).
• The initial amplitude of oscillation is \( A \), and the angular frequency is \( \omega' \).

Step 1: Governing Equation of Damped Oscillations

The motion of a damped harmonic oscillator is given by:

\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \]

Step 2: Expressions for Different Quantities

• Angular frequency of damped oscillations:
• Total energy of the system:
• Displacement as a function of time:

Step 3: Identifying the Incorrect Expression

The given option:

\[ x = A e^{-\frac{b}{m}} \cos(\omega' t + \phi) \]

is incorrect because the exponent should be \( -\frac{b}{2m} t \), not \( -\frac{b}{m} \).

Answer: The incorrect option is D.

Speed of Sound and RMS Velocity

Given:

• Speed of sound in an ideal gas at temperature \( T \) is \( v \).
• RMS speed of gas molecules at the same temperature is \( v_{rms} \).
• Ratios of these velocities for helium and oxygen are \( X \) and \( X' \) respectively.

Step 1: Expression for the Ratio

The ratio \( \frac{X}{X'} \) is given by:

\[ \frac{X}{X'} = \frac{5}{\sqrt{21}} \]

Answer: The correct option is A.