A cylindrical object of cork of mass `15 g` and cross-sectional area `A_(1) = 10 cm^(2)` floats in a pan of water as shown in the figure. An aluminum, cylinder of mass `25 g` and cross-sectional area `A_(2) = 2 cm^(2)` is attached `4 cm` below the cork and slides through a watertight frictionless hole in the bottom of the pan. Take density of the cork, `rho = 0.2 g//cm^(3), rho_("aluminium") = 2.7 g//cm^(3), g = 10 m//s^(2)`. Based on the above information, answer the following questions.

The length of the cork cylinder inside the water in equilibrium is

Options

A. `6 cm`
B. `4 cm`
C. `8 cm`
D. `3 cm`

Answer

A

Explanation

Let `x` cm be the length of cork cylinder inside the water and `T` be the tension in the string.
The free body diagram of the two cylinders are as shown in the following figures.

Here `m_(1)=15g, m_(2)=25 g, h=4cm, v_("in")=A_(1)xcm^3`
For cork cylinder `T+m_(1)g=V_("in")rhog`
For aluminium cylinder
`T+rho_(0)A_(2)=m_(2)g(p_(0)+rhogx+rhogh)A_(2)`
`impliesx=6cm`
`T=0.45N`