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 by pressure is 141855 Joules.

Step 1 - Convert Pressure from atm to Pascals

First, convert the pressure difference from atmospheres to pascals since SI units are required for the calculation. \[ 1 ext{ atm} = 101325 ext{ Pa} \] So, the pressure difference is \( 1.0 \text{ atm} = 101325 \text{ Pa} \).

Step 2 - Identify the Volume

The volume of water that is being forced through the pipe is already given as \( V = 1.4 \text{ m}^3 \).

Step 3 - Calculate Work Done by Pressure

The work done by the pressure in moving the water through the pipe is calculated using the formula \( W = P \times V \), where \( P \) is the pressure difference and \( V \) is the volume. Substituting the given values, we have: \[ W = 101325 \text{ Pa} \times 1.4 \text{ m}^3 = 141855 \text{ J} \] (Joules).

Step 4 - Review the Units and Concept

Since the internal diameter of the pipe does not affect the calculation of work done by pressure (which depends only on pressure difference and volume in this context), we have used only pressure and volume in the calculation. Make sure units are consistent and the process is understood.

Key Concepts

Understanding Pressure

Pressure is a key concept in physics, especially when calculating how forces are transmitted through fluids like water. In simpler terms, pressure is the force exerted by a substance per unit area on a surface. For example, in our exercise, pressure helps us understand how much work is needed to push water through a pipe.

Pressure is measured in various units, but the International System of Units (SI) uses Pascals (Pa). It's essential to convert any given pressure, like atmospheres in this case, into Pascals when doing calculations. This is because other units might make the calculations less intuitive and harder to compare. For example, the problem gives the pressure in atmospheres, which we convert to Pascals as follows:

• 1 atmosphere (atm) is 101325 Pascals (Pa).

By converting units carefully, we ensure all calculations are accurate. Remember, when the pressure difference is applied across a certain volume of liquid, it results in useful work being done, which we'll explore further.

Exploring Volume

Volume refers to the amount of three-dimensional space occupied by a substance or an object. In the context of liquids and gases, volume is often crucial in understanding how much of a substance is being dealt with. In our exercise, we have a volume of water being pushed through a pipe.

The volume of a substance is typically measured in cubic meters (m³) in SI units. This is particularly useful in calculations involving work because the volume directly influences how much work is done by pressure. For instance:

• 1.4 m³ of water is flowing through the pipe as stated in the exercise.

This large volume indicates how much space the water takes up as it travels through the pipe. When applying pressure to drive this volume, the result is a measurable work output, showing how crucial volume is in such calculations.

Importance of Joules in Measuring Work

When discussing work in physics, the term "joules" (J) often comes into play as the unit of measurement. Joules give us a way to quantify energy transfer or work done when a force moves an object over a distance. In our exercise, we use joules to understand the work done by pressure on the water.

Work done can be calculated by using the formula: \( W = P \times V \), where \( P \) is pressure and \( V \) is volume. Once we've performed the necessary calculations, we obtain the result in joules. In the previous problem:

• The work done was found to be 141855 J.

By using joules, we have a standard way to express the work done, making it easier to communicate and compare results. Remember, one joule is equivalent to one newton-meter, which links directly to how forces and distances relate in physics.