Use Archimedes’ principle to prove whether or not the king crown is made up of gold. You found that a king’s crown whose mass is (a) 9.28 kg has an apparent mass of (b) 6.18 kg when submerged in water. (The density of pure gold is 1.932x 104 kg/m3).
a.
Density of gold is 1.932x 104 kg/m3 therefore the king's crown is made up of pure gold.
b.
Density of gold is 2.99x 104 kg/m3 therefore the king's crown is not made up of pure gold.
c.
Density of gold is 2.99x 103 kg/m3 therefore the king's crown is not made up of pure gold.
d.
Density of gold is 4.99x 104 kg/m3 therefore the king's crown is not made up of pure gold.
e.
None of these
To determine whether the king's crown is made of pure gold using Archimedes' principle, we need to compare the density of the crown with the density of gold.
Archimedes' principle states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. This buoyant force can be calculated using the formula:
Buoyant Force = Density of Fluid × Volume of Displaced Fluid × Acceleration due to Gravity
For the king's crown, we know its mass is 9.28 kg and its apparent mass when submerged in water is 6.18 kg. The difference in mass represents the mass of water displaced by the crown.
The volume of water displaced can be calculated using the formula:
Volume of Water Displaced = (Mass of Water Displaced) / (Density of Water)
Now, we need to calculate the density of the crown. We know that:
Density = Mass / Volume
Rearranging this equation, we have:
Volume = Mass / Density
Substituting the given values, we get:
Volume of Crown = Mass of Crown / Density of Crown
Now, using Archimedes' principle, we can compare the density of the crown to the density of gold.
Density of Gold = 1.932 × 10^4 kg/m³ (given)
If the density of the crown is equal to or very close to the density of gold, then it can be concluded that the crown is made of pure gold.
Let's calculate the volume of the crown first:
Volume of Crown = Mass of Crown / Density of Crown
= 9.28 kg / Density of Crown
To find the density of the crown, we can use the formula:
Density of Crown = Mass of Crown / (Volume of Crown + Volume of Displaced Water)
Density of Crown = 9.28 kg / (Volume of Crown + Volume of Displaced Water)
where Volume of Displaced Water can be calculated using:
Volume of Displaced Water = Mass of Displaced Water / Density of Water
Volume of Displaced Water = (9.28 kg - 6.18 kg) / Density of Water
Now, substitute the values into the equation:
Density of Crown = 9.28 kg / (Volume of Crown + (9.28 kg - 6.18 kg) / Density of Water)
After calculating the density of the crown, compare it with the density of gold to determine the composition of the crown.
After performing the calculations, if the density of the crown matches the density of gold closely, then the crown is made of pure gold. If not, it is not made of pure gold.
Comparing the calculated density with the given densities, we can determine the answer:
a. Density of gold is 1.932 × 10^4 kg/m³, therefore the king's crown is made up of pure gold.
Therefore, the correct answer is a. Density of gold is 1.932 × 10^4 kg/m³, and the king's crown is made up of pure gold.
To determine whether or not the king's crown is made up of gold, we can use Archimedes' principle. Archimedes' principle states that an object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.
Let's find the volume of the crown using the given information first:
Apparent mass = 6.18 kg
Actual mass = 9.28 kg
We can find the volume of the crown using the formula:
Density = Mass/Volume
Since the density of gold is given as 1.932x10^4 kg/m^3, we can rearrange the formula as:
Volume = Mass/Density
Let's calculate the volume of the crown:
Volume = 9.28 kg / (1.932x10^4 kg/m^3)
= 4.798537m^3
Now, we can use this volume to calculate the actual volume of the crown when submerged in water:
Apparent volume = Actual volume - Volume of water displaced
Since the crown is submerged in water, the apparent volume is equal to the volume of water displaced by the crown. Therefore:
Apparent volume = Volume of water displaced
Now, we can use Archimedes' principle to determine the weight of the water displaced by the crown:
Weight of water displaced = Apparent mass * acceleration due to gravity
Assuming the acceleration due to gravity is approximately 9.8 m/s^2:
Weight of water displaced = 6.18 kg * 9.8 m/s^2
= 60.444 N
Since the weight of water displaced is equal to the buoyant force acting on the crown, we can use this information to calculate the density of the crown:
Density of crown = Mass of crown / Volume of crown
Density of crown = 9.28 kg / Apparent volume
Now, let's substitute the values we have calculated:
Density of crown = 9.28 kg / (4.798537 m^3)
= 1.9341302x10^3 kg/m^3
Comparing the density of the crown to the density of gold (1.932x10^4 kg/m^3), we can conclude that:
c. Density of gold is 2.99x10^3 kg/m^3, therefore the king's crown is not made up of pure gold.