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.

Answer 2

In the case of damped oscillations, the general form of the displacement \(x(t)\) of the block is given by: \[ x(t) = Ae^{-\frac{b}{2m} t} \cos(\omega' t + \phi) \] where:
\(A\) is the initial amplitude,
\(b\) is the damping constant,
\(m\) is the mass of the block,
\(\omega'\) is the angular frequency of the damped oscillations,
\(\phi\) is the phase constant.

Now let's analyze each option: -

Option (A) is correct: The angular frequency \(\omega'\) of damped oscillations is given by \(\omega' = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2}\), which is the correct expression.
Option (B) is correct: The energy \(E\) in damped oscillations decreases exponentially over time, as shown by \(E = \frac{1}{2} kA^2 e^{-\frac{b}{m} t}\).
Option (C) is correct: This is the equation of motion for the damped harmonic oscillator.
Option (D) is incorrect: The expression \(x = Ae^{-\frac{b}{m}} \cos(\omega' t + \phi)\) is incorrect because the exponential decay factor should be in terms of time \(t\) rather than just a constant factor. The correct form is \(x(t) = Ae^{-\frac{b}{2m} t} \cos(\omega' t + \phi)\).

Thus, the correct answer is: \[{\text{(D) } x = Ae^{-\frac{b}{m}} \cos(\omega' t + \phi)} \]