Question

A solid sphere of radius 5 cm floats in water. If a maximum load of 0.1 kg can be put on it without wetting the load, find the specific gravity of the material of the sphere.

Answer 1

Given:
Radius of the sphere, r = 5 cm
Mass of the maximum load, m = 0.1 kg
Let the weight of the sphere be W₁ and the weight of the load be W₂.
Now,
W₁ + W₂ = U
Here, U is the upward thrust.
Let V be the volume of the sphere. 

\[\text{ Thus, we have: } \]

\[\text{mg + V }\times \rho_\text{s} \times \text{g = v } \times \rho_\text{w } \times \text{g}\]

\[\text{Here, }\]

\[ \rho_\text{ s }=\text{ Density of the sphere in gm/cc }\]

\[ \rho_\text{w }=\text{ Density of water }\]

\[\text{ On substituting the respective values in the above equation, we get: } \]

\[ (0 . 1) \times {10}^3 + \left( \frac{4}{3} \right) \times \pi \times (5 )^3 \times \rho_s = \left( \frac{4}{3} \right) \times \pi \times (5 )^3 \times 1\]

\[ \Rightarrow 100 = \left( \frac{4}{3} \right) \times \pi \times 125 \times (1 - \rho_\text{s} )\]

\[ \Rightarrow 1 - \rho_\text{s} = \frac{3 \times 100}{4 \times \pi \times 125} = 0 . 19\]

\[ \Rightarrow \rho_\text{s} = 1 - (0 . 19)\]

\[ = 0 . 81 \text{ gm/cc = 0 . 8 gm/cc}\]