Archimedes purportedly used his principle to verify that the king's crown was pure gold, by weighing the crown while it was submerged in water. Suppose the crown's actual weight was 14.0 N. The densities of gold, silver, and water are 19.3 g/cm^3, 10.5 g/cm^3, and 1.00 g/cm^3, respectively.

(a) What would be the crown's apparent weight if it were made of pure gold?

(b) What would be its apparent weight if it were made of 80% gold and 20% silver, by volume?

Gold, being denser than silver, will have less volume and less buoyant force acting upon it.

(a) Crown mass = W/g = 14.0/9.8
= 1.428 kg
Crown volume = V = 1428 g/19.3 g/cm^3
= 74.0 cm^3 = 74.0*10^-6 m^3
Buoyant force = V*g*(water density)
= 74*10^-6*9.8*1000 = 0.725 N
Apparent weight = 14.0 N - 0.725 N
= 13.3 N

(b) density of alloy = 0.8*19.3 + 0.2*10.5 = 17.54 g/cm^3
Volume will be 1.10 times as large to achieve the same crown weight.
Buoyancy force will be 10% greater. or 0.798 N
Apparent weight = 13.2 N

(a) If the crown was made of pure gold, we can determine its apparent weight by using Archimedes' principle. According to the principle, the apparent weight of an object submerged in fluid is equal to the weight of the fluid displaced by the object. Since the crown is submerged in water, the weight of the water displaced by the crown will be equal to the apparent weight of the crown.

To calculate the weight of the water displaced, we need to determine the volume of the crown. Since the crown's density would be the same as that of gold (19.3 g/cm^3), we can use the given weight (14.0 N) to calculate the volume of the crown.

First, we need to convert the weight of the crown from newtons to grams:
Weight (in grams) = Weight (in newtons) * 1000 g/kg
Weight (in grams) = 14.0 N * 1000 g/kg = 14000 g

Next, we can use the density of gold to find the volume of the crown:
Volume = Mass / Density
Volume = 14000 g / 19.3 g/cm^3 ≈ 725.39 cm^3

Since the volume of the water displaced is equal to the volume of the crown (725.39 cm^3), the weight of the water displaced is:
Weight of water displaced = Volume of water displaced * Density of water = 725.39 cm^3 * 1.00 g/cm^3 = 725.39 g

Therefore, the apparent weight of the crown if it were made of pure gold would be 725.39 g or 725.39 N.

(b) If the crown were made of 80% gold and 20% silver by volume, we need to modify the calculations slightly. First, let's determine the volume of each metal.

Volume of gold = 80% of total volume = 0.8 * 725.39 cm^3 = 580.31 cm^3
Volume of silver = 20% of total volume = 0.2 * 725.39 cm^3 = 145.08 cm^3

Next, we need to find the mass of each metal.

Mass of gold = Volume of gold * Density of gold = 580.31 cm^3 * 19.3 g/cm^3 = 11205.83 g
Mass of silver = Volume of silver * Density of silver = 145.08 cm^3 * 10.5 g/cm^3 = 1523.74 g

Now, we can calculate the total weight of the crown.
Total weight = Weight of gold + Weight of silver + Weight of displaced water
Total weight = Mass of gold * g + Mass of silver * g + Weight of water displaced
Total weight = 11205.83 g * 9.8 m/s^2 + 1523.74 g * 9.8 m/s^2 + 725.39 g * 9.8 m/s^2
Total weight ≈ 109942.54 N

Therefore, the apparent weight of the crown if it were made of 80% gold and 20% silver, by volume, would be approximately 109942.54 N.

To answer these questions, we need to apply Archimedes' principle, which states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. We can calculate the apparent weight of the crown using the densities provided.

(a) If the crown were made of pure gold, its density would be 19.3 g/cm^3. Since the density of water is 1.00 g/cm^3, the crown would displace its own weight in water. Therefore, the apparent weight of the crown would be zero when submerged in water.

(b) If the crown were made of 80% gold and 20% silver by volume, we need to calculate the density of this alloy. We can use the weighted average formula:

Density of the alloy = (Volume gold * Density gold + Volume silver * Density silver) / Total volume

To calculate the total volume, we assume the crown has a volume of 1 cm^3. The volume of gold would be 0.8 cm^3 (80% of 1 cm^3), and the volume of silver would be 0.2 cm^3 (20% of 1 cm^3).

