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Chapter 7: Problem 865

A wire elongates by \(1 \mathrm{~mm}\) when a load \(\mathrm{W}\) is hanged from it. If the wire goes ever a pulley and two weight \(\mathrm{W}\) each are hang at the two ends. What will be the elongation of the wire? (in \(\mathrm{mm}\) ) (A) \(2 \ell\) (B) zero (C) \(C / 2\) (D) \(\ell\)

Short Answer

Expert verified
The wire elongates by \(\ell\) under the force of weight W. When the wire goes over a pulley and two weights of W each are hung at its ends, the effective force acting downwards remains the same (W) on each side of the wire. Therefore, the elongation of the wire in the new configuration is unchanged and remains \(\ell\). The correct answer is (D) \(\ell\).

Step by step solution

01

Understand the setup

Initially, we have a wire experiencing elongation \(\ell\) when weight W is hung from it. Now, the wire goes over a pulley, with two weights of W each hung from its ends. We need to determine the elongation in the new setup.
02

Analyze the forces on the wire

When the single weight W was hung on the wire, let's denote its tension by T. In the new setup, tension T will be equal to the weight W (Tx = W), as the weight will act downwards and tension will act upwards. Now, when the wire goes over the pulley and two weights are hung at the ends. Since each weight W is evenly distributed among the two sides of the wire, the total effective force acting downwards is still W on each side of the wire (as the sum of half W on each side).
03

Calculate the elongation

In the new setup, the tension on each side of the wire remains unchanged. Thus, the elongation experienced by the wire remains unchanged as well. Therefore, the elongation of the wire in the new configuration is also \(\ell\). So, the correct answer is (D) \(\ell\).

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