Problem 2 - College Physics

Question

Suppose you use the tip of one finger to support a 1.0-kg object. If your finger has a diameter of \(2.0 \mathrm{~cm}\), what is the stress on your finger?

Step-by-Step Solution

Answer

The stress on the finger is approximately 31210 N/m².

Step 1: Determine the Force

First, identify the force acting on the finger. The weight of the object, which is supported by the finger, acts as the force. Weight can be calculated using the formula: \( F = m \cdot g \), where \( m = 1.0 \text{ kg} \) is the mass of the object and \( g = 9.8 \text{ m/s}^2 \) is the acceleration due to gravity. Therefore, \( F = 1.0 \times 9.8 = 9.8 \text{ N} \).

Step 2: Calculate the Area

Now, calculate the cross-sectional area of the finger which is circular. The area, \( A \), can be calculated using the formula for the area of a circle: \( A = \pi \cdot r^2 \). First, find the radius, \( r \), which is half the diameter: \( r = \frac{2.0 \text{ cm}}{2} = 1.0 \text{ cm} = 0.01 \text{ m} \). Thus, \( A = \pi \times (0.01)^2 \approx 3.14 \times 10^{-4} \text{ m}^2 \).

Step 3: Compute the Stress

With the force and area known, stress is calculated by dividing the force by the area: \( \text{Stress} = \frac{F}{A} = \frac{9.8 \text{ N}}{3.14 \times 10^{-4} \text{ m}^2} \approx 31210 \text{ N/m}^2 \).

Key Concepts

Force Calculation

When calculating force, it's important to remember that force is essentially the weight of the object you're dealing with. In physics, force is a vector quantity—it has both magnitude and direction.
To find the force exerted by an object, we use the formula:

• \( F = m \cdot g \)

Here, \( F \) represents force, \( m \) is the mass of the object and \( g \) is the gravitational acceleration, typically \( 9.8 \text{ m/s}^2 \) on Earth.
In this context, the force applied by a 1.0 kg object would be its weight, calculated by multiplying its mass by Earth's gravitational pull (9.8 m/s²). So, for our problem:

• \( F = 1.0 \text{ kg} \times 9.8 \text{ m/s}^2 = 9.8 \text{ N} \)

The unit "N" stands for Newton, which is the standard unit of force.

Stress Analysis

Stress refers to the internal distribution of forces within a solid object. It's an indicator of how much force is experienced per unit area within a material. Stress is pivotal in determining the strength and resilience of materials under load.
Let's delve into the formula for calculating stress:

• \( \text{Stress} = \frac{F}{A} \)

Here, \( F \) is the force applied to the surface and \( A \) is the area over which the force is distributed.
In our exercise, we calculate stress by dividing the force, which is 9.8 N, by the cross-sectional area of the finger supporting the weight. Therefore:

• \( \text{Stress} = \frac{9.8 \text{ N}}{3.14 \times 10^{-4} \text{ m}^2} \approx 31210 \text{ N/m}^2 \)

This tells us the intensity of the force distributed over the area. Here, the stress experienced by the finger is expressed in pascals (Pa), which is the standard unit for stress in the International System of Units.

Cross-Sectional Area Calculation

The cross-sectional area is a crucial factor in understanding the distribution of force in materials. It represents the area of the cut surface when an object is sliced across a specific plane—picture cutting a cucumber into slices.
In cylindrical objects or fingers, calculating the cross-sectional area requires the formula for the area of a circle:

• \( A = \pi \times r^2 \)

To find the radius, halve the diameter. For our scenario:

• \( r = \frac{2.0 \text{ cm}}{2} = 1.0 \text{ cm} = 0.01 \text{ m} \)

Then, you substitute this radius into the formula to find the area:

• \( A = \pi \times (0.01)^2 \approx 3.14 \times 10^{-4} \text{ m}^2 \)

Understanding the concept of cross-sectional area is key in fields like engineering and physics, where it helps predict how materials will behave under different forces.