Question

A leak proof cylinder of length \(1~m\), made of a metal which has very low coefficient of expansion is floating vertically in water at \(0^{\circ}~C\) such that its height above the water surface is \(20~cm\). When the temperature of water is increased to \(4^{\circ}~C\), the height of the cylinder above the water surface becomes \(21~cm\). The density of water at \(T=4^{\circ}C\), relative to the density at 𝑇=0°C is close to

Answer 1

Law of floatation, \(B=mg\)

\(V_i\rho_\text{liquid}g=V\rho_\text{body}g\)

\(\dfrac{V_i}{V}=\dfrac{\rho_\text{body}}{\rho_\text{liquid}}\Rightarrow\dfrac{h_i}{h_0}=\dfrac{\rho_\text{body}}{\rho_\text{liquid}}\)

in given case

\(\dfrac{h_1}{h_2}=\dfrac{(\rho_{4^{\circ}C})\text{water}}{(\rho_{0^{\circ}C})\text{water}}\)

\([\rho_\text{body}=\text{constant}]\)

\(\dfrac{\rho_{4^{\circ}C}}{\rho_{0^{\circ}C}}=\left(\dfrac{100-20}{100-21}\right)=\dfrac{80}{79}=1.01\)