# University of Wisconsin Green Bay

A royal treasurer commissions a craftsman to make an elaborate gold crown. When the crown is delivered, she suspects that it may actually be made of brass covered in gold. She has the crown tested to make sure. The crown weighs 24.5 N in air, and has an apparent weight in fresh water of 22.1 N. Is the treasurer correct?

• Even though most text books put buoyant force problems in a separate (fluids) chapter from Newton’s Laws, this is a Newton’s 2nd Law problem. You are told about forces (weight, apparent weight) on a crown and have the extra knowledge that we typically weight objects when they are at rest. In other words, you also know about the crown’s motion.

Force and motion of a single object are always related through Newton’s Second Law, so this is a force or 2nd Law problem regardless of what information is requested in the problem.

• The key equation for any problem that relates forces and motion is Newton’s Second Law. Regardless of what quantity you are asked to find, begin with the Second Law. If additional information is needed, it will become apparent as you proceed.

• Step 1

Newton’s 2nd Law allows you to solve for the volume of the crown, but does not yet tell you if it is pure gold. Continue to step 2 for the final step to answer the question.

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Step 2

At this point in the problem, you know that the crown weighs 24.5 N and has a volume of 2.45 x 10-4 m3. These two properties are related through the definition of density, and density can be used to identify a metal.

ρcrown = masscrown/volumecrown
ρcrown = (24.5 N/9.81 m/s2)/( 2.45 x 10-4 m3)
ρcrown = (2.50 kg)/( 2.45 x 10-4 m3) = 10,200 kg/m3

Using a table from your text book or from the web, you can find that the density of gold is 19,300 kg/m3, so the crown is definitely not pure gold. The density of brass ranges around 8300-8700 kg/m3 (remember brass is an alloy) and so it is entirely possible that this is largely a brass crown covered in gold. The treasurer is right that the crown isn’t gold, and her guess of its actual composition is reasonable.

• ρ = mass/volume
ρcrown = (2.50 kg)/( 2.45 x 10-4 m3) = 10,200 kg/m3≠ ρgold

Regardless of what you were asked to find, this is a Newton’s Second Law problem, because the interactions that are described in the problem are forces acting on the crown. When the crown is submerged, water acts to push up on it. (It doesn’t float because the density of the crown is so much greater than that of water.) Therefore, it appears to weigh less than its actual value of mg. The downward force of gravity is exactly balanced by the upward force of the water plus that of the scale.

This information allows us to solve for the only unknown in the problem—that of the volume of the crown. The problem, however, asked us to determine if the crown is made of pure gold, and that identification can be made through density. Density is directly related to volume through a simple definition: