A steel wire W has a length / and a circular cross-section of radius r. When W hangs vertically and a load is attached to the bottom end, it extends by e. Another wire X made from the same material has the same load attached to it. Which length and radius for X will produce an extension of $rac{e}{4}$?

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

Question 24

A steel wire W has a length / and a circular cross-section of radius r. When W hangs vertically and a load is attached to the bottom end, it extends by e. Another wi... show full transcript

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

Step 1

Length of X and Radius of X

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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.

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

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

Length of X and Radius of X

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