Problem 22 - Master Resource Book in JEE Main Physics

Question

A wire of length \(L\) suspended vertically from a rigid support is made to suffer extension \(l\) in its length by applying a force \(F\). The work is (a) \(\frac{F l}{2}\) (b) \(F l\) (c) \(2 \mathrm{Fl}\) (d) \(\mathrm{Fl}\)

Step-by-Step Solution

Answer

Option (a) \( \frac{F l}{2} \) is correct.

Step 1: Understanding the Problem

We need to calculate the work done when a force is applied to stretch a wire by a certain extension. The options are given in terms of force applied and the extension achieved.

Step 2: Identifying Work Done Formula

The work done when a force is used to stretch an object can be calculated using the formula for elastic potential energy. Since work done is equal to the elastic potential energy stored in the wire, it is expressed as \( W = \frac{1}{2} F l \), where \( F \) is the force applied and \( l \) is the extension.

Step 3: Applying the Formula

Using the formula \( W = \frac{1}{2} F l \), if \( F \) is the force and \( l \) is the extension, plug these values into the equation. This leads to the expression for the work done as \( \frac{F l}{2} \).

Step 4: Finding the Correct Option

The calculated work done \( \frac{F l}{2} \) matches option \(a\). Thus, the correct choice from the given options is \( \frac{F l}{2} \).

Key Concepts

Work Done

When you apply a force to an object, causing it to move, you've performed what we call "work." The concept of work is central in physics, especially when it comes to understanding how energy is transferred or transformed. The formula for work done is given by:

• Work Done (W) = Force (F) × Distance (d)

When the force applied stretches or compresses an object, like a wire, the distance is referred to as the extension (or the change in length). But remember, the direction of the force and the movement must align. If they are not perfectly parallel, you will need to consider only the component of the force that acts in the direction of the movement.
The work done in stretching a wire, like in our exercise, is also a measure of the energy stored in the wire, which is called elastic potential energy. We use the formula:

• \( W = \frac{1}{2} F l \)

This equation tells us that only half of the product of the force and the extension contributes to the work done. This is because the force increases gradually from zero to a maximum value as the extension happens.

Force and Extension

The relationship between force and extension is an essential part of studying elasticity. When a material like wire is stretched, a force is applied, resulting in the material extending by a certain length. This relationship can often be described by Hooke's Law, as long as the material remains within its elastic limits.
Hooke's Law states:

• Force (F) = Spring Constant (k) × Extension (x)

This tells us that force is directly proportional to the extension, under the condition that the material does not surpass its elastic limit. The constant of proportionality, \(k\), is a property of the material itself and indicates its stiffness.
When you plot force against extension, you get a straight line for values within the elastic limit, showing this direct proportional relationship. However, if the applied force surpasses the elastic limit, the material won't return to its original shape or length, losing its elasticity.

Elasticity in Physics

Elasticity is the property of a material that causes it to return to its original shape after being deformed (stretched, compressed, etc.). When a material like a wire is elastic, it will recover its shape once the external force is removed, provided the limits of elasticity are not exceeded.
When considering elasticity, two crucial points are:

• Elastic Limit: The maximum extent to which a material can be deformed and still return to its original shape once the force is removed.
• Elastic Potential Energy: The energy stored within a material due to its deformation.

Materials behave elastically until they reach their elastic limit. Beyond this point, they may become permanently deformed. Elastic potential energy is stored in an object (such as a stretched wire) and is released when the object returns to its original state. This is described by the equation \( W = \frac{1}{2} F l \), showcasing how elastic potential energy is pivotal in understanding the work done on elastic materials.