Question

Suppose that a woman weighing \(130 \mathrm{lb}\) and wearing highheeled shoes momentarily places all her weight on the heel of one foot. If the area of the heel is 0.50 in. \(^{2}\), calculate the pressure exerted on the underlying surface in (a) kilopascals, (b) atmospheres, and (c) pounds per square inch.

Short answer

The pressure exerted on the underlying surface by the woman's heel is approximately 260 psi, 1790.6 kPa, and 17.69 atm.

Step-by-step solution

Finding Pressure in Pounds per Square Inch

First, we will find the pressure exerted by the heel in pounds per square inch (psi). All our given values are already in the right units (lbs for weight and in² for area). Given: Weight (force) F = 130 lb Area A = 0.50 in² Calculate pressure P = F / A P (in psi) = (130 lb) / (0.50 in²) P (in psi) ≈ 260 psi

Conversion Factors

We need to convert the pressure from psi to kilopascals (kPa) and atmospheres (atm). The conversion factors are as follows: 1 psi = 6.89476 kPa 1 atm = 101.325 kPa 1 psi = 0.068046 atm (calculated by dividing 6.89476 kPa by 101.325 kPa)

Finding Pressure in Kilopascals

Now we will convert the pressure we found in psi to kilopascals (kPa) using the conversion factor: P (in kPa) = P (in psi) × 6.89476 kPa/psi P (in kPa) = 260 psi × 6.89476 kPa/psi P (in kPa) ≈ 1790.6 kPa

Finding Pressure in Atmospheres

Finally, we will convert the pressure we found in psi to atmospheres (atm) using the conversion factor: P (in atm) = P (in psi) × 0.068046 atm/psi P (in atm) = 260 psi × 0.068046 atm/psi P (in atm) ≈ 17.69 atm So, the pressure exerted on the underlying surface by the woman's heel is approximately 260 psi, 1790.6 kPa, and 17.69 atm.

Key concepts

Conversion Factors

Understanding conversion factors is crucial when dealing with different units of measurement. A conversion factor is a number used to change one set of units to another, by multiplying or dividing. In the context of pressure, we often need to convert between units like pounds per square inch (psi), kilopascals (kPa), and atmospheres (atm) to communicate measurements consistently and accurately.

For example, to convert psi to kPa, you would use the conversion factor of 1 psi equals approximately 6.89476 kPa. These factors are derived from the definitions of the units in terms of the base SI or metric units. Thus, knowing the correct conversion factor allows for quick and easy calculations across different measurement systems.

Kilopascals

The kilopascal (kPa) is a unit of pressure in the International System of Units (SI). One kilopascal equals 1,000 pascals and is based on the pascal, which is defined as one newton per square meter. The prefix 'kilo-' simply means 'thousand', so a kilopascal is one thousand pascals.

Common Uses

Kilopascals are commonly used in scientific contexts, as well as in industries like automotive to measure tire pressure, or meteorology for atmospheric pressure readings. When converting from the more commonly used psi, remember that it is a straightforward multiplication using the conversion factor. This importance in various fields emphasizes why understanding how to calculate and convert to kilopascals is a valuable skill.

Atmospheres

An atmosphere (atm) is another unit for measuring pressure. It is based on the average atmospheric pressure at sea level on Earth. One atmosphere is defined as the pressure that can support a column of mercury 760 millimeters high in a mercury barometer at sea level and at a temperature of 0 degrees Celsius.

Relation to Other Units

One atmosphere equals 101.325 kPa, which is derived from precise scientific measurement of air pressure. In practical calculations, we may use the conversion factor of 1 psi being equal to 0.068046 atm to change psi into atmospheres or vice versa. These figures are immensely important when working with systems where the atmospheric pressure reference is the standard, such as in scuba diving and aviation.

Pounds per Square Inch

Pounds per square inch, or psi, is a unit of pressure widely used in the United States and some other parts of the world. It reflects the force in pounds exerted on one square inch of area.

Calculating Pressure in PSI

In the given problem, we calculated the pressure exerted by the heel by dividing the weight, in pounds, by the area of the heel in square inches. This calculation gives us a direct reading in psi, a unit that is very intuitive for day-to-day use in various applications like monitoring car tire pressure or the pressure in hydraulic systems.

Being able to switch between psi and metric units ensures that pressures can be compared and understood in multiple contexts, expanding the understanding and communication of pressure across different industries and countries.