CBSE Class 11th Work done by a variable force Details & Preparations Downloads
In the intricate dance of forces and motions that govern the physical world, the concept of work done plays a central role. While the understanding of work done by a constant force is wellestablished, the narrative becomes more nuanced when we venture into the realm of variable forces. In this exploration, we will unravel the intricacies of work done by a force that changes along the path of motion.
Defining the Work Done:
Work done (W) is a fundamental concept in physics, representing the energy transferred to or from an object as a result of a force acting on it, causing it to move a certain distance. When the force acting on an object remains constant, as is often assumed in introductory physics, calculating work is a straightforward multiplication of force and displacement (W=F⋅d). However, realworld scenarios frequently involve forces that vary along the path of motion, requiring a deeper dive into the dynamics of the system.
Mathematical Expression of Variable Work
For a variable force
F(x) acting over an infinitesimally small displacement dx, the infinitesimal work done (dW) is expressed as the product of force and displacement:
dW=F(x)⋅dx
To find the total work done, the process involves integrating this expression over the entire displacement:
W=∫F(x)dx
This integration, a cornerstone of calculus, allows us to sum up the work done at each infinitesimally small segment along the path, taking into account the changing force.
Graphical Insight: Tracing the ForceDisplacement Curve
Visualising the work done by a variable force often involves examining the forcedisplacement graph. The area under this curve represents the total work done. Regions where the force is positive contribute positively to the work, while regions with negative force values contribute negatively. This graphical representation offers an intuitive understanding of how the force varies and the impact on the overall work done.
Realworld Applications: A Glimpse into Dynamics
Understanding work done by variable forces is not just an abstract concept but a key player in various realworld scenarios. Consider a scenario where an object is pulled or pushed with a force that changes at different points in its journey. Examples include:
1. Spring Systems:

Work done in stretching or compressing a spring involves a variable force, dictated by Hooke's Law ( F=−kx), where k is the spring constant and x is the displacement from equilibrium. Integrating this force over displacement reveals the total work done on the spring.
2. Frictional Forces:

Work done against friction, a force that often varies depending on factors like the nature of surfaces in contact, showcases the dynamic nature of forces in motion.
Conservative and Nonconservative Forces: A Dichotomy
Variable forces can be categorised into conservative and nonconservative forces. Conservative forces, such as gravity, allow for a potential energy function. In such cases, the work done is pathindependent, and the total work done is the change in potential energy. On the other hand, nonconservative forces, like friction, may lead to energy dissipation and require a more intricate analysis.
Insights from the WorkEnergy Theorem
The workenergy theorem is a guiding principle in understanding the interplay between work and kinetic energy. For a variable force, the theorem states that the net work done ( W net) is equal to the change in kinetic energy (ΔKE) of the object:
W net=ΔKE
This succinctly captures the essence of how the work done by variable forces influences the motion and energy state of an object.
Challenges and Complexities: Realising the Dynamic Nature
While the mathematical expressions and graphical insights provide a structured framework, the analysis of work done by variable forces introduces challenges. Integrating complex force functions and navigating through intricate forcedisplacement curves require a solid grasp of mathematical tools and a keen eye for graphical interpretation.
CBSE Class 11th Downloadable Resources:
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SAMPLE PRACTICE QUESTIONS OF SIGNIFICANT FIGURES:
Q1 What is work done by a force?
Answer 1 Work done by a force is the product of the force applied to an object and the displacement of the object in the direction of the force. Mathematically, it is given by the formula W=∫Fdx, where W is work, F is the force, and dx is the displacement.
Q2 How is work done by a variable force different from constant force?
Answer 2 When the force acting on an object is constant, the calculation is straightforward (W=F⋅d). In the case of a variable force, the force may change at different points along the displacement, requiring integration to find the total work done.
Q3 What is the significance of a variable force in physics?
Answer 3 Variable forces are common in realworld scenarios. Understanding work done by variable forces is crucial for analyzing systems where forces change over time or position, such as springs, gravitational fields, or electromagnetic forces.
Q4 How is work done by a variable force calculated graphically?
Answer 4 The work done by a variable force can be calculated graphically by finding the area under the forcedisplacement curve. The integral of the forcedisplacement function over the given range provides the total work done.
Q5 What is the unit of work?
Answer 5 The unit of work is the joule (J). One joule is equal to one newtonmeter (N·m), which represents the amount of work done when a force of one newton displaces an object by one meter in the direction of the force.
Class 11th CBSE Physics Chapters 
Chapter1: UNITS AND MEASUREMENTS 
Chapter2: MOTION IN A STRAIGHT LINE 
Chapter3: MOTION IN A PLANE 
Chapter4: LAWS OF MOTION 
Chapter5: WORK, ENERGY AND POWER 
> Introduction 
> Notions of work and kinetic energy: The workenergy theorem 
> Work 
> Kinetic energy 
> The concept of potential energy 
> The conservation of mechanical energy 
> The potential energy of a spring 
> Power 
> Collisions 
Chapter6: SYSTEM OF PARTICLES AND ROTATIONAL MOTION 
Chapter7: GRAVITATION 
Chapter8: MECHANICAL PROPERTIES OF SOLIDS 
Chapter9: MECHANICAL PROPERTIES OF FLUIDS 
Chapter10: THERMAL PROPERTIES OF MATTER 
Chapter12: KINETIC THEORY 
Chapter13: OSCILLATIONS 
Chapter14: WAVES 
Class 11th CBSE Chemistry Chapters 
Chapter1: SOME BASIC CONCEPTS OF CHEMISTRY 
Chapter2: STRUCTURE OF ATOMS 
Chapter3: CLASSIFICATION OF ELEMENTS AND PERIODICITY IN PROPERTIES 
Chapter4: CHEMICAL BONDING AND MOLECULAR STRUCTURE 
Chapter5: THERMODYNAMICS 
Chapter6: EQUILIBRIUM 
Chapter7: REDOX REACTIONS 
Chapter8: ORGANIC CHEMISTRY  SOME BASIC PRINCIPLE AND TECHNIQUES 
Chapter9: Hydrocarbons HYDROCARBONS 
Class 11th CBSE Mathematics chapter 
Chapter1: SETS 
Chapter2: RELATIONS AND FUNCTIONS 
Chapter3: TRIGONOMETRIC FUNCTIONS 
Chapter4: COMPLEX NUMBER AND QUADRATIC EQUATIONS 
Chapter5: LINEAR INEQUALITIES 
Chapter6: PERMUTATIONS AND COMBINATIONS 
Chapter7: BINOMIAL THEOREM 
Chapter8: SEQUENCES AND SERIES 
Chapter9: STRAIGHT LINES 
Chapter10: CONIC SECTIONS 
Chapter11: INTRODUCTION TO THREEDIMENSIONAL GEOMETRY 
Chapter12: LIMITS AND DERIVATIVES 
Chapter13: STATISTICS 
Chapter14: PROBABILITY 
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