AIIMS 2018 Physics Damped Oscillations MCQ Question

To find $n$, we start with the amplitude equation $A = A_0 e^{-bt/2m}$. Given that $A = \frac{A_0}{3}$ at $t = 2 \, \text{s}$, we can express this as $\frac{A_0}{3} = A_0 e^{-b(2)/2m}$, leading to $e^{-b(2)/2m} = \frac{1}{3}$. At $t = 6 \, \text{s}$, we have $A = \frac{A_0}{n} = A_0 e^{-b(6)/2m}$, which simplifies to $e^{-b(6)/2m} = \frac{1}{n}$. Using the relationship $e^{-b(6)/2m} = (e^{-b(2)/2m})^3$ gives $\frac{1}{n} = \left(\frac{1}{3}\right)^3$, resulting in $n = 27$.

Options A (9), C (81), and D (3) are incorrect because they do not satisfy the exponential decay relationship derived from the given amplitude conditions.