Density of the alloy = (0.8 cm^3 * 19.3 g/cm^3 + 0.2 cm^3 * 10.5 g/cm^3) / 1 cm^3

Density of the alloy = (15.44 g + 2.10 g) / 1 cm^3

Density of the alloy = 17.54 g/cm^3

Now, using Archimedes' principle, we can calculate the apparent weight of the crown when submerged in water.

Apparent weight = Actual weight - Weight of water displaced

Weight of water displaced = Volume of the crown * Density of water

Since the crown's volume is 1 cm^3, the weight of water displaced is:

Weight of water displaced = 1 cm^3 * 1.00 g/cm^3 = 1.00 g

Apparent weight = 14.0 N - 1.00 g

Note: To convert grams(g) to Newtons(N), we need to use the conversion factor 1 N = 0.10197 g.

Apparent weight = 14.0 N - (1.00 g * 0.10197 N/g)

Apparent weight = 14.0 N - 0.10197 N = 13.9 N

Therefore, the apparent weight of the crown made of 80% gold and 20% silver, by volume, would be 13.9 N when submerged in water.

To determine the crown's apparent weight, we need to use Archimedes' principle, which states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.

Let's break down the problem step by step:

(a) If the crown were made of pure gold, we need to find the weight of the gold crown and the weight of the water it displaces.

1. First, we need to convert the density of gold to kilograms per cubic meter (kg/m^3). Since the given density is in grams per cubic centimeter (g/cm^3), we can multiply it by 1000 to get the density in kg/m^3.

Density of gold = 19.3 g/cm^3 * (1000 g/kg) = 19,300 kg/m^3

2. Next, we need to find the volume of the crown. Since the crown's weight is given as 14.0 N, we can use the formula for weight:

Weight = mass * acceleration due to gravity

Applying this to the crown:

14.0 N = mass * 9.8 m/s^2

Solving for mass:

mass = 14.0 N / 9.8 m/s^2 = 1.43 kg

3. Now we can calculate the volume of the crown. Since density is defined as mass per unit volume, we can rearrange the formula:

Volume = mass / density

Using the mass we found and the density of gold:

Volume = 1.43 kg / 19,300 kg/m^3 = 0.000074 m^3

4. Finally, we can calculate the weight of the water displaced by the crown using its volume.

Weight of the water displaced = Volume * density of water * acceleration due to gravity

Weight of the water displaced = 0.000074 m^3 * 1000 kg/m^3 * 9.8 m/s^2 = 0.725 N

The crown's apparent weight, if it were made of pure gold, would be the weight of the gold crown minus the weight of the water displaced:

Apparent weight = Weight of the crown - Weight of the water displaced = 14.0 N - 0.725 N = 13.275 N

Therefore, the crown's apparent weight, if it were made of pure gold, would be approximately 13.275 N.

(b) To find the apparent weight of the crown if it were made of 80% gold and 20% silver, by volume, we follow a similar process.

1. First, we calculate the total volume of the crown, taking into account the different densities of gold and silver.

Assuming the volume is 100 units, 80% of it would be gold and 20% would be silver.

Volume of gold = 80 units
Volume of silver = 20 units

2. Next, we calculate the weights of the gold and silver components.

Weight of gold = Volume of gold * density of gold * acceleration due to gravity
Weight of silver = Volume of silver * density of silver * acceleration due to gravity

Using the given densities of gold and silver:

Weight of gold = 80 units * 19.3 g/cm^3 * (1000 g/kg) * 9.8 m/s^2 = 150,080 N
Weight of silver = 20 units * 10.5 g/cm^3 * (1000 g/kg) * 9.8 m/s^2 = 20,580 N

3. Finally, we calculate the total weight of the crown by summing the weights of the gold and silver components.

Total weight = Weight of gold + Weight of silver = 150,080 N + 20,580 N = 170,660 N

Therefore, the crown's apparent weight, if it were made of 80% gold and 20% silver, by volume, would be approximately 170,660 N.