Question

A ripple is created in water. The amplitude at a distance of $5$cm from the point where the sound ripple was created is $4$cm. Ignoring damping, what will be the amplitude at the distance of $10$cm.

• A. $\sqrt{16}$cm
• B. $\sqrt{8}$cm
• C. $\sqrt{4}$cm
• D. $\sqrt{2}$cm

Answer 1

The correct option is B $\sqrt{8}$cm

Let's assume that the ripple was created in a pool of water at the center so that with time the ripple moves away.
Let the energy of the ripple be $E$ .
At a distance of $5cm$ from the point of origin the amplitude is $4cm$.
Energy distributed in circle of circumference $2\pi 5cm$.
Energy per unit length, $E_1=\frac{E}{10\pi }$
When it reaches $10cm$, $E_2=\frac{E}{2\pi \times 10}$ $=\frac{E_1}{2}$
Now energy is proportional to square of amplitude ($A$).
Thus if per unit energy decreases by $\frac{1}{2}$, then amplitude will decrease by $\frac{1}{\sqrt{2}}$.
New amplitude, $A_1=\frac{4}{\sqrt{2}}=\sqrt{8}$