Question

When a rubber band is stretched by a distance $x$, it exerts a restoring force of magnitude $F=a x+$ $b x^{2}$, where $a$ and $b$ are constants. The work done in stretching the unstretched rubber band by $L$ is: [2014] (A) $a L^{2}+b L^{3}$ (B) $\frac{1}{2}\left(a L^{2}+b L^{3}\right)$ (C) $\frac{a L^{2}}{2}+\frac{b L^{3}}{3}$ (D) $\frac{1}{2}\left(\frac{2 L^{2}}{2}+\frac{b L^{2}}{3}\right)$

Name: Problem 131, Chapter 4: A Complete Resource Book in Physics for JEE Main
Uploaded: 2021-10-05T15:08:15-08:00
Duration: 1 min 36 s
Channel: Vysakh M
Description: When a rubber band is stretched by a distance x, it exerts a restoring force of magnitude F=a x+ b x^2, where a and b are constants. The work done in stretching the unstretched rubber band by L is: [2014] (A) a L^2+b L^3 (B) (1)/(2)(a L^2+b L^3) (C) (a L^2)/(2)+(b L^3)/(3) (D) (1)/(2)((2 L^2)/(2)+(b L^2)/(3))

   When a rubber band is stretched by a distance $x$, it exerts a restoring force of magnitude $F=a x+$ $b x^{2}$, where $a$ and $b$ are constants. The work done in stretching the unstretched rubber band by $L$ is:
[2014]
(A) $a L^{2}+b L^{3}$
(B) $\frac{1}{2}\left(a L^{2}+b L^{3}\right)$
(C) $\frac{a L^{2}}{2}+\frac{b L^{3}}{3}$
(D) $\frac{1}{2}\left(\frac{2 L^{2}}{2}+\frac{b L^{2}}{3}\right)$

A Complete Resource Book in Physics for JEE Main

Sanjeev Kumar 1st Edition

Chapter 4, Problem 131

Instant Answer

Solved by Expert Vysakh M

10/05/2021

Step 1

In mathematical terms, this is represented as: \[W = \int F dx\] Show more…

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When a rubber band is stretched by a distance $x$, it exerts a restoring force of magnitude $F=a x+$ $b x^{2}$, where $a$ and $b$ are constants. The work done in stretching the unstretched rubber band by $L$ is: [2014] (A) $a L^{2}+b L^{3}$ (B) $\frac{1}{2}\left(a L^{2}+b L^{3}\right)$ (C) $\frac{a L^{2}}{2}+\frac{b L^{3}}{3}$ (D) $\frac{1}{2}\left(\frac{2 L^{2}}{2}+\frac{b L^{2}}{3}\right)$

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Key Concepts

Nonlinear Elasticity

This refers to the behavior of materials whose restoring force does not follow a simple linear relationship (as per Hooke's law). In such cases, additional terms (like a quadratic term) are used to more accurately model the force, which can depend on higher powers of displacement.

Integration of a Variable Force

This concept involves calculating work by integrating a force that changes with displacement. Instead of the constant force work formula (force × distance), the work done when the force varies is found by integrating the force function with respect to displacement over the distance of interest.

Work-Energy Principle

This principle states that work done on a system is stored as potential energy. For elastic materials, when a displacement is applied, the potential energy accumulated is equal to the area under the force versus displacement curve, which is determined through integration.

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