Figure shows a uniform metal ball suspended by thread of negligible mass from an upright cylinder that floats partially submerged in water. The cylinder has height 6 cm, face area 11 cm2 on the top and bottom and density 0.5 g/`cm^3`. 4 cm of cylinder’s height is inside the water surface. Density of the metal ball is 8gm/`cm^3`. R is the radius of the ball. It is found that `R^(3)-3/(alpha) cm^(3)` where `alpha` is an integer. Find `alpha.(rho_(w)=1 gm//cm^(3))` (system is in equilibrium)

Explanation

Taking cylinder and the ball as system
`4/3 piR^(3). rho_(2).g +Ah.rho_(1)g=4/3 piR^(3).rho_(w).g+ Ah_(1).rho_(w)g`
`toR=[(3A(h_(1)rho_(w)-hrho_(1)))/(4pi(rho_(2)-rho_(w)))]^(1//3)`
using values
`A=11 cm^(2), h_(1)=4 cm, rho_(w)=1 gm//cm^(3)`,
`rho_(1)=0.5 gm//cm^(3), rho_(2)=8 gm//cm^(3)`
`R=[(3xx11(4xx1-6xx0.5))/(4xx(22/7)xx(8-1))]^(1//3) =(3/8)^(1//3) cmrArr R^(3)=3//8`