Problem 55 - Fundamentals of Physics

Question

How much work is done by pressure in forcing \(1.4 \mathrm{~m}^{3}\) of water through a pipe having an internal diameter of \(13 \mathrm{~mm}\) if the difference in pressure at the two ends of the pipe is \(1.0 \mathrm{~atm}\) ?

Step-by-Step Solution

Answer

The work done is 141855 Joules.

Step 1: Convert Pressure to Pascals

First, convert the pressure from atm to Pascals. We know that 1 atm is equivalent to 101325 Pascals. Thus, the pressure difference \( \Delta P \) is \( 1.0 \text{ atm} = 101325 \text{ Pa} \).

Step 2: Calculate Cross-Sectional Area

Next, find the cross-sectional area \( A \) of the pipe with a diameter of 13 mm. Convert the diameter to meters (13 mm = 0.013 m) and use the area formula for a circle: \[ A = \pi \left( \frac{d}{2} \right)^2 = \pi \left( \frac{0.013}{2} \right)^2 \approx 1.327 \times 10^{-4} \text{ m}^2. \]

Step 3: Calculate Work Done

The work done by the pressure when moving the volume \( V \) of liquid is given by the formula \( W = \Delta P \times V \). We have \( \Delta P = 101325 \text{ Pa} \) from Step 1 and \( V = 1.4 \text{ m}^3 \). Thus, the work done is: \[ W = 101325 \times 1.4 = 141855 \text{ Joules}. \]

Key Concepts

Work Done by Pressure

In everyday life, we often hear about work being done, but in physics, it's a very specific concept. Here, work is done when a force moves an object over a distance. In the context of fluid mechanics, and more specifically this exercise, the pressure difference between two ends of a pipe does the work to move a fluid (like water) through that pipe.

The amount of work done by pressure in moving a volume of fluid can be calculated using the formula:

• \[ W = \Delta P \times V \]

where:

• \( W \) is the work done (in Joules),
• \( \Delta P \) is the pressure difference (in Pascals), and
• \( V \) is the volume of the fluid moved (in cubic meters).

To complete this calculation, it's crucial to ensure all units are consistent and correctly converted, particularly the pressure as illustrated further on.

Pressure Conversion

Pressure is a critical factor in work calculations and can be expressed in numerous units such as atmospheres (atm), Pascals (Pa), or bar. To simplify calculations and ensure accuracy, it's vital to convert all pressure measurements into Pascals, which is the standard SI unit for pressure.

One atmosphere is commonly known to be equivalent to 101325 Pascals. Thus, to convert atmospheric pressure to Pascals, simply multiply the number of atmospheres by 101325.

Here's the conversion process:

• Given pressure in atm: 1.0 atm
• Converted to Pascals:\[ 1.0 \, \text{atm} = 1.0 \times 101325 \, \text{Pa} = 101325 \, \text{Pa} \]

This conversion is a key step before applying any formulas related to pressure and makes integrating different equations much more straightforward.

Cross-Sectional Area Calculation

The cross-sectional area of a pipe determines how much fluid can flow through it at a given pressure difference. By calculating this area, we enable more precise predictions about fluid movement.

The cross-sectional area \( A \) of a circular pipe is determined using the formula for the area of a circle:

• \[ A = \pi \left( \frac{d}{2} \right)^2 \]

where:

• \( d \) is the diameter of the circle (in meters)

To convert the diameter from millimeters to meters, simply divide by 1000. For instance, a diameter of 13 mm translates to 0.013 meters. Therefore, the area can be computed as follows:

• \[ A = \pi \left( \frac{0.013}{2} \right)^2 \approx 1.327 \times 10^{-4} \, \text{m}^2 \]

This calculated area is then used to further determine flow-related properties and predictions within calculations involving fluid dynamics.