# Inspired by Dish-washing!

**Classical Mechanics**Level 4

Consider the following diagram-

L is a murky ideal liquid of refractive index \(n=2.23\) filled in a container **C** of cross section **Z** to a height of **H**= 64cm. Container **C** has an orifice **O** (near the bottom) of cross section **z** from which the liquid flows out starting at a time t = 0. L has a unique property that the intensity of light transmitted through it varies as \(I=I_{initial}e^{-10d}\) where **d** is the distance travelled by light through the the liquid.The bottom **R** of **C** is reflecting (perfectly). **A** (of mass 50g) consists of a floating light emitter (it emits a narrow beam of light) **a** and receiver **b** which lie very close to each other. **a** emits light of intensity \(I_{0}\) and \(b\) responds only when it receives light of intensity greater than \(0.04076I_{0}\). If **t**(to the nearest positive integer) is the time after which **b** responds and if U is the loss in potential energy of **A** in the time interval between the emission of the light ray which produces the first response and the reception of the same such that \(U=m10^{-x}\),

find |t| + x.

Given: - \(\frac{z}{Z} =10^{-4}\)

\(c= 3 \times 10^{8}ms^{-1}\)

\(g= 10ms^{-2}\)