Question

A cylindrical glass tube \(12.7 \mathrm{~cm}\) in length is filled with mercury (density \(=13.6 \mathrm{~g} / \mathrm{mL}\) ). The mass of mercury needed to fill the tube is \(105.5 \mathrm{~g}\). Calculate the inner diameter of the tube (volume of a cylinder of radius \(r\) and length \(h\) is \(V=\pi r^{2} h\) ).

Short answer

The inner diameter of the tube is approximately 1.396 cm.

Step-by-step solution

Understand the problem

We need to find the inner diameter of a cylindrical tube given its length, the density of mercury, and the mass of mercury filling the tube. We will use the relationship between volume, density, and mass, as well as the formula for the volume of a cylinder.

Relate mass, volume, and density

We know that mass is equal to density times volume. Given the density of mercury is 13.6 g/mL, and the mass of mercury is 105.5 g, we can find the volume \( V \) of mercury that fills the tube: \( V = \frac{\text{mass}}{\text{density}} = \frac{105.5}{13.6} \). Compute \( V \) to get the volume in mL, which is equivalent to cm³.

Calculate the volume of mercury

Using the calculation from Step 2, \( V = \frac{105.5}{13.6} \approx 7.7574 \) cm³. This is the volume of the cylinder, which is filled with mercury.

Use the volume formula for a cylinder

The volume of a cylinder \( V \) is given by \( V = \pi r^2 h \), where \( h \) is the height (or length) and \( r \) is the radius. We know \( V = 7.7574 \) cm³ and \( h = 12.7 \) cm. Substitute these values into the formula: \( 7.7574 = \pi r^2 \times 12.7 \).

Solve for the radius \( r \)

Rearrange the equation from Step 4 to solve for \( r^2 \): \( r^2 = \frac{7.7574}{\pi \times 12.7} \). Calculate \( r^2 \) and then take the square root to find \( r \).

Calculate and interpret

After performing the calculation in Step 5, \( r \approx 0.698 \) cm. This is the radius of the tube. To find the diameter, use \( d = 2r \), so \( d \approx 2 \times 0.698 \approx 1.396 \) cm.

Key concepts

Density and Volume Relationship

Density is a fundamental concept that links mass and volume. It is defined as the mass of an object divided by its volume. The formula for density is: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] In our problem, density helps us understand how much mercury is needed to occupy a given volume. It's given that mercury has a density of 13.6 g/mL. This suggests that each milliliter of mercury has a mass of 13.6 grams. Understanding this relationship allows us to calculate how much space mercury takes up when we know its mass. By rearranging the formula to find volume, we have: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \] This computation is crucial for converting the mass of mercury into volume, which is essential for determining further dimensions of the cylinder.

Mass to Volume Conversion

To solve many physics problems, converting mass into volume is a key step. With the mass of mercury being 105.5 grams and its density 13.6 g/mL, we can calculate the volume of mercury needed to fill the cylindrical tube. Using the formula: \[ V = \frac{\text{Mass}}{\text{Density}} = \frac{105.5}{13.6} \] After the calculations, the volume (\( V \)) is approximately 7.7574 cm³. This number tells us how much space the mercury actually occupies in the tube. It's equivalent to the total internal volume of the cylindrical tube.

Calculation of Radius from Volume

Once we have the volume, calculating the radius of the cylinder becomes straightforward using the formula of cylinder volume: \[ V = \pi r^2 h \] Here, \( r \) is the radius, \( h \) is the height (or length in this problem), and \( V \) is the volume already known (approximately 7.7574 cm³). As the height is 12.7 cm, we rearrange the formula for the radius (\( r \)) as: \[ r^2 = \frac{V}{\pi h} \] Substituting the values gives: \[ r^2 = \frac{7.7574}{\pi \times 12.7} \] Solving this will provide \( r \), which is the measure we need for further calculations.

Inner Diameter Determination

Once the radius (\( r \)) of the cylinder is determined from our calculations, finding the diameter is next. The formula for the diameter is twice the radius, given as: \[ d = 2r \] For a cylinder with a radius \( r \approx 0.698 \text{ cm} \), the diameter would be: \[ d \approx 2 \times 0.698 \approx 1.396 \text{ cm} \] This diameter gives us the tube's inner circle span from one side to the other. Understanding how to calculate the inner diameter is crucial because it's necessary for determining the capacity and ensuring mercury fills the tube completely.