91Ó°ÊÓ

Force is one of the fundamental concepts in physics, describing any interaction that, when unopposed, will change the motion of an object. In this particular exercise, force is represented by the stretching force applied to the cables. The force on the first cable is given as 270 newtons (N). For the second cable, we need to calculate the force so both cables experience the same stress. Understanding force is essential, as it helps us analyze how much effort is exerted on the cables, leading to either their elongation or compression depending on the direction of the force applied.
Force is often expressed in newtons (N), which is the standard international unit of force. To establish a common ground for comparison or for integration into the calculation of other physical properties like stress, identifying and calculating the force is crucial. In this exercise, we find that when the concept of stress is tied to force, knowing the applied force and the resulting stress can help us tackle forces acting on different structures, like cables.

The cross-sectional area of an object like a cable is crucial when calculating stress. It refers to the area of the particular section exposed when a solid is cut through along a plane. In this exercise, we analyze the cross-sectional area of two cables. The formula for the cross-sectional area of a circle is given by \( A = \pi r^2 \).
For the first cable, using a radius of \(3.5 \times 10^{-3} \) meters, we derive the cross-sectional area \( A_1 \approx 3.85 \times 10^{-5} \ \text{m}^2 \). The second cable, having a slightly larger radius of \(5.1 \times 10^{-3} \) meters, results in a larger cross-sectional area of \( A_2 \approx 8.17 \times 10^{-5} \ \text{m}^2 \).
Recognizing the significance of the cross-sectional area is key, as it directly influences the stress experienced by an object when a force is applied. A larger cross-sectional area, under constant force, tends to distribute the stress across a bigger surface, effectively reducing the stress experienced at any given point.

Stress is a measure of the internal forces acting within a deformable body, such as a cable, when an external force is applied. Mathematically, it is determined using the formula: \( \text{Stress} = \frac{F}{A} \), where \( F \) represents the force applied and \( A \) signifies the cross-sectional area.
In our given exercise, we calculated the stress on the first cable. With a force \( F_1 \) of 270 N and a cross-sectional area \( A_1 \approx 3.85 \times 10^{-5} \ \text{m}^2 \), the stress is\( 7.01 \times 10^6 \ \text{N/m}^2 \). This same stress value applies to the second cable due to the problem stating equal stress conditions.

• Calculate the cross-sectional areas.
• Use the stress formula.
• Maintain equal stress conditions.

Considering stress is paramount, as it tells us how much internal pressure the cable can endure before deforming or breaking. In applications, keeping stress within permissible limits prevents structural failures.

The radius of a cable or any circular object is the distance from the center of its cross-section to its edge. The given radii in this exercise, \(3.5 \times 10^{-3} \) m for the first cable and \(5.1 \times 10^{-3} \) m for the second, are vital for calculating areas and stress on each cable.
A small change in radius profoundly affects the cross-sectional area because the area formula involves squaring the radius. As a rule of thumb:

• Smaller radius leads to a smaller area.
• Larger radius results in a larger area.

Understanding this subtlety clarifies why, with equal stress, the force requirement differs. For example, doubling the radius results in a fourfold increase in area since area is proportional to the square of the radius. Hence, in engineering and physics applications, precise measurement and understanding of radius are critical.