A hydrometer floats in water of density 1.0g/cm³ with 6.0cm of its graduated stem above the water level, and in oil of density 0.8g/cm³ with 4.0cm above the oil level. Calculate the length of stem above the liquid level when the hydrometer placed in a liquid of density 0.9g/cm³.
Let x be the length of the stem above the liquid level when placed in a liquid of density 0.9g/cm³.
Using the principle of flotation:
Weight of hydrometer in water = Weight of water displaced
Weight of hydrometer in oil = Weight of oil displaced
Weight of hydrometer in liquid of density 0.9g/cm³ = Weight of liquid displaced
The weight of the hydrometer is the same in all three cases. Let's assume it is w grams.
In water:
Weight of hydrometer = w
Weight of water displaced = volume of water displaced x density of water x gravity
= (total volume of hydrometer - submerged volume of hydrometer) x 1.0g/cm³ x 9.8 m/s²
= (6 + x) cm x 1.0g/cm³ x 9.8 m/s²
Therefore:
w = (6 + x) cm x 1.0g/cm³ x 9.8 m/s²
In oil:
Weight of hydrometer = w
Weight of oil displaced = volume of oil displaced x density of oil x gravity
= (total volume of hydrometer - submerged volume of hydrometer) x 0.8g/cm³ x 9.8 m/s²
= (4 cm) x 0.8g/cm³ x 9.8 m/s²
Therefore:
w = 4 cm x 0.8g/cm³ x 9.8 m/s²
Equating the weight of the hydrometer in water and in oil:
(6 + x) cm x 1.0g/cm³ x 9.8 m/s² = 4 cm x 0.8g/cm³ x 9.8 m/s²
Simplifying:
(6 + x) x 1 = 3.2
6 + x = 3.2
x = -2.8
This result doesn't make sense, since x represents a length and cannot be negative. Therefore, the hydrometer would sink in the liquid of density 0.9g/cm³.
To solve this problem, we can use the principle of buoyancy.
Step 1: Calculate the volume of the hydrometer stem submerged in water.
The hydrometer floats in water with 6.0 cm of its stem above the water level. The density of water is 1.0 g/cm³. Since the hydrometer is floating, the weight of the hydrometer is balanced by the buoyant force acting on it.
We can calculate the volume of the submerged stem using the formula:
Volume = (Weight of the hydrometer) / (Density of water) = (Buoyant force) / (Density of water)
The weight of the hydrometer is equal to the weight of the volume of water displaced by the submerged stem:
Weight of the hydrometer = (Density of water) x (Volume of submerged stem)
The buoyant force is equal to the weight of the hydrometer:
Buoyant force = (Weight of the hydrometer) = (Density of water) x (Volume of submerged stem)
Since the weight of the hydrometer and the buoyant force are equal, we can rewrite the equation:
(Density of water) x (Volume of submerged stem) = (Density of water) x (Volume of submerged stem)
Volume of submerged stem in water = Volume of hydrometer
The volume of the hydrometer stem is equal to the volume of a cylinder:
Volume of hydrometer = π x (radius)² x (height)
In this case, the height is 6.0 cm, and the radius can be determined by measuring the diameter of the stem.
Step 2: Calculate the length of the stem above the liquid level for the liquid with a density of 0.9 g/cm³.
Now, we need to calculate the length of the stem above the liquid level when the hydrometer is placed in a liquid with a density of 0.9 g/cm³. We can use a similar approach as in step 1.
Let's denote the length of the stem above the liquid level as x cm.
Using the same logic, we can calculate the volume of the hydrometer stem submerged in the liquid:
Volume of submerged stem in liquid = Volume of hydrometer
The volume of the hydrometer stem is still a cylinder, but with a different height:
Volume of hydrometer = π x (radius)² x (height)
In this case, the height is (4.0 cm + x) since there are 4.0 cm of the stem above the oil level and x cm above the liquid level. The radius can be determined by measuring the diameter of the stem.
Now we have the equation:
(Density of liquid) x (Volume of submerged stem in liquid) = (Density of water) x (Volume of hydrometer)
We know the densities of the liquids and the volume of the hydrometer from step 1, so we can solve for the volume of the submerged stem in the liquid.
Step 3: Solve for x.
Now we have two equations:
(Density of water) x (Volume of submerged stem in water) = (Density of water) x (Volume of hydrometer)
(Density of liquid) x (Volume of submerged stem in liquid) = (Density of water) x (Volume of hydrometer)
Since the volume of the hydrometer is the same in both cases, we can equate the two equations:
(Density of water) x (Volume of submerged stem in water) = (Density of liquid) x (Volume of submerged stem in liquid)
Substituting the values we have, we can solve for x, which is the length of the stem above the liquid level.
With these calculations, we can find the length of the stem above the liquid level when the hydrometer is placed in a liquid of density 0.9 g/cm³.