Question

How much work is done by pressure in forcing \(1.4 \mathrm{~m}^{3}\) of water through a \(13-\mathrm{mm}\) intemal diameter pipe if the difference in pressure at the two ends of the pipe is \(1.2 \mathrm{~atm} ?\)

Short answer

The work done by the pressure in forcing 1.4 m^3 of water through a 13-mm internal diameter pipe with a pressure difference of 1.2 atm is 170226 Joules.

Step-by-step solution

Convert pressure difference from atm to Pa

The pressure difference is given in atm. To convert it to pascal, multiply by 101325 (since 1 atm = 101325 Pa). So, pressure difference = \(1.2 * 101325 = 121590\) Pa.

Convert volume from liters to cubic meters

The volume is given in cubic meters, so it can be used as it is.

Calculate the Work

The work done by the pressure \(W\) can be calculated using the formula \(W = P \Delta V\), where \(P\) is the pressure difference and \(\Delta V\) is the volume of the fluid displaced. So, plug in the values: \(W = 1.2 * 101325 * 1.4 = 170226\) Joules.

Key concepts

Pressure to Pascal Conversion

Understanding pressure and its unit conversions is crucial in physics, especially when calculating work done by pressure. Pressure, which is defined as the force exerted per unit area, is commonly measured in atmospheres (atm) or Pascals (Pa). The atmosphere is a unit that is based on the average atmospheric pressure at sea level. To convert pressure from atmospheres to Pascals, we use the equivalence: 1 atm = 101,325 Pa.

As seen in the exercise provided, a pressure difference of 1.2 atm was converted to Pascals by multiplying it by this factor, resulting in a value of 121,590 Pa. Performing such conversions accurately is essential for ensuring correct calculations in problems involving work done by pressure in various applications, from plumbing to meteorology.

Volume Conversion

Volume conversion is a fundamental aspect of solving problems across many areas of science and engineering. Volume can be expressed in multiple units, such as liters, cubic meters (m³), or cubic centimeters (cm³), depending on the context. In physics problems, such as the one dealing with work done by pressure, it is often necessary to use the SI unit of volume, which is the cubic meter.

In the example problem, the volume of water being moved through the pipe is already given in cubic meters, so no conversion was necessary. However, if the volume were provided in liters, one would need to convert it to cubic meters by using the relation that 1 cubic meter equals 1000 liters. Being fluent in these conversions is important for accurate problem-solving and calculations in physics.

Calculating Work in Physics

The concept of work in the realm of physics is distinct and more quantifiable compared to its colloquial use. In physics, work is defined when a force causes a displacement. To calculate the work done by a constant force, one must know the magnitude of the force, the displacement caused, and the angle between the force and the direction of displacement. Mathematically, this is expressed as:

Work Done (W) = Force (F) * Displacement (d) * cos(θ)

However, when dealing with fluids, work is commonly calculated from pressure and volume change rather than force and displacement. Using the formula:

Work Done (W) = Pressure Difference (P) * Volume Change (ΔV)

In the exercise, the work done to move water through a pipe is a direct product of the pressure difference and the volume of water. By substituting the provided values, we find the work done by the pressure to be 170,226 Joules. This context-specific calculation of work helps in analyzing and designing systems involving fluid movement, like piping and hydraulic systems.