Problem 36 - Master Resource Book in JEE Main Physics

Question

An elastic string of unstretched length \(L\) and force constant \(k\) is stretched by a small length \(x .\) It is further stretched by another small length \(y\). The work done in the second stretching is (a) \(\frac{1}{2} k y^{2}\) (b) \(\frac{1}{2} k\left(x^{2}+y^{2}\right)\) (c) \(\frac{1}{2} k(x+y)^{2}\) (d) \(\frac{1}{2} k y(2 x+y)\)

Step-by-Step Solution

Answer

The work done in the second stretching is \(\frac{1}{2} k y(2x + y)\). Correct answer: (d).

Step 1: Understand the Problem

We need to find the work done in stretching an elastic string that has already been stretched once. Initially, the string is unstretched with a length \(L\) and is first stretched by a length \(x\). Then, it is further stretched by a length \(y\). We are asked to calculate the work done specifically in the second stretching.

Step 2: Apply Work Done Formula

The work done when stretching an elastic material is given by the formula for elastic potential energy change: \(W = \frac{1}{2} k (\Delta x)^2\), where \(\Delta x\) is the change in length. For the second stretch, the change in length is \(y\), since it stretches an additional length of \(y\) beyond the initial stretch of \(x\).

Step 3: Calculate the Work Done for Second Stretch

To calculate the work done for the second stretch from length \(x\) to \(x+y\), we use the expression:\[ W = \frac{1}{2} k [(x+y)^2 - x^2] \].This represents the change in potential energy from the initial extension to the final extension, essentially \((x+y)^2\) since \(x\) is already reached, minus \(x^2\).

Step 4: Simplify the Expression

Simplify \((x+y)^2 - x^2\) to find the work done:\[ (x+y)^2 - x^2 = x^2 + 2xy + y^2 - x^2 = 2xy + y^2 \].Thus, substituting back, we have the work done as:\[ W = \frac{1}{2} k (2xy + y^2) \],which simplifies to \(\frac{1}{2} k y(2x + y)\).

Step 5: Select the Correct Option

From the calculations, the work done in the second stretching matches option (d) \(\frac{1}{2} k y(2x + y)\).

Key Concepts

Elastic String

An elastic string is a type of string that returns to its original length after being stretched.
It behaves according to Hooke's Law, which states that the force needed to stretch the string is directly proportional to the extension of the string from its natural length.
In simple terms, the more you stretch it, the more force it will take. The force constant, often denoted as \(k\), measures the stiffness of the string. The greater the \(k\), the more force you need to stretch the string by a certain amount. For instance:

• If a string has a force constant \(k = 50 \, \text{N/m}\), it takes 50 newtons to stretch it 1 meter.
• This relationship holds as long as the string is not stretched beyond its elastic limit, where permanent deformation occurs.

This concept is crucial for problems involving elastic potential energy and work done in stretching.

Elastic Potential Energy

Elastic potential energy is the energy stored in elastic materials as a result of their stretching or compression. This energy represents the potential to do work as the material returns to its original shape.
In the context of an elastic string, when the string is stretched, it stores energy given by the formula:\[U = \frac{1}{2} k x^2\]where:

• \(U\) is the elastic potential energy,
• \(k\) is the force constant of the string,
• \(x\) is the amount of stretch from the natural length.

For example, if a string with a force constant of 100 N/m is stretched by 0.1 m, the elastic potential energy stored will be:\[U = \frac{1}{2} \times 100 \times (0.1)^2 = 0.5 \times 100 \times 0.01 = 0.5 \, \text{J}\]Thus, understanding how elastic potential energy works helps determine the energy changes involved when stretching an elastic string.

Work Done Formula

The work done on an elastic string during stretching can be calculated using the change in elastic potential energy formula. The work done is an expression of this energy shift as the string is stretched from one length to another.
Specifically, if you are stretching the string by an additional length \(y\) after it has already been stretched by \(x\), the work done in this additional stretch is calculated as:\[W = \frac{1}{2} k [(x+y)^2 - x^2]\]Let's explain the formula:

• \((x+y)^2\) is the potential energy at the new stretched length.
• \(x^2\) is the potential energy before the additional stretch.
• The difference \((x+y)^2 - x^2\) gives the change in potential energy caused by stretching it an additional \(y\) meters.

Once simplified, this becomes:\[W = \frac{1}{2} k (2xy + y^2)\]This streamlined expression helps to compute the work done during the second stretch of the string. It shows the relationship between additional stretch and work required, providing essential insights into elastic dynamics.