Problem 13 - Chemistry: The Central Science

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 \(0.50 \mathrm{in} .^{2}, \mathrm{cal}-\) culate the pressure exerted on the underlying surface in kilopascals.

Step-by-Step Solution

Answer

To find the pressure exerted on the underlying surface in kilopascals, first convert the woman's weight to newtons: \(Weight \ (Newtons) = 130 \times 4.44822\). Then, convert the heel area to square meters: \(Area = 0.50 \times 0.00064516\). Next, calculate the pressure in pascals: \(Pressure \ (Pascals) = \frac{Weight \ (Newtons)}{Area \ (m^{2})}\). Finally, convert the pressure to kilopascals: \(Pressure \ (Kilopascals) = \frac{Pressure \ (Pascals)}{1000}\).

Step 1: Convert weight from pounds to newtons

To convert the woman's weight from pounds to newtons, we can use the conversion factor 1 pound = 4.44822 newtons. So, her weight in newtons is: \(Weight \ (Newtons) = 130 \times 4.44822\)

Step 2: Calculate the pressure exerted in pascals

To find the pressure exerted on the surface in pascals, we can use the formula for pressure: \(Pressure = \frac{Force}{Area}\) In this case, the pressure is the force exerted by the woman's weight, which we calculated in step 1, divided by the area of the heel 0.50 in², which we need to convert to m². There are 0.00064516 m² in 1 in², so: \(Area = 0.50 \times 0.00064516\) Now, we can find the pressure in pascals by dividing the weight in newtons by the area in m²: \(Pressure \ (Pascals) = \frac{Weight \ (Newtons)}{Area \ (m^{2})}\)

Step 3: Convert pressure from pascals to kilopascals

To convert the pressure from pascals to kilopascals, we can use the conversion factor 1 kilopascal = 1000 pascals. So, the pressure in kilopascals is: \(Pressure \ (Kilopascals) = \frac{Pressure \ (Pascals)}{1000}\) Now, perform the calculations and find the pressure exerted on the underlying surface in kilopascals.

Key Concepts

Unit Conversion

When solving problems involving pressure, one initial step often involves converting units to ensure consistency. In the exercise, we started by converting the woman's weight from pounds to newtons, a common requirement in physics. Understanding unit conversion is crucial because it allows us to use the same unit types across a calculation.

To convert pounds to newtons, we utilize the conversion factor where 1 pound is equal to approximately 4.44822 newtons. This factor helps us express the weight, which is a force, in terms of newtons, the standard unit of force in the International System of Units (SI).
For example:

• Weight in newtons = 130 pounds \( 130 \times 4.44822 = 578.2686 \text{ newtons} \)

It's also important to convert the area from square inches to square meters because we need compatible units for calculating pressure using pascals or kilopascals. Remember, 1 square inch is 0.00064516 square meters:

Pressure Formula

Once the units are properly converted, calculating pressure involves using the fundamental formula: pressure is equal to force divided by area. This is succinctly expressed as:\[Pressure = \frac{Force}{Area}\]Understanding this formula helps us determine how much force is exerted over a specific area, and it is essential in various fields, from engineering to everyday situations like wearing different types of shoes.

In our scenario, the force equates to the woman's weight in newtons, and the area corresponds to the sole of her heel in square meters:

• Converted Area = 0.50 in² \(0.50 \times 0.00064516 = 0.00032258 \text{ m}^2 \)

By plugging the force in newtons and the converted area into the formula, we are able to find pressure in pascals:

• Pressure (Pascals) = \(\frac{578.2686 \, ext{newtons}}{0.00032258 \, ext{m}^2} = 1792.63 \, ext{pascals}\)

Force and Area Relationship

The relationship between force and area is pivotal in understanding pressure. Essentially, pressure rises when a certain force is applied over a smaller area. This idea explains why high heels exert more pressure on the ground than a flat shoe does, given that the force (i.e., body weight) is concentrated over a much smaller surface.

The concept can be understood with a few simple ideas:

• **Larger Area**: Force distributed over a larger area results in lower pressure.
• **Smaller Area**: With the same force concentrated on a smaller area, pressure increases.

In our example, the entire body weight focuses on the heel's small area, and this concentration magnifies the pressure exerted. After calculating the pressure in pascals, converting it to kilopascals involves dividing by 1,000, since 1 kilopascal equals 1,000 pascals:

• Pressure (Kilopascals) = \(\frac{1792.63}{1000} = 1.79263 \, ext{kPa} \)

Observing the outcomes in different shoes or surfaces vividly illustrates the impact of this force and area relationship on pressure.