When researchers find a reasonably complete fossil of a...

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?

Key Concepts

Scale Models and Geometric Scaling

Scale models are reduced-size representations that maintain the proportional relationships of the original object. In scaling, lengths are multiplied by a scale factor, areas by the square of the scale factor, and volumes by the cube of the scale factor. This concept is crucial when using dimensions of a model to deduce properties such as volume and mass of the actual object.

Archimedes’ Principle and Buoyancy

Archimedes’ Principle states that a submerged object experiences an upward buoyant force equal to the weight of the fluid it displaces. This principle provides a method for determining the volume of an object by measuring the change in weight when it is immersed in a fluid, as the weight difference equals the weight of the displaced fluid.

Density and Mass Relationship

Density is defined as the mass per unit volume of a substance. When the density is known (for instance, being approximately equal to that of water), the mass of an object can be determined by multiplying its volume by its density. This relationship is fundamental in connecting volumetric measurements back to the mass of the object.

Transcript

0:00 Hi there.

00:01 So for this problem, we are told that when researchers find a reasonably complete fossil of a dinosaur, they can determine the mass and weight of the living dinosaur with an scale model sculpted from plastic and based on the dimensions of the fossil bonds.

00:19 So the scale of the model is 1 over 20, that is lens or 1 over 20, that is lens, are 1 over 20 actual lens, areas are 1 over 20 elevated to the square, and volumes are, of course, 1 over 20 elevated to the 3.

00:44 Actual volumes.

00:46 First, the model is suspended from one arm.

00:48 Often balancings and weights are added to the other arm until the equilibrium is rich, as is shown in this figure.

00:57 Then the model is fully submershing water, and enough weight are removed from the second.

01:03 Arm to establish the equilibrium.

01:06 So for part a of this problem we need to determine what was the volume of the model for part a of this problem.

01:21 So when the model is suspended in earth, the reading we know that it's going to be the, it's true weight neglecting any buoyant effort caused by the air.

01:33 When the model is submerged in water, the reading is lessen it because of the buoyant force.

01:40 We are going to have a reading of its weight minus the buoyant force.

01:46 We denote the difference in readings as the difference in mass.

01:50 So we will have that this is equal to the weight minus the weight minus the buoyant force.

02:00 This is the difference in written and this is equal to the change in mass times the acceleration, due to gravity.

02:09 This led us that, as you can see, the weights cancel, so we obtained that the buoyant force is equal to the change in mass times the acceleration due to gravity.

02:21 Now we can write the mass as the product between the density of water times the acceleration due to gravity times the weight, that this is going to be the weight of water displaced by the more.

02:40 So if we solve for the volume of the water displays, we can obtain the volume of the model.

02:54 So that is the change in the mass divided by the density of water.

03:00 Now the change in the mass is given because it's 637 .76 grams...

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