I've seen volume of a hollow sphere mostly defined (in books) as volume of its equivalent solid sphere minus the volume of the hollow region/cavity. But often a solid object of some material (density) sinks in water while hollow object of the same material stays afloat. If density of both the objects is same, then why does this happen. I always assumed that in hollow bodies the volume of the hollow body is also taken into consideration, which significantly decreases it's density. Is that not the case. Is volume defined differently in physics and maths?
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1$\begingroup$ Think in terms of average density. The average density of a solid sphere is definitely higher than the average density of a hollow sphere. $\endgroup$– David WhiteCommented Aug 25 at 19:11
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4 Answers
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... often a solid object of some material (density) sinks in water while hollow object of the same material stays afloat. If density of both the objects is same, then why does this happen ?
The average density of the two objects is not the same - and it is the average density that matters when considering buoyancy. By introducing a hollow inside the object we reduce its average density, and if we reduce its average density below the density of water then the hollow object will float.
This is why it is possible to construct a floating concrete ship, whereas a solid object made from concrete and steel would sink.
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You have made a mistake in the concept. The density you are talking about is not the density of the object, but the density of the substance. However, the density that we actually use in physics is the density of the object, which is the total mass divided by the total volume. Therefore, the density of the object is less than the density of the substance (if it is hollow).
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$\begingroup$ My confusion was more rooted in the volume part of the problem. Do solid and hollow sphere have same volume or not? $\endgroup$ Commented Aug 28 at 16:07
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1$\begingroup$ Of course, it's relative. The question is which part interacts with the water. If the hole inside is completely blocked and water cannot enter, the hole inside does not play any role, so the volume of the two spheres should be considered the same. However, if water can enter the hole inside, the volume is different and the hollow sphere will sink. $\endgroup$– user424518Commented Aug 28 at 16:26
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If we pu an object into water there are two forces, which act onto the body:
- The gravitational force, $F = m_{object} \cdot g$, which is acting "downwards", and
- The buoyant force, $F = \rho_{water} \cdot V_{object} \cdot g$, which is acting upwards.
For a solid sphere the density is constant and we can write the downwards force as $m_{object} = \rho_{object}\cdot V_{object}$. This results in the effective force difference $$\Delta F = (\rho_{object} - \rho_{water}) \cdot V_{object} \cdot g$$ In contast, if the sphere is hollow and possesses a radial length $dR$, the mass of the object is given by \begin{align} m_{object} &= \rho_{object} \frac{4\pi }{3} \left( (R+dR)^3 - R^3 \right) \\ &\approx \rho_{object} 4\pi R^2\cdot dR \\ &= \rho_{object} V_{object} \frac{3}{R}dR \end{align} Thus, the effective force difference is given by $$ \Delta F = (\rho_{object} \frac{3dR}{R} - \rho_{water}) \cdot V_{object} \cdot g $$
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Any real hollow sphere will have a shell of non zero thickness. If the density of the material the shell is made out of is p>0 and the shell thickness is d>0, the mass of the shell would be $p(4/3\pi R^3-4/3\pi (R-d)^3)$. If the hollow interior is a vacuum, the average density of the hollow sphere would be: $$\frac{\text{mass}}{\text{volume}} = \frac{p(4/3\pi R^3-4/3\pi (R-d)^3)}{4/3 \pi R^3} = p\left(1-\frac{(R-d)^3}{R^3}\right)$$
If density of both the objects is same, then why does this happen.
the density of a solid sphere made of the same material is simply p and the average density of the hollow sphere calculated above is less than p, so your assumption that the density of the two objects is the same is not correct.
Does a hollow sphere and solid sphere (of same outer radius) have different volumes?
No, they have the same volume, but the hollow sphere has lower density and lower mass than the solid sphere when the shell of the hollow sphere is made out of the same material.
