You need to determine the density of a ceramic statue. If you suspend it from a spring scale, the scale reads 35.4 N. If you then lower the statue into a tub of water, so that it is completely submerged, the scale reads 12.4 N. What is the density?
There is a bouyancy force when submerged that is equal to 35.4 - 12.4 = 23.0 N
The mass of displaced water is 23.0/9.8 = 2.347 kg= 2347 g
The volume of displaced water (and the statue) is 2347/1.0 = 2347 cm^3
The mass of the statue is 35.4/9.8 = 3.612 kg = 3612 g
The density of the statue is 3612/2347 = ___ g/cm^3
To determine the density of the ceramic statue, we need to use the concept of buoyancy.
Step 1: Calculate the weight of the statue in air.
The scale reading when the statue is suspended in air is 35.4 N. Therefore, the weight of the statue is also 35.4 N.
Step 2: Calculate the buoyant force when the statue is submerged in water.
The scale reading when the statue is completely submerged in water is 12.4 N. This reading represents the difference between the weight in air and the buoyant force acting on the statue.
Buoyant force (B) = weight in air - scale reading in water
B = 35.4 N - 12.4 N = 23 N
Step 3: Calculate the weight of water displaced by the statue.
According to Archimedes' principle, the buoyant force exerted on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
Weight of water displaced = Buoyant force
= 23 N
Step 4: Use the density formula to calculate the statue's density.
Density (ρ) = Mass (m) / Volume (V)
We know that the statue's mass stays constant regardless of its position, so we can write:
Density = Mass / Volume
From Step 3, we know that the weight of water displaced is equal to the buoyant force, which is equal to the weight of the statue in air.
Weight of water displaced = Weight in air = 35.4 N
Using the formula Weight = Mass * Gravity, where the acceleration due to gravity is approximately 9.8 m/s²:
Mass = Weight / Gravity = 35.4 N / 9.8 m/s² ≈ 3.61 kg
Now, we can calculate the density:
Density = Mass / Volume
Density = 3.61 kg / Volume
However, we still need to determine the volume of the statue. To do this, we can use the fact that the volume of water displaced by the statue is equal to its volume.
Step 5: Calculate the volume of the statue.
Volume of statue = Volume of water displaced
The density of water is 1000 kg/m³, so the volume of water displaced by the statue is:
Volume of water displaced = Weight of water displaced / Density of water
Volume of water displaced = 23 N / (1000 kg/m³) ≈ 0.023 m³
Therefore, the volume of the statue is approximately 0.023 m³.
Step 6: Calculate the density of the statue.
Density = Mass / Volume
Density = 3.61 kg / 0.023 m³ ≈ 157.4 kg/m³
Hence, the density of the ceramic statue is approximately 157.4 kg/m³.
To determine the density of the ceramic statue, we can use Archimedes' principle, which states that an object immersed in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the object.
First, let's find the weight of the statue in air. The scale reading in air is 35.4 N, which represents the downward force due to gravity acting on the statue. Therefore, the weight of the statue in air is 35.4 N.
Next, let's find the weight of the statue when it is submerged in water. The scale reading in water is 12.4 N. However, in water, the reading also includes the force of buoyancy acting upward on the statue. The buoyant force is equal to the weight of the water displaced by the statue.
Using Archimedes' principle, we can say that the weight of the statue in air (35.4 N) is equal to the weight of the statue in water (12.4 N) plus the buoyant force.
Buoyant Force = Weight in Air - Weight in Water
Buoyant Force = 35.4 N - 12.4 N
Buoyant Force = 23 N
Now we can calculate the weight of the water displaced. Since water has a density of 1000 kg/m^3, the weight of the water displaced is equal to the volume of the statue submerged in water multiplied by the density of water and the acceleration due to gravity (9.8 m/s^2).
Weight of Water Displaced = Volume of Statue Submerged * Density of Water * g
To find the volume of the statue submerged, we need to use the formula:
Density = Mass / Volume
Since the mass of the statue is not given, we need another equation to solve for the mass. We can use the equation:
Weight = Mass * g
Given that the weight of the statue in air is 35.4 N, we can solve for the mass:
Mass = Weight / g
Mass = 35.4 N / 9.8 m/s^2
Mass = 3.61 kg (approx.)
Now we have the mass of the statue. Let's substitute this value into the equation to calculate the volume submerged:
Density = Mass / Volume
3.61 kg / Volume = Density
Next, let's calculate the volume submerged by dividing the weight of the water displaced by the product of the density of water and the acceleration due to gravity:
Weight of Water Displaced = Volume of Statue Submerged * Density of Water * g
23 N = Volume of Statue Submerged * 1000 kg/m^3 * 9.8 m/s^2
Solving for the volume of the statue submerged:
Volume of Statue Submerged = 23 N / (1000 kg/m^3 * 9.8 m/s^2)
Volume of Statue Submerged = 0.00235 m^3
Now we can calculate the density of the statue:
Density = Mass / Volume
Density = 3.61 kg / 0.00235 m^3
Density = 1536 kg/m^3
Therefore, the density of the ceramic statue is approximately 1536 kg/m^3.