##### Question

In: Mechanical Engineering

# Archimedes measured the weight of the crown in air to be 11.8 N and its weight in water to be 10.9 N. Was it pure gold?

It is said that Archimedes discovered the buoyancy laws when asked by King Hiero of Syracuse to determine whether his new crown was pure gold (SG = 19.3). Archimedes measured the weight of the crown in air to be 11.8 N and its weight in water to be 10.9 N. Was it pure gold?

## Solutions

##### Expert Solution

3509-2-105P SA Code: 0670

SR Code: 9421

Given Data

Specific gravity of pure gold crown $$\mathrm{SG}_{\mathrm{GC}}=19.3$$

Weight of the crown in air, $$W_{air}=11.8 \mathrm{~N}$$

Weight of the crown in water, $$W_{\text {water }}=10.9 \mathrm{~N}$$

Let $$\mathrm{SG}_{\mathrm{c}}$$ be the specific gravity of crown

Let $$V$$ be the volume of the crown.

Let $$\gamma$$ be the Specific weight of water.

We know that

Weight of the crown in air, $$W_{a i r}=\left(\mathrm{SG}_{\mathrm{C}} \times \gamma\right) \times V$$

$$=\mathrm{SG}_{\mathrm{c}} \gamma V$$

Buoyancy force acts on the crown, $$B=\gamma \times V$$

The buoyancy force is the difference between the weight of the body in the air and the weight of the body in water.

Therefore

Buoyancy, $$\quad B=W_{\text {air}}-W_{\text {water }}$$

$$=11.8 \mathrm{~N}-10.9 \mathrm{~N}$$

$$=0.9 \mathrm{~N}$$

Now from equation $$(1),$$ we get

$$W_{\text {air}}=\mathrm{SG}_{\mathrm{c}} \gamma V$$

$$W_{a i r}=\mathrm{SG}_{\mathrm{c}} \times B(\because$$ from equation (2)$$)$$

$$\mathrm{SG}_{\mathrm{C}}=\frac{W_{\text {air}}}{B}$$

$$\mathrm{SG}_{\mathrm{GC}}=\frac{11.8 \mathrm{~N}}{0.9 \mathrm{~N}}$$

$$\mathrm{SG}_{\mathrm{C}}=13.11 \neq 19.3$$

Therefore, Crown is not a pure gold.

Therefore, Crown is not a pure gold.