Problem 43 - Fundamentals of Physics

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 \(1 / 20 ;\) 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 second arm to reestablish equilibrium (Fig. 14-42). For a model of a particular T. rex fossil, \(637.76 \mathrm{~g}\) 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?

Step-by-Step Solution

Answer

(a) Model volume: 637.76 cm³, (b) T. rex volume: 509,612,800 cm³, (c) T. rex mass: 509,612.8 kg.

Step 1: Understanding the Problem

We have a scale model of a T. rex with a scale of \( 1/20 \). When the model is submerged in water, it displaces a certain volume of water, allowing us to determine the volume of both the model and the actual dinosaur from which the model is scaled. We also need to find the mass of the T. rex, assuming its density is roughly the same as water.

Step 2: Determining Volume of the Model

When the model is submerged in water, it displaces a volume of water equivalent in mass to the amount of weight removed from the balance. Thus, the weight removed (637.76 g) is equal to the mass of the water displaced, and since the density of water is 1 g/cm³, the volume of water displaced, and hence the volume of the model, is 637.76 cm³.

Step 3: Scaling to Find Volume of Actual Dinosaur

Since the scale of the model is \( 1/20 \), the ratio for volumes is \((1/20)^3\). Therefore, to find the volume of the actual dinosaur, multiply the model's volume by \((20)^3\). \[ \text{Volume of T. Rex} = 637.76 \times 20^3 = 509,612,800 \text{ cm}^3 \]

Step 4: Calculating the Mass of the T. rex

Given that the density of T. rex is approximately the same as water (1 g/cm³), the mass of T. rex is equivalent to its volume in cubic centimeters. Thus, \[ \text{Mass of T. rex} = 509,612,800 \text{ g} \] Converting to kilograms (1 kg = 1000 g): \[ \text{Mass of T. rex} = 509,612.8 \text{ kg} \]

Key Concepts

Scale Modeling

Scale modeling is a fascinating method used to create smaller, manageable representations of much larger objects, such as a dinosaur like the T. rex. By using a specific scale, researchers can work with a replica that is proportionally accurate to the original specimen.

For example, in the case of a T. rex scale model with a scale of \(1/20\), each measurement of the model—like length, area, and volume—corresponds to a fraction of the actual dinosaur's size. Lengths are reduced to \(1/20\) of their true size. This means if the real T. rex was 20 meters long, the model would measure just 1 meter.

• Area scales by \((1/20)^2\), meaning the area of any section of the model is smaller by the square of the scale factor.
• Volume scales by \((1/20)^3\), making the model identical in proportion but significantly smaller in volume.

Understanding scale modeling allows researchers to synthesize findings from small-scale replicas, effectively bridging the gap between miniature and life-size realities. By mastering this concept, students can appreciate the precision and creativity found in this scientific and artistic technique.

Volume Displacement

The concept of volume displacement is a practical way to determine the volume of an object through the water it displaces when submerged. Archimedes' principle states that the volume of displaced water will equal the volume of the submerged object, provided the object does not absorb any liquid.

In our T. rex model scenario, when the scale model is submerged in water, it forces water out of the way, equal to its own volume. This action allows researchers to measure the volume of the model. The weight that was removed from the balance, corresponding to the displaced water, was 637.76 grams.

• Since the density of water is 1 g/cm³, the volume of the displaced water is exactly equal to this mass: 637.76 cm³.
• This volume represents the full volume of the scale model itself.

Volume displacement is a critical method for calculating the volume of irregularly shaped objects, which is particularly beneficial in paleontology, where precise shapes can be hard to measure directly. This technique also lays the foundation for further calculations, like scaling up to determine the volume of the actual dinosaur.

Density and Mass Calculation

Calculating density and mass is integral in understanding physical properties of objects. In our T. rex model, the concept is used to scale up the volume of the model to determine the mass of the real dinosaur.

If the density of the actual T. rex was about equal to that of water, then its density is 1 g/cm³. The scaling factor from the model to the actual dinosaur uses the cube of the scale factor (\((20)^3 = 8000\)), making the real volume much larger than that of the model.

• Knowing that the model's volume was 637.76 cm³, the dinosaur's volume would be \(637.76 \times 8000\) cm³, which is 509,612,800 cm³.
• Given the approximate water-equivalent density, the mass is directly equivalent to the volume using 1 g/cm³. Therefore, the T. rex’s mass is 509,612,800 grams.

Converting this mass into kilograms by dividing by 1000 results in a mass of 509,612.8 kg. This calculation provides an insight into the large scale of historical creatures and demonstrates the art and science of using models and measurements to unlock historical mysteries.