When researchers find a reasonably complete fossil of a dinosaur, they can deter

Question

When researchers find a reasonably complete fossil of a dinosaur, they can determine the mass and weight of the living dinosaur with a scale model sculpted from plastic and based on the dimensions of the fossil bones. The scale of the model is 1/30; that is, lengths are 1/30 actual length, areas are (1/30)2 actual areas, and volumes are (1/30)3 actual volumes. First, the model is suspended from one arm of a balance and weights are added to the other arm until equilibrium is reached. Then the model is fully submerged in water and enough weights are removed from the second arm to reestablish equilibrium (the figure). For a model of a particular T. rex fossil, 599 g had to be removed to reestablish equilibrium, what was the volume of (a) the model and (b) the actual T. rex ? (c) If the density of T. rex was approximately the density of water, what was its mass?

Explanation / Answer

According to archimedes's principle,

Buoyant force, F = Weight of the fluid displaced

F = Volume displaced x density of water x g = V x 1000 x 9.8 = 9800.V

(Taking density of water = 1000 kg/m3)

a.

The weights removed must be equal to the buoyant force as this is the only new force when the model is put in water.

So F = 9800.V = .599 x 9.8 = 5.87 Newton

Solving,

V = 0.000599 m3.= 599 cm3.

b.

As the volume of this model is (1/30)3 the volume of T. rex, Volume of T. rex VT will be (30)3 times V.

So, VT = 0.000599 x (30)3 = 0.000599 x 27000 = 16.17 m3.

c.

Mass = Volume x Density

Mass = 16.17 x 1000 = 16170 kg

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