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प्रश्न
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.
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उत्तर
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.
\[\text{ Thus, we have: } \]
\[\text{mg + V }\times \rho_\text{s} \times \text{g = v } \times \rho_\text{w } \times \text{g}\]
\[\text{Here, }\]
\[ \rho_\text{ s }=\text{ Density of the sphere in gm/cc }\]
\[ \rho_\text{w }=\text{ Density of water }\]
\[\text{ On substituting the respective values in the above equation, we get: } \]
\[ (0 . 1) \times {10}^3 + \left( \frac{4}{3} \right) \times \pi \times (5 )^3 \times \rho_s = \left( \frac{4}{3} \right) \times \pi \times (5 )^3 \times 1\]
\[ \Rightarrow 100 = \left( \frac{4}{3} \right) \times \pi \times 125 \times (1 - \rho_\text{s} )\]
\[ \Rightarrow 1 - \rho_\text{s} = \frac{3 \times 100}{4 \times \pi \times 125} = 0 . 19\]
\[ \Rightarrow \rho_\text{s} = 1 - (0 . 19)\]
\[ = 0 . 81 \text{ gm/cc = 0 . 8 gm/cc}\]
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