Problem 2 - College Physics

Question

A petite young woman distributes her \(500 \mathrm{~N}\) weight equally over the heels of her high-heeled shoes. Each heel has an area of \(0.750 \mathrm{~cm}^{2}\). (a) What pressure is exerted on the floor by each heel? (b) With the same pressure, how much weight could be supported by two flat- bottomed sandals, each of area \(200 \mathrm{~cm}^{2} ?\)

Step-by-Step Solution

Answer

(a) 3,333,333.33 Pa, (b) 133,333.33 N.

Step 1: Understand the Known Values

We know the weight of the woman is 500 N, and it's equally distributed over two heels. The area of each heel is given as 0.750 cm squared.

Step 2: Convert Units for Area to m²

First, convert the area from cm² to m², because the standard unit of pressure is Pascal (Pa), which is N/m². 1 cm² = 0.0001 m², so 0.750 cm² is 0.750 * 0.0001 = 0.000075 m².

Step 3: Calculate Force on Each Heel

Since the weight is equally distributed over two heels, each heel supports half the total weight. Thus, each heel supports 500 N / 2 = 250 N.

Step 4: Calculate Pressure Exerted by Each Heel

Pressure is calculated as force divided by the area over which the force is applied. Therefore, the pressure exerted by each heel is \( \frac{250 \text{ N}}{0.000075 \text{ m}^2} \).Calculate this to find the pressure: \(\frac{250}{0.000075} \approx 3,333,333.33 \text{ Pa}\).

Step 5: Understand Pressure for the Two Sandals

We will use the same pressure exerted by each heel to find out how much weight can be supported by the two sandals, each of area 200 cm².

Step 6: Convert Units for Sandal Area to m²

Convert the area of each sandal from cm² to m². 200 cm² = 200 * 0.0001 = 0.02 m².

Step 7: Calculate Total Area for Two Sandals

The total area for the two sandals is 2 * 0.02 m² = 0.04 m².

Step 8: Calculate Maximum Weight Supported by Sandals

Using the pressure of 3,333,333.33 Pa (from the heel calculation), apply the formula\( \text{Weight} = \text{Pressure} \times \text{Total Area} \).Thus, \( \text{Weight} = 3,333,333.33 \times 0.04 \) N.Calculate this to find the maximum weight:133,333.33 N.

Key Concepts

Pressure Calculation

To understand pressure calculation, let's explore how pressure is defined. Pressure is the amount of force applied over a particular area, and it's calculated using the formula:

• Pressure = Force / Area

The unit of pressure is the Pascal, symbolized as Pa. One Pascal corresponds to one Newton per square meter. In scenarios like the one with the high heels, we are curious about how much force is spread out over the small area of the heel. The smaller the area with the same force, the higher the pressure applied. In our exercise, each heel exerts a pressure of about 3,333,333.33 Pa, illustrating how concentrated force can amplify pressure significantly when distributed over a tiny area.

Unit Conversion

Unit conversion is a fundamental skill, especially in physics, where different systems of measurement are used. In our exercise, the area was initially given in square centimeters (cm²). However, pressure calculations require the area in square meters (m²) because pressure is expressed in Pascals (N/m²). This necessitates a conversion:

• 1 cm² = 0.0001 m²

Hence, converting 0.750 cm² to square meters becomes:

• 0.750 cm² = 0.000075 m²

Performing these conversions ensures accurate calculations, adhering to standard scientific units.

Force Distribution

Force distribution describes how a force is spread across different areas or points. In the given problem, the woman’s weight of 500 N is distributed evenly over two heels. This means each heel supports half of the total weight, calculated as:

• Weight per heel = Total Weight / Number of heels
• 250 N = 500 N / 2

Understanding this distribution is crucial because it affects the pressure exerted on the floor. With each heel bearing 250 N, this distributed force becomes the starting point for calculating pressure.

Area and Pressure Relationship

The relationship between area and pressure is inverse. That means when you apply the same force over an area, the smaller the area, the higher the pressure - and vice versa. This principle becomes clear through the heel and sandal scenario:

• Small heel area = High pressure
• Larger sandal area = Lower pressure for the same force

For instance, the sandals, covering more area, could support a substantial weight when maintaining the same pressure calculated for the heels. With each sandal having a larger area (200 cm²), converted into 0.02 m², and a total area for two sandals as 0.04 m², we assess that these sandals could support a heavier load. This illustrates how varying the area impacts the weight a surface can support under constant pressure.