A sphere of diameter 6 cm and mass 250 g is floating in a beaker containing a liquid. When the temperature is raised, the sphere just begins to sink at a temperature of 30^@C . The density of the liquid at 0^@Cis 2.92 g//cm^3. If the expansion of the sphere is ignored, the coefficient of cubical expansion of the sphere is gamma , then (10^9 gamma)/(13837) = 2^a 3^b 11^c . Compute the value of (ab)/(c )

A sphere of diameter 6 cm and mass 250 g is floating in a beaker conta

1.	A sphere of diameter 6 cm and mass 250 g is floating in a beaker containing a liquid. When the temperature is raised, the sphere just begins to sink at a temperature of 30^@C . The density of the liquid at 0^@Cis 2.92 g//cm^3. If the expansion of the sphere is ignored, the coefficient of cubical expansion of the sphere is gamma , then (10^9 gamma)/(13837) = 2^a 3^b 11^c . Compute the value of (ab)/(c )
Answer» Solution :Radius of sphere, R = 3 cm Volume of the sphere, `V = 4/3 pi r^3 = 4/3 xx 22/7 xx 3^2` ` = 264/3 cm^3` Mass of the sphere, m = 250 g Density of the sphere, `rho = m/V = (250 xx 3)/(264) = 2.84 g//cm^3` Given, density of liquid at `0^@C, rho_0 = 2.92 g//cm^3` When the temperature of the BEAKER is raised, the liquid expands and its density decreases. When the sphere just floats, Density of liquid at `30^@C`= Density of the body ` therefore rho_1 = 1.84 g//cm^3 , rho_0 2.92 g//cm^3` ` Delta T = 30^@C` Coefficient of cubical EXPANSION is `gamma = (rho_0 - rho_t)/(rho_0 Delta T) = (2.92 - 2.84)/(2.92 xx 30)` = 0.000913242 ` 10^9 gamma = 913242` `(10^9 gamma)/(13837) = (913242)/(13837) = 66 = 2 xx 3 xx 11` ` therefore a = b = c = 1`