A hollow spherical shell at the outer radius \(R\) floats just submerged under the water surface. The inner radius of the shell is \(r\). If the specific gravity of the shell material is \(\frac{27}{8}\) with respect to water, the value of \(r\) is:\(\left [\text{Given:} \left(19^{\frac{1}{3}}= \frac{8}{3}\right )\right]\)
1. \(\frac{4}{9}R\)
2. \(\frac{8}{9}R\)
3. \(\frac{1}{3}R\)
4. \(\frac{2}{3}R\)

A wooden block is initially floating in a bucket of water with \(\frac{4}{5}\)​ of its volume submerged. When a certain amount of oil is poured into the bucket, the block is found to be just under the oil surface with half of its volume submerged in water and half in oil. What is the density of the oil relative to that of water?
1. \(0.7\)
2. \(0.5\)
3. \(0.8\)
4. \(0.6\)

A cubical block of side \(0.5\) m floats on water with \(30\)% of its volume under water. What is the maximum weight that can be put on the block without fully submerging it underwater?
(take, density of water \(=10^3\) kg/m³⁾
1. \(30.1\) kg
2. \(87.5\) kg
3. \(65.4\) kg
4. \(46.3\) kg

A air bubble of radius \(1\) cm in water has an upward acceleration \(9.8~\text{cm} \text{s}^{-2}\). The density of water is \(1~\text{gm} \text{cm}^{-3}\) and water offers negligible drag force on the bubble. The mass of the bubble is: (\(g = 980\) cm/s² )
1. \(3.15 ~\text{gm}\)
2. \(1.52 ~\text{gm}\)
3. \(4.51 ~\text{gm}\)
4. \(4.15~\text{gm}\)