Take the density of the crown to be ρc.

Question

According to legend, the following challenge led Archimedes to the discovery of his famous principle: Hieron, king of Syracuse, was suspicious that a new crown that he had received from the royal goldsmith was not pure gold, as claimed. Archimedes was ordered to determine whether the crown was in fact made of pure gold, with the condition that only a nondestructive test would be allowed. Rather than answer the problem in the politically most expedient way (or perhaps extract a bribe from the goldsmith), Archimedes thought about the problem scientifically. The legend relates that when Archimedes stepped into his bath and caused it to overflow, he realized that he could answer the challenge by comparing the volume of water displaced by the crown with the volume of water displaced by an amount of pure gold equal in weight to the crown. If the crown was made of pure gold, the two volumes would be equal. If some other (less dense) metal had been substituted for some of the gold, then the crown would displace more water than the pure gold.

A similar method of answering the challenge, based on the same physical principle, is to compute the ratio of the actual weight of the crown, WactualWactual, and the apparent weight of the crown when it is submerged in water, WapparentWapparent. See whether you can follow in Archimedes' footsteps. The figure shows what is meant by weighing the crown while it is submerged in water.

(Figure 1)

Take the density of the crown to be ρcρc. What is the ratio of the crown's apparent weight (in water) WapparentWapparent to its actual weight WactualWactual?

Express your answer in terms of the density of the crown ρcρc and the density of water ρwρw.

Answer 1

Ok, EURICA!

At first, I want to remind you the difference between the weight and apparent weight. Weight mg is force of gravity, i.e. the force that the Earth exerts on the object. At a given location it stays the same. Apparent weight W_apt is a force that the body exerts on a scale. If the body at rest or moves with constant speed, the magnitudes of those two forces are equal to each other. If the body moves with acceleration or not on horizontal surface, or submerged in a liquid, the apparent weight changes its value.

Let’s now talk about the measuring of the weight of the crown. When it is suspended in the air and is not moving, the scale is showing its actual weight W_act = mg, where m – mass of the body, g – acceleration due to gravity.

If the crown is submerged in the water, we will see that the reading of the scale is less. The reason is the presence of the buoyant force F_B that acts on the body vertically up.

In this case 3 forces acts on the crown:

1.       weight mg – direction down
2.       tension of the rope T– direction up
3.   buoyant force F_B – direction up.

The system is in equilibrium; hence we can write:

                                                                 mg = F_B + T

We need to find apparent weight W_apt. It is easy, if we recall the Newton’s third law: in our case crown acts on the rope (apparent weight) and rope acts on the crown (tension force). Hence their magnitudes are the same, i.e. W_apt = T.  So,

                                                             F_B = mg - W_apt  = W_act - W_apt.              (1)

We see that buoyant force is the difference between the weight of the body in the air and in the liquid.

We can rewrite equation (1) as

                                                                 W_apt / W_act = 1 - F_B / W_act

Knowing that the buoyant force F_B = ρ_wgV (here ρ_w – density of the liquid, in our case water, V – submerged volume of the body), W_act = mg and mass of the crown m = ρ_cV ( ρ_c – density of the crown) we get:

                                                               W_apt / W_act = 1 - ρ_wgV / ρ_cgV = (ρ_c - ρ_w / ρ_c

                                                                     

                                                                                Answer: W_apt / W_act = (ρ_c - ρ_w / ρ_c