A steel wire W has a length / and a circular cross-section of radius r - AQA - A-Level Physics - Question 24 - 2018 - Paper 1

To solve this problem, we can use the relationship between the extension of a wire and its dimensions.

Step 1: Understanding Extension The extension (e) of a wire is directly proportional to the load applied and the length of the wire, while inversely proportional to the cross-sectional area. Given by: e ext{ (extension)} = rac{F L}{A Y} where:

• $F$ is the force (load),
• $L$ is the length of the wire,
• $A$ is the cross-sectional area (for a circular section: A = rac{ ho^2 imes ext{π}}{4}),
• $Y$ is Young's modulus (a property of the material).

Step 2: Relating W and X For wire W with length $L_W$ and radius $r$, the extension is given as: e_W = rac{F L_W}{rac{ ext{π} r^2}{4} Y}.

Now, let's denote the length and radius of wire X as $L_X$ and $r_X$. The extension for wire X is: e_X = rac{F L_X}{rac{ ext{π} r_X^2}{4} Y}.

Step 3: Setting the Extensions We need the extension $e_X$ to be rac{e}{4}. From the proportional relation, we can derive: rac{e_X}{e_W} = rac{L_X / (r_X^2)}{L_W / (r^2)} = rac{1}{4}.

Step 4: Solving for Length and Radius From the above equation, we can express: $L_X = L_W ext{ and } r_X = 2r$. Thus, for the answers provided in the options, we arrive at:

• Length of X = $L_W$ (which can be $2L_W$, depending on the context)
• Radius of X = $2r$

So the correct answer is D: Length of X = 2L/ and Radius of X = 4r.