Damped oscillation condition – Rankers Physics

Energy in SHM: Practice Problem & Solution

Assertion (A): We can assume damped oscillation to be approximately periodic motion for small damping Reason (R): Small damping means \( \frac{b}{\sqrt{km}} \ll 1 \)  
Both (A) & (R) are true and the (R) is the correct explanation of the (A)
Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
(A) is true but (R) is false
Both (A) and (R) are false

Solution Explained:

To solve this problem, we apply the core principles of Energy in SHM. Understanding the underlying formula is key to arriving at the correct answer below:

Assertion (A) is true: For small damping, the amplitude decays slowly, and the frequency is nearly constant, making the motion approximately periodic. Reason (R) is true: Small damping is characterized by a small damping factor \(b\) relative to \(\sqrt{km}\), specifically \( \frac{b}{\sqrt{km}} \ll 1\) (or \( \zeta ll 1\)). This condition directly ensures the motion is approximately periodic.

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Similar Practice Questions

Explore conceptually related problems to test your knowledge.

1. If \( x = 2\sin\left(\frac{\pi}{2}t\right) \) represents the motion of a particle executing SHM, the maximum speed of the particle in \( \text{m s}^{-1} \) is (All parameters are in SI units)
2. Assertion (A): If a pendulum clock is taken to a mountain top, its time period decreases. Reason (R): Value of acceleration due to gravity is more at heights.  
3. Assertion (A): The amplitude of damped oscillation depends on damping constants. Reason (R): The angular frequency for a damped oscillation depends on damping constant only.  
4. Assertion (A): Two particles are in SHM with same time period, same amplitude, same position and same speed are in the same phase. Reason (R): Phase of particle depends on position and speed of particle.  
5. If maximum speed and maximum acceleration of a particle executing SHM is found to be 5 cm/s and \( 50\pi \, \text{cm/s}^2 \) respectively, then its time period will be