The stress experienced by a wire of diameter $\frac{\text{y}}{\pi}$m loaded by a weight y$^2$N is

Analysis

To solve this problem, we need to calculate the stress applied to the wire. Stress is defined as the force applied per unit cross-sectional area of the material.

Concept

The formula for stress ($\sigma$) is:

Where:

• $F$ is the force (weight) applied.
• $A$ is the cross-sectional area of the wire.

Since the wire is cylindrical, its cross-section is a circle. The area of a circle is given by:

(where $r$ is the radius and $d$ is the diameter).

Step-by-Step Solution

1. Identify the given values:

• Force ($F$) = $y^2$ N
• Diameter ($d$) = $\frac{y}{\pi}$ m

2. Calculate the cross-sectional area ($A$): Using the diameter formula $A = \frac{\pi d^2}{4}$:

Simplify by canceling one $\pi$:

3. Calculate the stress ($\sigma$): Substitute $F$ and $A$ into the stress formula:

When you divide by a fraction, you multiply by its reciprocal:

The $y^2$ terms cancel out:

Conclusion

Comparing our result to the given options, the calculated stress is $4\pi \, \text{Nm}^{-2}$.

Exam Tip: Always be careful with the diameter vs. radius. Many students forget to divide by 4 when using the diameter formula ($A = \frac{\pi d^2}{4}$), which often leads to choosing the wrong option. If you prefer using the radius, remember to divide the diameter by 2 first ($r = \frac{y}{2\pi}$) and then use $A = \pi r^2$.