Question

In a damped oscillation with damping constant \(b\). The time taken for amplitude of oscillation to drop to half what is its initial value? (A) \((\mathrm{b} / \mathrm{m}) \ln 2\) (B) (b / \(2 \mathrm{~m}) \ln 2\) (C) \((\mathrm{m} / \mathrm{b}) \ln 2\) (D) \((2 \mathrm{~m} / \mathrm{b}) \ln 2\)

Short answer

The time taken for the amplitude of a damped oscillation to drop to half its initial value is \((2 \mathrm{~m} / \mathrm{b}) \ln 2\).

Step-by-step solution

Write down the decay constant formula for damped oscillations

The decay constant for damped oscillations can be written as: \(λ = \cfrac{b}{2m}\) where \(b\) is the damping constant and \(m\) is the mass of the object.

Find the time taken for amplitude to drop to half of its initial value

Considering the definition of half-life, the amplitude will reduce to half when the time elapsed is equal to the half-life \(t_{1/2}\). The general formula for half-life is: \(t_{1/2} = \cfrac{\ln 2}{λ}\)

Substitute the decay constant into the half-life formula

Now we will substitute the decay constant \(λ\) from Step 1 into the half-life equation from Step 2. \(t_{1/2} = \cfrac{\ln 2}{\cfrac{b}{2m}}\)

Simplify the equation

We will now simplify the equation to find the final formula. \(t_{1/2} = \cfrac{2m}{b} \ln 2\)

Compare the derived formula with the given options

Now compare the derived formula to the given choices: (A) \((\mathrm{b} / \mathrm{m}) \ln 2\) (B) (\(\mathrm{b} / 2 \mathrm{~m}) \ln 2\) (C) \((\mathrm{m} / \mathrm{b}) \ln 2\) (D) \((2 \mathrm{~m} / \mathrm{b}) \ln 2\) The derived formula matches option (D). Therefore, the time taken for the oscillation's amplitude to drop to half its initial value is \((2 \mathrm{~m} / \mathrm{b}) \ln 2\).