A solid aluminum cylinder, ρ = 2700 kg/m3 , “weigh” 67 g in air and 45 g when immersed in turpentine. Determine the density of turpentine.
0.22kg/m3
To determine the density of turpentine, we can use the concept of buoyancy.
The buoyancy force acting on the cylinder when immersed in turpentine can be calculated using Archimedes' principle, which states that the buoyancy force is equal to the weight of the fluid displaced by the object.
Let's begin by calculating the weight of the aluminum cylinder in air and in turpentine:
Weight in air = 67 g
Weight in turpentine = 45 g
Next, we need to calculate the buoyant force acting on the cylinder when it is immersed in turpentine.
Buoyant force = Weight in air - Weight in turpentine
Buoyant force = 67 g - 45 g
Buoyant force = 22 g
Since we need to determine the density of turpentine, we can calculate it using the formula:
Density of turpentine = Buoyant force / Volume of turpentine displaced
Now, we need to relate the buoyant force to the volume of turpentine displaced by the cylinder.
Buoyant force = Density of turpentine x Volume of turpentine displaced x gravity
The volume of turpentine displaced by the cylinder is equal to the volume of the cylinder itself. The volume of a cylinder can be calculated using the formula:
Volume of a cylinder = π x (radius)^2 x height
We need to know the dimensions of the cylinder to calculate its volume. If you have the dimensions of the cylinder, please provide them so that we can proceed with the calculation.
To determine the density of turpentine, we need to use the principle of buoyancy. The difference in weight between the cylinder in air and when immersed in turpentine is equal to the weight of the fluid displaced by the cylinder. The weight of the fluid displaced is determined by the buoyant force acting on the cylinder, which is equal to the weight of the fluid it displaces.
Let's denote:
- W_air: Weight of the cylinder in air
- W_turpentine: Weight of the cylinder in turpentine
- V_cylinder: Volume of the cylinder
- ρ_cylinder: Density of the cylinder
- ρ_turpentine: Density of turpentine
Given:
W_air = 67 g
W_turpentine = 45 g
ρ_cylinder = 2700 kg/m3
The volume of the cylinder can be calculated using the formula:
V_cylinder = W_air / (ρ_cylinder * g)
where g is the acceleration due to gravity (around 9.8 m/s^2).
Substituting the given values:
V_cylinder = (67 g) / (2700 kg/m3 * 9.8 m/s^2)
= 0.002 m3
Now, since the weight of the fluid displaced is the difference in weight between the cylinder in air and in turpentine, we have:
W_fluid_displaced = W_air - W_turpentine
The weight of the fluid displaced can also be calculated using the formula:
W_fluid_displaced = V_fluid_displaced * ρ_turpentine * g
where V_fluid_displaced is the volume of the fluid displaced.
Since the volume of the cylinder is equal to the volume of the fluid displaced, we have:
V_fluid_displaced = V_cylinder
Therefore, we can rewrite the formula for the weight of the fluid displaced as:
W_fluid_displaced = V_cylinder * ρ_turpentine * g
Now, we can substitute the given values and solve for ρ_turpentine:
V_cylinder * ρ_turpentine * g = W_air - W_turpentine
0.002 m3 * ρ_turpentine * 9.8 m/s^2 = 67 g - 45 g
0.002 m3 * ρ_turpentine * 9.8 m/s^2 = 22 g
To convert the weight from grams to kilograms, we divide by 1000:
0.002 m3 * ρ_turpentine * 9.8 m/s^2 = 0.022 kg
Simplifying the equation:
ρ_turpentine = 0.022 kg / (0.002 m3 * 9.8 m/s^2)
= 1122.45 kg/m3
Therefore, the density of turpentine is approximately 1122.45 kg/m3.