Problem 13 - Chemistry 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 \(0.50\) in. \(^{2}\), calculate the pressure exerted on the underlying surface in kilopascals.

Step-by-Step Solution

Answer

The pressure exerted on the underlying surface by the woman wearing high-heeled shoes is approximately \(1794.215 \ \text{kPa}\).

Step 1: Convert weight to force

First, we need to convert the woman's weight (130 lb) to force using the acceleration due to gravity (9.81 m/s^2). 1 lb is approximately equal to 4.448 N, and 1 inch is equal to 0.0254 meters. Let's convert her weight to N and the area to square meters. Weight (N) = 130 lb × 4.448 (N/lb) Weight (N) = 578.24 N

Step 2: Calculate the pressure

Now that we have the weight in newtons, we can calculate the pressure exerted on the heel area. First, let's convert the given area to square meters: Area = 0.50 in^2 × (0.0254 m/in)^2 Area ≈ 0.000322259 m^2 The formula for pressure is: Pressure = Force / Area So, using the weight as force and the heel area in square meters: Pressure = 578.24 N / 0.000322259 m^2 Pressure ≈ 1794215.12 Pa

Step 3: Convert pressure to kilopascals

Finally, we need to convert the pressure from pascals (Pa) to kilopascals (kPa). 1 kPa = 1000 Pa Pressure (kPa) = Pressure (Pa) / 1000 Pressure (kPa) = 1794215.12 Pa / 1000 Pressure (kPa) ≈ 1794.215 kPa Therefore, the pressure exerted on the underlying surface is approximately 1794.215 kPa.

Key Concepts

Force Conversion: Understanding the Basics

To understand how to convert force, it's important to start with the concept that force is a push or pull on an object. In this exercise, we needed to convert weight from pounds (lbs) to newtons (N), because newtons are the standard unit of force in the International System of Units (SI).

Here’s a simple way to do it:

• Know that 1 pound is approximately equal to 4.448 newtons. This is because newton is a unit that combines mass (in kilograms) and acceleration (due to gravity, which is approximately 9.81 m/s² on Earth).
• By multiplying the weight in pounds by 4.448, we convert the weight to a force measured in newtons.

In the exercise, we calculated a weight of 130 lbs, leading to a force conversion to 578.24 N. This conversion is essential for calculating pressure in SI units.

Area Conversion: From Inches to Meters

Another important conversion in our task was changing area from square inches to square meters. This is necessary because we are using SI units for calculations, which rely on meters for measuring length.

To convert:

• Start by understanding that 1 inch is equal to 0.0254 meters.
• When converting an area, which is a squared measurement, you square the conversion factor. Thus, 1 in² is equal to (0.0254 m/in)² square meters.
• In this case, an area of 0.50 in² converts to approximately 0.000322259 m².

Only after this area conversion could we accurately use it alongside force to determine pressure in SI units.

Kilopascals: A Key Pressure Metric

Pressure is often expressed in pascals (Pa) or kilopascals (kPa) in scientific contexts. To understand kilopascals, it's helpful to know that pressure is the force applied per unit area.

Steps to convert pressure to kilopascals:

• First, calculate the pressure using the formula: Pressure = Force / Area.
• In our example, the force was 578.24 N, and the area was approximately 0.000322259 m², leading to a pressure calculation of about 1794215.12 Pa.
• Since 1 kilopascal is equivalent to 1000 pascals, you convert by dividing by 1000.
• Thus, 1794215.12 Pa becomes approximately 1794.215 kPa.

This conversion to kilopascals measures how much force is exerted over a certain area, allowing for a comprehensible, standardized way to convey pressure values in the SI unit system.