# Calculate the density of an object by the amount of displaced mass in water

• sagigever
In summary, the volume of the crown is 50 cm^3 because the crown displaced an amount of water equal to 50 grams-force.
sagigever
Homework Statement
King Hiero II of Syracuse suspected he was being cheated by the goldsmith to whom he
had supplied the gold to make a crown. He asked Archimedes to find out if pure gold has
been substituted by the same weight of electrum (alloy of gold and silver). Archimedes
solved this problem by weighing the crown first in air and then in water. Suppose the
weight of crown in air was 740 g and in water 690 g. What should Archimedes have told
the king? Density of gold is 19.3 g/cm
Relevant Equations
##\rho = \frac{m}{v}##
I have the solution for this problem but I did not understand the following statement:

The mass of water the crown displaced is ##m = 740- 690= 50 g##. Therefore the volume of the crown is ## 50 cm^3##

how can I conclude the volume of the crown from that displacement?

The mass of water the crown displaced is ##m = 740- 690= 50 g##. Therefore the volume of the crown is ## 50 cm^3##

how can I conclude the volume of the crown from that displacement?

How much does water weigh ?

There's probably a mistake here. You are using weight and mass interchangeably which is incorrect.
An object's mass wouldn't change in water.

etotheipi
archaic said:
There's probably a mistake here. You are using weight and mass interchangeably which is incorrect.
An object's mass wouldn't change in water.

And to add to the potential confusion, an object's apparent weight would change but its weight wouldn't. And when we say something is 'weightless' we are implicitly referring to apparent weight

The solution is true for 100% however I do not understand the conclusion about the volume of the crown from the equation I mentioned

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An immersed object displaces its own volume of water.

sagigever said:
The solution is true for 100% however I do not understand the conclusion about the volume of the crown from the equation I mentioned
Let us start with the givens of the problem. The crown "weigh" 740 grams in air and "weighs" only 690 grams when immersed in water.

To establish this, one can imagine that an equal arm balance scale was employed. The crown was set on one pan and 740 standard grams on the other. The balance balanced. [Fortunately, King Hiero II had the foresight to acquire a set of properly calibrated SI measuring equipment].

Then a beaker of distilled water was procured and chilled to 4 degrees Celsius. The crown was hung from the balance pan by a slender thread so that it dangled, fully submerged in the water, not touching the sides or bottom. 690 standard grams were placed on the opposite pan and again the balance balanced.

Now we can dispense with much of the banter about whether this amounts to a measure of mass, of force, or of some bastardized hybrid of the two. Suffice it to say that the buoyancy provided by the water amounted to 50 grams-force.

How much buoyant force would there be on an object that displaced one cubic centimeter of water? Answer in grams-force for convenience.

How many cubic centimeters need to be displaced to result in 50 grams-force of buoyant force?

What does that say about the volume of the crown?

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Lnewqban
sagigever said:
The solution is true for 100% however I do not understand the conclusion about the volume of the crown from the equation I mentioned

When fully summerged, that solid crown has occupied the volume that certain mass of liquid had been previously occupying.
If you know the relation between mass and volume of that specific liquid, which is more or less a fixed value for each substance (density), you can estimate the compromised volume of both, the liquid and the fully summerged solid.

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## What is the purpose of calculating the density of an object by the amount of displaced mass in water?

The purpose of this calculation is to determine the density of an object, which is a measure of how much mass is contained within a certain volume. This can provide valuable information about the composition and properties of the object.

## How do you calculate the density of an object using the amount of displaced mass in water?

To calculate the density, you need to measure the mass of the object and the volume of water it displaces. Then, divide the mass by the volume to get the density. The formula is density = mass/volume.

## Why is water used to calculate the density of an object?

Water is used because it has a known density of 1 gram per cubic centimeter. This makes it a convenient and accurate substance to use for measuring the density of other objects.

## What are the units for density?

The units for density are typically grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3). However, other units such as pounds per cubic inch (lbs/in3) or ounces per cubic inch (oz/in3) may also be used.

## What factors can affect the accuracy of the calculated density?

The accuracy of the calculated density can be affected by factors such as measurement errors, irregularities in the shape of the object, and temperature changes in the water. It is important to take multiple measurements and ensure the object is fully submerged in the water to minimize these sources of error.

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