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Chapter 3: Problem 94

Quantify the experiment performed by Archimedes to identify the material content of King Hiero's crown. Assume you can measure the weight of the king's crown in air, \(W_{a}\) and the weight in water, \(W_{w^{*}}\) Express the specific gravity of the crown as a function of these measured values.

Short Answer

Expert verified
The specific gravity of the crown can be calculated using the formula: Specific Gravity = \( \frac{W_{a}}{W_{a} - W_{w^*}}\).

Step by step solution

01

Understand the Specific Gravity Concept

Specific gravity is a dimensionless quantity representing the ratio of density of a substance to the density of a reference substance. Here, the reference substance is water. It can be calculated using the formula: Specific Gravity = \( \frac{Density_{substance}}{Density_{water}} \) .
02

Utilize Archimedes' Principle

According to Archimedes' principle, the buoyant force on an object is equal to the weight of the fluid displaced by the object. Therefore, we can say that the weight of the crown in air ( \(W_{a}\) ) minus its weight in water ( \(W_{w^*}\) ) equals the weight of the water displaced by the crown ( \(W_{w}\). Hence, we derive: \(W_{w} = W_{a} - W_{w^*}\).
03

Derive the Formula for Specific Gravity

The weight of an object is given by the density of the object times its volume times the acceleration due to gravity. Therefore, if \(v\) represents the volume of the crown and \(g\) the gravity,the density of the crown is \( \frac{W_{a}}{v \times g}\) and the density of the water is \( \frac{W_{w}}{v \times g}\). Substituting these values into the specific gravity formula gives us: Specific Gravity = \( \frac{W_{a}}{W_{w}}\).
04

Finalize the Formula

Replacing \(W_{w}\) with the derived value from step 2 in the formula from step 3: Specific Gravity = \( \frac{W_{a}}{W_{a} - W_{w^*}}\). This is the final formula to calculate the specific gravity of the crown as a function of the measured weights in air and in water.

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