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A leak proof cylinder of length $$1m$$, made of a metal which has low coefficient of expansion is floating vertically in water at $$0^oC$$ such that its height above the water surface is $$20cm$$. When the temperature of water is increased to $$4^oC$$, the height of the cylinder above the water surface becomes $$21cm$$. The density of water at $$T = 4^oC$$, relative to the density at $$T = 0^oC$$ is close to:
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Correct option is D. $$1.01$$
Let $$A$$ be the area of cross section of the cylinder.
$$p_0$$ be the density of water at $$0^oC$$
$$p_4$$ be the density of water at $$4^oC$$.
Total length of the cylinder is $$100cm$$
Given that the length of the cylinder above the surface of water is $$20cm$$
At $$0^oC$$, the weight of the cylinder is balanced by the buoyant force.
$$p_0A \times 0.8 \times g = w$$ [Buoyant force $$= pA lg$$] ...(i)
At 4^oC$$, thef weight of the cylinder is balanced by the buoyant force.
$$p_4A\times 0.79\times g = w$$ ...(ii)
Dividing (i) by (ii)
$$\dfrac{p_0\times 0.8}{p_4 \times 0.79} = 1$$, $$p_4 = \dfrac{p_0\times 0.8}{0.79} = 0.012p_0$$
$$\dfrac{p_4}{p_0} = 1.012 \simeq 1.01$$
$$p_0$$ be the density of water at $$0^oC$$
$$p_4$$ be the density of water at $$4^oC$$.
Total length of the cylinder is $$100cm$$
Given that the length of the cylinder above the surface of water is $$20cm$$
At $$0^oC$$, the weight of the cylinder is balanced by the buoyant force.
$$p_0A \times 0.8 \times g = w$$ [Buoyant force $$= pA lg$$] ...(i)
At 4^oC$$, thef weight of the cylinder is balanced by the buoyant force.
$$p_4A\times 0.79\times g = w$$ ...(ii)
Dividing (i) by (ii)
$$\dfrac{p_0\times 0.8}{p_4 \times 0.79} = 1$$, $$p_4 = \dfrac{p_0\times 0.8}{0.79} = 0.012p_0$$
$$\dfrac{p_4}{p_0} = 1.012 \simeq 1.01$$
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