In dampled oscillation , the amplitude of oscillation is reduced to half of its initial value of 5 cm at the end of 25 osciallations. What will be its amplitude when the oscillator completes 50 oscillations ?
Hint `: A= A_(0) e^((-bt)/(2m))` , let T be the time period of oxcillation
Case -I `: (A_(0))/(2) = A_(0)e^(-bx(25T)/(2m))`
or `(1)/(2)= e^(-25(bT)/(2m))` ......(i)
Case -II` A=A_(0)e^(-bxx50(T)/(2m))`
`A _(0)(e^(-25(bT)/(2m)))^(2)`
Use euation (i) to find a .

Explanation

To solve the problem of finding the amplitude of a damped oscillator after 50 oscillations, we can follow these steps:

### Step 1: Understand the given information
We know that the initial amplitude \( A_0 \) is 5 cm, and after 25 oscillations, the amplitude is reduced to half of its initial value. Therefore, the amplitude after 25 oscillations is:
\[ A = \frac{A_0}{2} = \frac{5 \, \text{cm}}{2} = 2.5 \, \text{cm} \]

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