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

To determine the elongation of the wire when two weights are hung from either end, we can start by analyzing the situation step by step. Initially, when a single weight W is applied to the wire, it elongates by A mm. Now, when we hang two weights W, one on each end of the wire, the total load on the wire effectively doubles. This change in load will directly affect the elongation of the wire.

Understanding the Basics of Elongation

Elongation in a wire is governed by Hooke's Law, which states that the extension of a material is directly proportional to the load applied to it, as long as the material remains within its elastic limit. Mathematically, this can be expressed as:

F = k * x

Where:

• F is the force applied (in this case, the weight W),
• k is the spring constant of the wire, and
• x is the elongation (or extension) of the wire.

Applying the Concept to Our Problem

When a single weight W is applied, the elongation is A mm. This means:

W = k * A

Now, when we hang two weights W, the total force acting on the wire becomes:

F_total = W + W = 2W

Using Hooke's Law again, we can express the new elongation (let's call it x) as:

2W = k * x

Relating the Two Situations

We can substitute the expression for W from the first scenario into the second scenario. From the first equation, we have:

W = k * A

Substituting this into the equation for the total force gives:

2(k * A) = k * x

Now, simplifying this equation:

2kA = kx

Assuming k is not zero (which it can't be for a physical wire), we can divide both sides by k:

2A = x

Final Result

Thus, the elongation of the wire when two weights W are hung from either end will be:

x = 2A mm

In summary, when you double the load on the wire by hanging weights on both ends, the elongation also doubles, resulting in an elongation of 2A mm. This illustrates the direct relationship between the load applied and the elongation of the wire, as described by Hooke's Law.