In an experiment to determine the Young's modulus, steel wires of five different lengths (1,2,3,4, and 5 m) but of same cross section (2 mm²) were taken and curves between extension and load were obtained. The slope (extension/load) of the curves were plotted with the wire length and the following graph is obtained. If the Young's modulus of given steel wires is x × 10¹¹ Nm⁻², then the value of x is __________.

Solution

Physics sir · 4 steps

Extension over load is just L/(AY), so the slope of this straight line is 1/(AY).

1. What is plotted. From Y = (F/A)/(Δ L/L), the extension per unit load is (Δ L)/F = L/AY. So extension/load grows in proportion to the wire length L, which is why the graph is a straight line through the origin.

   slope = (Δ L / F)/L = 1/AY

2. Read the slope. At L = 1 m the graph gives 0.25 × 10⁻⁵ m N⁻¹, so the slope is 2.5 × 10⁻⁶ N⁻¹. That means AY = 1/(2.5 × 10⁻⁶) = 4 × 10⁵ N.
3. Divide by area. The area is A = 2 mm² = 2 × 10⁻⁶ m². So Y = (4 × 10⁵)/(2 × 10⁻⁶) = 2 × 10¹¹ N m⁻².
4. So. Y = 2 × 10¹¹ N m⁻², so x = 2.

The common slip: 0.002. Converting 2 mm² as 2 × 10⁻³ m² gives Y = 2 × 10⁸ N m⁻², that is x = 0.002. Since 1 mm = 10⁻³ m, 1 mm² = 10⁻⁶ m².

The idea to keep. When a graph is given, write the physics law in the form of what is plotted. Then the slope is a known combination like 1/(AY), and one reading from the line gives the answer.

Answer from the NTA final answer key. Solution by PrepWiser.