A sphere of relative density and diameter D has concentric cavity of diameter d. The ratio of , if it just floats on water in a tank is:

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Q.
A sphere of relative density $σ$ and diameter D has concentric cavity of diameter d. The ratio of $\frac{D}{d}$ , if it just floats on water in a tank is: [2024]

1 ${(\frac{σ}{σ - 1})}^{\frac{1}{3}}$

2 ${(\frac{σ + 1}{σ - 1})}^{\frac{1}{3}}$

3 ${(\frac{σ - 1}{σ})}^{\frac{1}{3}}$

4 ${(\frac{σ - 2}{σ + 2})}^{\frac{1}{3}}$

Ans.
(1)

Weight of sphere, $W = \frac{4}{3} π (\frac{D^{3} - d^{3}}{8}) σ g$

Buoyant force $F_{b} = \frac{4}{3} π (\frac{D^{3}}{8}) ρ_{w} g$

For Just Float, $W = F_{b}$

   $\frac{4}{3} π (\frac{D^{3} - d^{3}}{8}) σg = \frac{4}{3} π (\frac{D^{3}}{8}) ρ_{w} g$

   $\Rightarrow 1 - \frac{d^{3}}{D^{3}} = \frac{ρ_{w}}{σ} (ρ_{w} = 1)$

   $1 - \frac{1}{σ} = {(\frac{d}{D})}^{3} \Rightarrow \frac{D}{d} = {(\frac{σ}{σ - 1})}^{1 / 3}$

Solution

Q.
A sphere of relative density $σ$ and diameter D has concentric cavity of diameter d. The ratio of $\frac{D}{d}$ , if it just floats on water in a tank is: [2024]

1 ${(\frac{σ}{σ - 1})}^{\frac{1}{3}}$

2 ${(\frac{σ + 1}{σ - 1})}^{\frac{1}{3}}$

3 ${(\frac{σ - 1}{σ})}^{\frac{1}{3}}$

4 ${(\frac{σ - 2}{σ + 2})}^{\frac{1}{3}}$

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Solution

Q. A sphere of relative density σ and diameter D has concentric cavity of diameter d. The ratio of Dd, if it just floats on water in a tank is: [2024]

1 (σσ-1)13

2 (σ+1σ-1)13

3 (σ-1σ)13

4 (σ-2σ+2)13

Ans. (1) Weight of sphere, W=43π(D3-d38)σg Buoyant force Fb=43π(D38)ρwg For Just Float, W=Fb 43π(D3-d38)σg=43π(D38)ρwg ⇒1-d3D3=ρwσ(ρw=1) 1-1σ=(dD)3⇒Dd=(σσ-1)1/3

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Q.
A sphere of relative density $σ$ and diameter D has concentric cavity of diameter d. The ratio of $\frac{D}{d}$ , if it just floats on water in a tank is: [2024]

1 ${(\frac{σ}{σ - 1})}^{\frac{1}{3}}$

2 ${(\frac{σ + 1}{σ - 1})}^{\frac{1}{3}}$

3 ${(\frac{σ - 1}{σ})}^{\frac{1}{3}}$

4 ${(\frac{σ - 2}{σ + 2})}^{\frac{1}{3}}$