Problem 1 - Applied Physics

Question

A packing crate \(2.50 \mathrm{~m} \times 0.80 \mathrm{~m} \times 0.45 \mathrm{~m}\) weighs \(1.41 \times 10^{5} \mathrm{~N}\). Find the stress (in kPa) exerted by the crate on the floor in each of its three possible positions.

Step-by-Step Solution

Answer

Stress in kPa: 70.5, 125.3, 391.7 for Positions 1, 2, and 3, respectively.

Step 1: Determine Area with Dimensions

To calculate stress, we need the area of contact, determined by the dimensions of the crate. Consider each position: - Position 1: Use the dimensions 2.50 m and 0.80 m. - Position 2: Use the dimensions 2.50 m and 0.45 m. - Position 3: Use the dimensions 0.80 m and 0.45 m.

Step 2: Calculate Contact Area for Each Position

Now, calculate the contact area for each of the three positions: - Position 1: Area = 2.50 m × 0.80 m = 2.00 m². - Position 2: Area = 2.50 m × 0.45 m = 1.125 m². - Position 3: Area = 0.80 m × 0.45 m = 0.36 m².

Step 3: Calculate Stress Using The Formula

Stress is calculated using the formula Stress = Force/Area. Since force (weight) is given as 141,000 N, plug the calculated areas into the formula for each respective position: - Position 1: Stress = 141,000 N / 2.00 m² = 70,500 N/m² - Position 2: Stress = 141,000 N / 1.125 m² = 125,333 N/m² - Position 3: Stress = 141,000 N / 0.36 m² = 391,667 N/m².

Step 4: Convert Stress to Kilopascals (kPa)

To convert stress from N/m² to kPa, remember that 1 kPa = 1,000 N/m². Thus, divide each stress value by 1,000: - Position 1: Stress = 70,500 N/m² ÷ 1,000 = 70.5 kPa - Position 2: Stress = 125,333 N/m² ÷ 1,000 = 125.3 kPa - Position 3: Stress = 391,667 N/m² ÷ 1,000 = 391.7 kPa.

Key Concepts

Force and Pressure

To grasp the concept of stress, it's important to understand what force and pressure mean. **Force** is a push or pull that can change the motion of an object. It is measured in newtons (N). In this problem, the weight of the crate, given as a force, is 141,000 N.

**Pressure** is when a force is distributed over an area. The formula for pressure or stress is:\[\text{Pressure (Stress)} = \frac{\text{Force}}{\text{Area}}\]This equation tells us that the smaller the area, the greater the stress if the force remains constant. In practical terms, think about how wearing high heels exerts more pressure on the ground compared to flat shoes because the force is the same, but the area is smaller. This idea is central to the exercise, as the same crate exerts different pressures when it rests on different surfaces.

Contact Area Calculation

Contact area plays a crucial role in calculating stress. It's all about the surface on which the object exerts force. For the given crate, different sides become the contact area depending on its position. The key is to calculate this area accurately for each possible position.
**Why contact area is important:**

• A larger surface area reduces the stress because the force spreads over a bigger space.
• A smaller surface area increases the stress, concentrating the force.

In each position the crate can be oriented, the contact area differs and thus affects the stress:

• Position 1: Contact area is calculated as 2.50 m by 0.80 m, resulting in 2.00 m².
• Position 2: Uses dimensions of 2.50 m by 0.45 m, providing an area of 1.125 m².
• Position 3: Involves dimensions 0.80 m by 0.45 m, which gives an area of 0.36 m².

Once these areas are known, we can determine the stress for each position of the crate.

Unit Conversion

Working with different units can be confusing, but it's crucial for precision. In this exercise, stress initially is calculated in newtons per square meter (N/m²), which is more commonly known as pascals (Pa).
**Converting to a Practical Unit:**
Often, we use kilopascals (kPa) for ease, especially with large values. 1 kPa = 1,000 N/m². This conversion simplifies larger numbers and makes them easier to interpret. For instance:

• Position 1 stress: 70,500 N/m² converts to 70.5 kPa by dividing by 1,000.
• Position 2: 125,333 N/m² becomes 125.3 kPa when divided.
• Position 3 goes from 391,667 N/m² to 391.7 kPa.

These conversions from N/m² to kPa make the stress values more understandable and are a common practice in scientific calculations.