A wire elongates by 1mm when a load w is hung from it. If the wire goes over a pulley and two weights W each are hung at two ends, what will be the elongation of the wire in mm?

To solve the problem of how much the wire elongates when two weights are hung from either end, we need to consider the principles of tension and elongation in materials. Let's break it down step by step.

Understanding the Setup

Initially, we have a wire that elongates by 1 mm when a single load \( w \) is applied to it. This elongation is due to the tension created in the wire by the weight. When we change the setup to have two weights \( W \) on either end of the wire, we need to analyze how this affects the tension and consequently the elongation.

Analyzing the Forces

When one weight \( W \) is hung from each end of the wire, the total tension in the wire is effectively doubled compared to when a single load \( w \) is applied. This is because each weight contributes to the overall tension in the wire. Therefore, the total tension \( T \) in the wire when two weights \( W \) are applied is:

• Total Tension \( T = W + W = 2W \)

Relating Tension to Elongation

The elongation of a wire is directly proportional to the tension applied to it, according to Hooke's Law, which states that \( \Delta L \propto T \). If we denote the elongation caused by a single weight \( w \) as \( \Delta L_w = 1 \) mm, we can express the elongation caused by the two weights as follows:

• Let \( \Delta L_{2W} \) be the elongation when two weights \( W \) are applied.
• Since the tension is doubled, we have: \( \Delta L_{2W} = 2 \times \Delta L_w \)

Calculating the Elongation

Substituting the known elongation for one weight into the equation gives us:

• \( \Delta L_{2W} = 2 \times 1 \text{ mm} = 2 \text{ mm} \)

Final Result

Thus, when two weights \( W \) are hung from either end of the wire, the total elongation of the wire will be 2 mm. This illustrates how the distribution of weight and the resulting tension in the wire directly influence its elongation.