Step-by-step solution
We begin by stating the condition for static equilibrium, and then we apply Archimedes’ principle.
Now we replace the masses in the previous equation by products of densities and volumes, solve for a ratio of volumes, and evaluate.
Since the volume VH2O of the displaced water equals the volume of the submerged portion of the iceberg, we have shown that 89.0% of the
iceberg is submerged.
For most substances, the solid phase is denser than the liquid phase, meaning a solid sinks when immersed in a liquid composed of the same
substance. Water is very unusual in this respect: the solid phase (ice) floats in liquid water. This proves crucial to life on Earth for a variety of
reasons.
Step Reason
1. equilibrium
2. mH2Og = miceg Archimedes
3. mH2O = mice simplify
Step Reason
4. ȡH2OVH2O = ȡiceVice definition of density
5. rearrange
6. evaluate
13.10 - Interactive problem: Eureka!
In this simulation you play the role of the ancient Greek mathematician and physicist
Archimedes. As the story goes, in olden Syracuse the tyrant Hieron suspected that
a wily goldsmith had adulterated one of his kingly crowns during manufacture by
adding some copper and zinc to the precious gold. His Royal Highness asked
Archimedes to discover whether this was indeed the case.
Archimedes pondered the problem for days. Then, one day, in the public bath, he
observed how the water level in the pool rose as he eased himself in for a good
soak, and suddenly realized that the answer was right in front of his eyes. Ecstatic,
he supposedly leapt from the bath and ran dripping through the streets of the city
crying “Eureka!” (I have found it!) The adulterated crown was quickly identified, and
the goldsmith roundly punished.
Your task is to determine the nature of Archimedes’ insight and apply it in the
simulation to the right. You have two crowns and a bar of pure gold. You have
already observed their masses using a balance scale. You can measure the volume
of a crown or bar in the simulation by dragging it to the bath and noting how much
water it displaces. You see the tube that measures the displaced liquid in the
illustration at the right. Decide which crown has been altered and drag it to the king’s palace. The appropriate consequences will be enacted.