JAMB 2023 Physics · Question 5

A wire of radius 0.2 mm is extended by 0.5% of its length when supported by a load of 1.5 kg. Determine the…

Question

A wire of radius 0.2 mm is extended by 0.5% of its length when supported by a load of 1.5 kg. Determine the Young's modulus for the material of the wire. [Take g = 10 ms $^{-2}$ ]

A. $2.4×10^{10}(Nm^{-2})$
B. $1.5×10^{10}(Nm^{-2})$
C. $2.4×10^9(Nm^{-2})$
D. $1.3×10^{10}(Nm^{-2})$

Answer: A. $2.4×10^{10}(Nm^{-2})$

Explanation written with AI and checked against the answer key

Analysis

To solve this problem, we need to calculate the Young's Modulus ( $Y$ ) of the wire material. Young's Modulus is defined as the ratio of tensile stress to tensile strain.

Concept

The formula for Young's Modulus is:

Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L}

Where:

$F$ = Force (Weight = $mg$ )
$A$ = Cross-sectional area of the wire ( $\pi r^2$ )
$\Delta L/L$ = Tensile strain (given as a percentage)

Step-by-Step Solution

1. Identify the given values:

Radius ( $r$ ) = $0.2 \text{ mm} = 0.2 \times 10^{-3} \text{ m} = 2 \times 10^{-4} \text{ m}$
Load ( $m$ ) = $1.5 \text{ kg}$
$g = 10 \text{ ms}^{-2}$
Strain ( $\frac{\Delta L}{L}$ ) = $0.5\% = \frac{0.5}{100} = 0.005 = 5 \times 10^{-3}$

2. Calculate Force ( $F$ ):

F = mg = 1.5 \times 10 = 15 \text{ N}

3. Calculate Cross-sectional Area ( $A$ ): Using $\pi \approx 3.142$ :

A = \pi r^2 = 3.142 \times (2 \times 10^{-4})^2

A = 3.142 \times 4 \times 10^{-8} \approx 1.2568 \times 10^{-7} \text{ m}^2

4. Calculate Young's Modulus ( $Y$ ):

Y = \frac{F}{A \times (\Delta L/L)}

Y = \frac{15}{(1.2568 \times 10^{-7}) \times (5 \times 10^{-3})}

Y = \frac{15}{6.284 \times 10^{-10}}

Y \approx 2.387 \times 10^{10} \text{ Nm}^{-2}

Rounding this value to two significant figures gives approximately $2.4 \times 10^{10} \text{ Nm}^{-2}$ .

Conclusion

Comparing our result to the given options, it matches option A.

Exam Tip

When dealing with units in elasticity questions, always convert millimeters (mm) to meters (m) immediately. A common mistake is forgetting to square the radius or forgetting to convert the percentage strain into a decimal. Keep your powers of 10 organized to avoid calculation errors!

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