Question

$\mathrm{A}$ and $\mathrm{B}$ are two wires. The radius of $\mathrm{A}$ is twice that of $\mathrm{B}$. They are stretched by the same load. Then what is the stress on $\mathrm{B}$ ? (A) Equal to that on $\mathrm{A}$ (B) Four times that on $\mathrm{A}$ (C) Two times that on $\mathrm{A}$ (D) Half that on $\mathrm{A}$

Short answer

The stress on B is 4 times the stress on A. The correct answer is (B) Four times that on A.

Step-by-step solution

Recall the formula for stress

To find the stress on the two wires, we need to recall the formula for stress. Stress is the force per unit area, which can be represented as: $$\text{Stress}=\frac{\text{Force}}{\text{Area}}$$

Given a relationship between the radii of A and B

We are given that the radius of wire A is twice the radius of wire B. Let the radius of wire B be r. Then, the radius of wire A will be 2r.

Find the cross-sectional area of A and B

To find the stress, we need the cross-sectional area of the wires. The cross-sectional area of a cylinder (in this case, our wires) is given by the formula: $$\text{Area}=\pi r^2$$ For wire A (radius = 2r): $$\text{Area of A}=\pi (2r{)}^2=4\pi r^2$$ For wire B (radius = r): $$\text{Area of B}=\pi r^2$$

Calculate the stress on A and B using the given formula

We know that both wires have the same load (Force). Let's denote the Force as F. Stress on A: $$\text{Stress on A}=\frac{\text{Force}}{\text{Area of A}}=\frac{F}{4\pi r^2}$$ Stress on B: $$\text{Stress on B}=\frac{\text{Force}}{\text{Area of B}}=\frac{F}{\pi r^2}$$

Find the relationship between stress on A and stress on B

Divide the stress on B by the stress on A: $$\frac{\text{Stress on B}}{\text{Stress on A}}=\frac{\frac{F}{\pi r^2}}{\frac{F}{4\pi r^2}}=\frac{1}{\frac{1}{4}}=4$$ Thus, the stress on B is 4 times the stress on A. The correct answer is (B) Four times that on A.

Key concepts

Force per Unit Area

Stress is an essential concept in physics, especially when analyzing materials under load. It is defined as the force applied to an area, and helps us understand how a material will react to being stretched, compressed, bent, or twisted. Stress is formally expressed with the formula $\text{Stress}=\frac{\text{Force}}{\text{Area}}$.
This simple ratio indicates how much force is exerted on each unit area of a material. When enough stress is applied, a material can deform or even break.
Understanding stress allows us to predict when these failures might occur and ensure that structures are safe and functional.

• Higher stress can lead to material failure.
• Stress levels need to be managed in engineering and construction projects.

Cross-sectional Area of a Wire

The cross-sectional area of a wire influences how stress is distributed across it. For cylindrical objects like wires, we calculate this area using the formula for the area of a circle, $\pi r^2$, where $r$ is the radius of the wire.
This tells us how large the slice through the wire is, perpendicular to its length.
In practical terms, knowing the cross-sectional area helps when calculating stress, because a larger area means the load is spread over more material, reducing the stress on any one part.

• Larger cross-sectional areas can handle more stress.
• This is a crucial factor in designing wires and cables for specific loads.

Radius and Stress Relationship

The relationship between the radius of a wire and the stress it experiences is a pivotal concept.
In the given problem, the radius of wire A is double that of wire B. Consequently, the cross-sectional area of wire A becomes four times greater than that of wire B, due to the area formula $\pi (2r{)}^2=4\pi r^2$.
This enlarged area for wire A means the same force is distributed across a larger surface, reducing the stress compared to wire B.
Hence, for the same applied load, wire A will experience less stress, while wire B will have four times the stress because its smaller area concentrates the load more densely.

• Doubling the radius results in quadrupling the area.
• Larger radii result in lower stress for the same force.