Problem 32 - College Physics

Question

Here is a demonstration Pascal used to show the importance of a fluid's pressure on the fluid's depth (vFig. 9.36): An oak barrel with a lid of area \(0.20 \mathrm{~m}^{2}\) is filled with water. A long, thin tube of cross- sectional area \(5.0 \times 10^{-5} \mathrm{~m}^{2}\) is inserted into a hole at the center of the lid, and water is poured into the tube. When the water reaches \(12 \mathrm{~m}\) high, the barrel bursts. (a) What was the weight of the water in the tube? (b) What was the pressure of the water on the lid of the barrel? (c) What was the net force on the lid due to the water pressure?

Step-by-Step Solution

Answer

(a) 5.88 N; (b) 117600 Pa; (c) 23520 N.

Step 1: Calculate the Volume of Water in the Tube

The volume of water in the tube can be calculated using the formula for volume of a cylinder: \( V = A \cdot h \). Here, the cross-sectional area \( A = 5.0 \times 10^{-5} \text{ m}^2 \) and the height \( h = 12 \text{ m} \). Plugging in these values, we find \( V = 5.0 \times 10^{-5} \text{ m}^2 \times 12 \text{ m} = 6.0 \times 10^{-4} \text{ m}^3 \).

Step 2: Calculate the Weight of Water in the Tube

The weight of the water can be found using the density \( \rho \) of water and gravitational acceleration \( g \). The mass is \( m = \rho V \) where \( \rho = 1000 \text{ kg/m}^3 \). Therefore, \( m = 1000 \text{ kg/m}^3 \times 6.0 \times 10^{-4} \text{ m}^3 = 0.6 \text{ kg} \). The weight \( W \) is then \( W = mg = 0.6 \text{ kg} \times 9.8 \text{ m/s}^2 = 5.88 \text{ N} \).

Step 3: Calculate the Pressure on the Lid

The pressure exerted by the water on the lid is given by \( P = \rho gh \). Substituting in \( \rho = 1000 \text{ kg/m}^3 \), \( g = 9.8 \text{ m/s}^2 \), and \( h = 12 \text{ m} \), we get \( P = 1000 \times 9.8 \times 12 = 117600 \text{ Pa} \).

Step 4: Calculate the Net Force on the Lid

The net force exerted by the water pressure on the lid can be calculated using \( F = PA \), where \( P = 117600 \text{ Pa} \) and \( A = 0.20 \text{ m}^2 \). Thus, \( F = 117600 \times 0.20 = 23520 \text{ N} \).

Key Concepts

Fluid Pressure

Fluid pressure is a fascinating concept that helps us understand how pressure behaves in liquids. It is primarily influenced by the depth of the fluid and the density of the liquid itself. When water is poured into a container of any shape, pressure increases with depth. This happens because of the weight of the fluid above pressing down on the fluid below.

In the context of Pascal's experiment with the oak barrel, the height of water in the tube created additional pressure on the water already in the barrel. This pressure is equal in all directions, which is why it can exert a force that could potentially burst the barrel.

This increase in pressure with depth is mathematically expressed by the formula \[ P = \rho gh \]Where:

• \( P \) is the pressure,
• \( \rho \) is the fluid density,
• \( g \) is the acceleration due to gravity, and
• \( h \) is the height of the fluid column.

As seen in Pascal's law, fluid pressure is a key component demonstrating the behavior of liquids under pressure.

Hydrostatics

Hydrostatics, a branch of fluid mechanics, deals with fluids at rest. It's critical for understanding how liquids like water behave under various pressures and forces without involving fluid motion.

In Pascal's demonstration, both the liquid in the barrel and the tube were stationary, providing insights into hydrostatics. The thin tube adds vertical pressure due to the height of the water column, which results in significant force on the barrel's lid.

This principle shows applications in everyday life, such as in dams, where the height of the water behind the dam plays a crucial role in producing pressure that the structure must withstand. Under Hydrostatic principles, knowing how forces act in still water can lead to better designs for containers and barriers that need to hold a certain amount of pressure.

Force Calculation

Force calculation is pivotal to understanding how pressure translates to physical interaction. In the case of the oak barrel, once the fluid pressure is known, we can determine the force that caused it to burst.

This is done using the formula: \[ F = PA \] Where:

• \( F \) is the force,
• \( P \) is the pressure calculated from the fluid's density and height,
• \( A \) is the area over which the force is acting.

Once we substitute the pressure and area, the resulting force indicates the total push against the lid due to the water. Understanding such computations helps students see the importance of pressure in causing structural changes, like the bursting of the barrel in this experiment.

Water Density

Water, a common fluid we encounter, has a density usually measured at about \(1000\ \text{kg/m}^3\). Density is a property that describes how much mass is packed into a given volume. It's instrumental in determining how fluids exert pressure.

In Pascal's experiment, the density of water was crucial for calculating how much pressure the water column in the tube exerted on the barrel. Knowing this density allows us to determine the weight of the water contained in the tube, as weight depends on both mass and gravity.

The formula used is:\[ m = \rho V \]Where:

• \( m \) is the mass,
• \( \rho \) is the density,
• \( V \) is the volume of the water.

Density ties directly into these calculations, demonstrating its importance in scientific experiments and real-world applications.