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# Moving Stonehenge

If transporting the stones dry, the minimum volume of wood required:

$(V_{stone}\rho_{stone} + V_{wood}\rho_{wood})g = V_{wood}\rho_{water}g$

$\therefore V_{wood}(\rho_{water} - \rho_{wood}) = V_{stone}\rho_{stone}$

$\therefore V_{wood} = \frac{V_{stone}\rho_{stone}}{\rho_{water} - \rho_{wood}} = 13.05m^3$

$V_{wood}/(\pi r_{tree}^2) = length_{tree} = 415.4m$

That's nearly half a kilometer of sizeable trees!

If the stones could be transported wet, which would of course require a river about 2 feet deeper, then less wood would have been required:

$(V_{stone}\rho_{stone} + V_{wood}\rho_{wood})g = (V_{wood} + V_{stone})\rho_{water}g$

$\therefore V_{wood}(\rho_{water} - \rho_{wood}) = V_{stone}\rho_{stone}$

$\therefore V_{wood} = \frac{V_{stone}(\rho_{stone} - \rho_{water})}{\rho_{water} - \rho_{wood}} = 8.55m^3$

$V_{wood}/(\pi r_{tree}^2) = length_{tree} = 272.2m$

That's still a lot of trees, but considerably fewer, maybe 25 large trees. That is the absolute minimum value though, at which the object will have neutral buoyancy, i.e. will have the overall density of water.

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If transporting the stones dry, the minimum volume of wood required:

$(V_{stone}\rho_{stone} + V_{wood}\rho_{wood})g = V_{wood}\rho_{water}g$

$\therefore V_{wood}(\rho_{water} - \rho_{wood}) = V_{stone}\rho_{stone}$

$\therefore V_{wood} = \frac{V_{stone}\rho_{stone}}{\rho_{water} - \rho_{wood}} = 13.05m^3$

$V_{wood}/(\pi r_{tree}^2) = length_{tree} = 415.4m$

That's nearly half a kilometer of sizeable trees!

If the stones could be transported wet, which would of course require a river about 2 feet deeper, then less wood would have been required:

$(V_{stone}\rho_{stone} + V_{wood}\rho_{wood})g = (V_{wood} + V_{stone})\rho_{water}g$

$\therefore V_{wood}(\rho_{water} - \rho_{wood}) = V_{stone}\rho_{stone}$

$\therefore V_{wood} = \frac{V_{stone}(\rho_{stone} - \rho_{water})}{\rho_{water} - \rho_{wood}} = 8.55m^3$

$V_{wood}/(\pi r_{tree}^2) = length_{tree} = 272.2m$

That's still a lot of trees, but considerably fewer, maybe 25 large trees. That is the absolute minimum value though, at which the object will have neutral buoyancy, i.e. will have the overall density of water.

A look at a fluid mechanics technique called the Steady Flow Momentum Equation.

Have you got the Mach knack? Discover the mathematics behind exceeding the sound barrier.