A hydrometer is an instrument for measuring the specific

gravity of a liquid. For example, producers of maple syrup use the hydrometer to find how much
sugar is in the collected sap. Markings along a stem are calibrated to indicate the specific gravity
for the level at which the hydrometer floats in a liquid. The weighted base ensures that the
hydrometer floats vertically. Suppose the hydrometer has a cylindrical stem of cross-sectional
area 0.400 cm
2.
The total volume of the bulb and stem is 8.8 cm
3
and the mass of the hydrometer
is 4.8 gm.
(a) How far from the top of the cylinder should a mark be placed to indicate a specific gravity of
1.00?
(b) When the hydrometer is placed in some oil, it floats with 7.25 cm of stem above the surface.
What is the specific gravity of the oil?
(c) What is the lowest specific gravity that can be measured with this hydrometer?

To answer these questions, we need to understand the concept of specific gravity and how it relates to the buoyancy of the hydrometer.

Specific gravity is the ratio of the density of a substance to the density of a reference substance, typically water. It is a dimensionless quantity, denoted as SG. For liquids and solids, the specific gravity is usually measured with respect to water, where SG of water is equal to 1.00.

Now let's solve each part of the question:

(a) To determine where the mark should be placed to indicate a specific gravity of 1.00, we need to consider the buoyancy of the hydrometer. When the hydrometer is floating in a liquid, it experiences an upward force known as buoyant force equal to the weight of the liquid it displaces.

In this case, we have the specific gravity equal to 1.00, which means the hydrometer would float entirely in the liquid without any part of the stem above the surface. So, the mark indicating SG=1.00 should be placed at the top of the cylinder.

(b) When the hydrometer is placed in the oil and floats with 7.25 cm of stem above the surface, we can determine the specific gravity of the oil using the principle of buoyancy and the dimensions of the hydrometer.

We know that the buoyant force is equal to the weight of the liquid displaced, and it balances the weight of the hydrometer. The weight of the hydrometer is equal to its mass multiplied by the acceleration due to gravity. In this case, the weight of the hydrometer is 4.8 gm (0.0048 kg) multiplied by 9.8 m/s^2.

The volume of the hydrometer is given as 8.8 cm^3. Since we are interested in the specific gravity of the oil, we can use the displacement volume of the hydrometer when it is floating with 7.25 cm of stem above the surface.

To find the volume of the oil displaced by the hydrometer, we subtract the volume of the stem (which is the cross-sectional area times the height of the stem above the surface). The cross-sectional area is given as 0.400 cm^2, and the height above the surface is 7.25 cm.

Once we have the volume of the oil displaced, we can use the formula for specific gravity: SG = (volume of oil displaced) / (volume of water displaced for SG = 1.00).

(c) To find the lowest specific gravity that can be measured with this hydrometer, we need to determine the maximum amount of liquid that the hydrometer can displace. This occurs when the entire volume of the hydrometer (bulb + stem) is submerged in the liquid.

The volume of the hydrometer is given as 8.8 cm^3. We can calculate the specific gravity by dividing this volume by the volume of water displaced for SG = 1.00.

By solving these calculations, we can obtain the answers to all three parts of the question.