Question

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

Short answer

The pressure exerted by the woman wearing high-heeled shoes on the underlying surface is (a) 260 psi, (b) approximately 1793 kPa, and (c) approximately 17.7 atm.

Step-by-step solution

Find Pressure in Pounds per Square Inch (psi)

First, we'll find the pressure in the unit of pounds per square inch (psi). To do so, we use the formula: Pressure = Force / Area Given the woman's weight is 130 pounds, which is the force, and the heel area is 0.50 square inches, we can plug in these values: Pressure (psi) = Force (lbs) / Area (in²) = 130 lbs / 0.50 in² Now, we compute the pressure in psi.

Calculate Pressure in psi

Pressure (psi) = 130 lbs / 0.50 in² = 260 psi So, the pressure exerted on the underlying surface in pounds per square inch is 260 psi.

Convert Pressure to Kilopascals (kPa)

To convert the pressure from psi to kilopascals (kPa), we can use the conversion factor: 1 psi ≈ 6.895 kPa So, to convert 260 psi to kPa, we can simply multiply the pressure value by the conversion factor: Pressure (kPa) = 260 psi * 6.895 kPa/psi Now, we calculate the resulting pressure in kPa.

Calculate Pressure in kPa

Pressure (kPa) = 260 psi * 6.895 kPa/psi ≈ 1793 kPa The pressure exerted on the underlying surface is approximately 1793 kPa.

Convert Pressure to Atmospheres (atm)

Finally, we will convert the pressure from psi to atmospheres (atm). To do this, we'll use the conversion factor: 1 psi ≈ 0.06805 atm So, to convert 260 psi to atmospheres, we'll multiply the pressure value by the conversion factor: Pressure (atm) = 260 psi * 0.06805 atm/psi Now, we compute the pressure in atmospheres.

Calculate Pressure in atm

Pressure (atm) = 260 psi * 0.06805 atm/psi ≈ 17.7 atm The pressure exerted on the underlying surface is approximately 17.7 atmospheres. In summary, the pressure exerted by the woman wearing high-heeled shoes on the underlying surface is (a) 260 psi, (b) approximately 1793 kPa, and (c) approximately 17.7 atm.

Key concepts

Pounds per Square Inch

Pounds per Square Inch (psi) is a unit of pressure commonly used in the United States. Pressure, in this context, represents the amount of force exerted on a given area.

To calculate the pressure in psi, we follow the formula:

$\text{Pressure}=\frac{\text{Force}}{\text{Area}}$.

Here, the force is 130 pounds which is the weight of the woman, and the area is 0.50 square inches, which is the area of the heel.

Inserting these into the formula, you get:

$\text{Pressure (psi)}=\frac{130}{0.50}=260\text{psi}$.

This calculation shows how much pressure the woman exerts on the ground each time she steps with her heel. Using psi is helpful in scenarios where force is distributed over small areas, such as a heel or a tire.

Kilopascals

Kilopascals (kPa) are a metric unit of pressure more commonly used internationally. One key advantage of this unit is its applicability in various scientific and engineering contexts.

To convert psi to kPa, we use the conversion factor where 1 psi is approximately 6.895 kPa.

For a pressure of 260 psi, the conversion is straightforward:

$\text{Pressure (kPa)}=260\text{psi}\times 6.895\text{kPa/psi}=1793\text{kPa}$.

This result tells us that the high pressure exerted by this small heel is quite substantial when measured in kilopascals. Understanding this conversion is essential for international contexts where kilopascals are preferred over psi.

Atmospheres

Atmospheres (atm) are an older unit of pressure still used in some scientific settings. An atmosphere represents the typical pressure at sea level on Earth, making it a useful natural reference.

To convert the pressure from psi to atmospheres, we consider that 1 psi is roughly 0.06805 atm.

So, for 260 psi, the conversion step would be:

$\text{Pressure (atm)}=260\text{psi}\times 0.06805\text{atm/psi}=17.7\text{atm}$.

This indicates the heel's pressure is 17.7 times greater than the pressure exerted by the Earth's atmosphere at sea level, highlighting the significant force involved. Each of these pressure units gives a different perspective on the magnitude of pressure the high heel imposes.