Problem 43 - Fundamentals of Physics

Question

When researchers find a reasonably complete fossil of a dinosaur, they can detcrmine 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 \(1 / 20 ;\) that is, lengths are \(1 / 20\) actual length, areas are \((1 / 20)^{2}\) actual areas, and volumes are \((1 / 20)^{3}\) 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 sccond arm to reestablish cquilibrium (Fig. \(14-42\) ). For a model of a particular \(T\), rex fossil, 637,76 g had to be removed to recstablish cquilibrium. 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?

Step-by-Step Solution

Answer

(a) Model volume: 637.76 cm³; (b) T. rex volume: 5102080 cm³; (c) T. rex mass: 5102.08 kg.

Step 1: Determine Volume of the Model

When the model is fully submerged in water, it experiences a buoyant force equal to the weight of the water displaced. Given that 637.76 g of weight had to be removed to rebalance, this is equivalent to the weight of the water displaced by the model. Since 1 g of weight corresponds to a volume of 1 cm³ in water, the volume of water displaced is 637.76 cm³. Therefore, the volume of the model is 637.76 cm³.

Step 2: Calculate Volume of the Actual T. rex

Using the scale for volumes, we know that the volume of the T. rex is the volume of the model multiplied by the cube of the inverse of the scale factor. Thus, the volume of the T. rex is given by \( V_{T. rex} = V_{model} \times \left( \frac{1}{1/20} \right)^3 = 637.76 \times 20^3 \). Calculate this to find \( V_{T. rex} = 637.76 \times 8000 = 5102080 \) cm³.

Step 3: Calculate Mass of the Actual T. rex

If the density of the T. rex is approximately the same as water, then the mass is equal to the volume (since density = mass/volume and the density of water is 1 g/cm³). Thus, the mass of the actual T. rex is 5102080 g or 5102.08 kg.

Key Concepts

Scale Model Calculations

Scale models are miniature representations used by researchers to estimate the characteristics of larger entities, like dinosaurs. In our problem, the scale model of a T. rex is crafted at a 1/20 ratio to the actual dinosaur. This scaling involves three main aspects:

• Length: Scaled down to 1/20 of the dinosaur's actual length.
• Area: Reduced to \(\left( \frac{1}{20} \right)^2\), or 1/400 of the actual area.
• Volume: Minimized to \(\left( \frac{1}{20} \right)^3\), or 1/8000 of the actual volume.

To find the volume of the full-sized T. rex, we begin with the volume of the model, which comes from buoyancy calculations. Knowing the scale, we then multiply the model's volume by 8000 to achieve the dinosaur's full volume.

Dinosaur Fossil Analysis

Analyzing dinosaur fossils involves intricate processes to understand these ancient creatures better. Once fossils are discovered, researchers can build scale models based on the fossils' dimensions. These models offer insights into the dinosaur's size, shape, and even their mass, which are crucial for understanding their biology and ecology.
Researchers use the fossil's dimensions to construct an accurate scale model. By weighing the model in air and while submerged, they can assess the displaced water's volume, which equals the model's volume. This method helps determine the dinosaur's volume indirectly through these fossil-derived models.
Thus, the analysis of dinosaur fossils is not just about identifying the bones, but also about using these findings to recreate and study the living dinosaurs more comprehensively.

Density and Mass Relationship

The relationship between density and mass is fundamental in understanding objects' physical properties. For dinosaurs like the T. rex, this relationship is key to estimating their mass accurately. Density is mathematically expressed as:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]In this problem, the dinosaur's density is equated to the density of water, which is 1 g/cm³. This equivalence greatly simplifies calculations, as the mass of the T. rex can be assigned directly from its volume, given that mass and volume are numerically equal when using water's parameters.
Hence, with the calculated volume of the T. rex determined from the scaled model, we can directly infer its mass in kilograms by converting from grams and applying the principles of density.