Problem 25 - Fundamentals of Physics

Question

In one observation, the column in a mercury barometer (as is shown in Fig. \(14-5 a\) ) has a measured height \(h\) of \(740.35 \mathrm{~mm}\). The temperature is \(-5.0^{\circ} \mathrm{C}\), at which temperature the density of mercury \(\rho\) is \(1.3608 \times 10^{4} \mathrm{~kg} / \mathrm{m}^{3} .\) The free-fall acceleration \(g\) at the site of the barom-eter is \(9.7835 \mathrm{~m} / \mathrm{s}^{2}\). What is the atmospheric pressure at that site in pascals and in torr (which is the common unit for barometer readings)?

Step-by-Step Solution

Answer

The atmospheric pressure is 9.8547×10⁴ Pa or 740.35 torr.

Step 1: Convert the height to meters

Since the measurement of the barometer's height is given in millimeters, we first convert it to meters because SI units are required for the calculation of pressure. To convert, use the formula: \(h = 740.35\, \text{mm} \times \frac{1\, \text{m}}{1000\, \text{mm}} = 0.74035\, \text{m}\).

Step 2: Use the hydrostatic pressure formula

To find the atmospheric pressure exerted by the column of mercury, we use the formula for hydrostatic pressure, which is given by: \[ P = \rho \cdot g \cdot h \]where \(\rho = 1.3608 \times 10^{4}\, \text{kg/m}^3\), \(g = 9.7835\, \text{m/s}^2\), and \(h = 0.74035\, \text{m}\).

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure is essential for understanding atmospheric pressure, especially when you use a liquid medium like mercury in barometers. It represents the pressure exerted by a fluid at rest due to the force of gravity. This pressure is calculated using the formula: \[ P = \rho \cdot g \cdot h \]where:

• \(P\) is the hydrostatic pressure.
• \(\rho\) represents the fluid density, in this case, it's mercury.
• \(g\) is the acceleration due to gravity.
• \(h\) is the height of the fluid column.

The hydrostatic pressure equation helps us determine how the height of a liquid column can reflect atmospheric pressure. In the context of a mercury barometer, the weight of the mercury column matches the atmospheric pressure at the top of the column.

Barometer

A barometer is a device used to measure atmospheric pressure. The pressure determines weather patterns and implications for altitude calculation. The mercury barometer is a traditional type composed of a column of mercury in a sealed tube. When atmospheric pressure rises, it pushes the mercury higher in the tube. Conversely, a decrease in atmospheric pressure allows the column to decline. This visible change in mercury level allows us to calculate the pressure by measuring the height of the column. Barometers are crucial for various fields:

• Weather forecasting relies on pressure readings to predict incoming weather changes.
• Aviation uses barometers for altitude measurements, crucial for flight safety.
• Scientific research utilizes precise pressure data for studies in physics and environmental sciences.

In this exercise, the measured height of the mercury column reflects the atmospheric pressure acting on it.

Mercury Density

Density plays a vital role in the functioning of a mercury barometer. It's defined as mass per unit volume and provides insight into a fluid's behavior under gravitational forces. The standard density of mercury is \(1.3608 \times 10^{4} \, \mathrm{kg/m}^{3}\) at \(-5.0 ^\circ \mathrm{C}\), the temperature at which this exercise takes place.The density directly influences the height of the mercury column. Denser liquids would support a shorter column of mercury for the same atmospheric pressure. Hence, knowing the density is crucial for calculating the hydrostatic pressure and, thus, atmospheric pressure accurately.If temperature changes occur, you will often need to consider potential changes in density, as it could affect the accuracy of the measurement. This comes especially into play in precise scientific conditions or locales with significant temperature fluctuations.

SI Units Conversion

Converting measurements to SI units is often necessary to ensure mathematical consistency and accuracy. In scientific contexts, the International System of Units (SI) provides a standard method of expressing the physical quantities used in mathematical calculations. In this exercise, the challenge was converting the height of the mercury column from millimeters to meters, a crucial step since the hydrostatic pressure formula requires measurements in meters. The conversion process is straightforward:\[ h = 740.35 \, \text{mm} \times \frac{1 \, \text{m}}{1000 \, \text{mm}} = 0.74035 \, \text{m} \]Using the SI units ensures all involved measurements—density, gravity, and height—remain standardized, leading to a more accurate calculation result. Converting so ensures consistency across scientific disciplines and aids in the efficient communication and comparison of results.