Question

A block of mass m is conducted to a light spring of force constant k. The system is placed inside a damping medium of damping constant b. The instantaneous values of displacement, acceleration and energy of the block are x, a and E respectively. The initial amplitude of oscillation is A and ω' is the angular frequency of oscillations. The incorrect expression related to the damped oscillations is

• A. \(\omega'=\sqrt{\frac{k}{m}-\frac{b^2}{4m^2}}\)
• B. \(E=\frac{1}{2}kA^2e^{-\frac{bt}{m}}\)
• C. \(m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0\)
• D. \(x=Ae^{-\frac{b}{m}}\cos(\omega't+\phi)\)

Answer 1

The Correct Option is D

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.