Section 5.3 provides the rigorous mathematical framework for understanding work in physics. While earlier sections introduced the idea of work conceptually and connected it to energy, this section formalizes the definition using vector algebra, calculus, and graphical interpretation. It also explores different physical situations where work may be positive, negative, or zero.
At Deeksha Vedantu, we emphasize that a deep understanding of work is essential because it forms the direct foundation for the Work–Energy Theorem, Conservation of Mechanical Energy, and Power. Many students lose marks in competitive exams due to weak clarity in sign conventions and direction-based reasoning. This section removes that confusion completely.
Definition of Work in Vector Form
In physics, work is defined as the scalar (dot) product of force and displacement vectors.
Using the definition of dot product:
Where:
= magnitude of applied force
= magnitude of displacement
= angle between force and displacement
Because the dot product is involved, work is a scalar quantity. It does not have direction, even though force and displacement are vectors.
This distinction is extremely important for competitive exams.
Physical Meaning of the Cosine Factor
The presence of 
tells us that only the component of force along displacement contributes to work.
If we resolve force into components:
Then work becomes:
Thus, perpendicular components do not contribute to work.
This explains why centripetal force does not work in circular motion.
Cases Based on Angle Between Force and Displacement
Case 1: 
(Force Parallel to Displacement)
(Force Parallel to Displacement)
Work is maximum and positive. The force increases the kinetic energy of the body.
Example: A person pushing a box forward.
Case 2: 
(Force Perpendicular to Displacement)
Even though force may be present, no work is done.
Example: Centripetal force in uniform circular motion.
Case 3: 
(Force Opposite to Displacement)
(Force Opposite to Displacement)
Work is negative. The force reduces kinetic energy.
Example: Friction acting on a moving body.
Understanding these three cases eliminates most conceptual mistakes.
Work Done by a Constant Force
If the force remains constant in magnitude and direction, work is simply:
If motion occurs along the x-axis:
This formula applies when force does not change during displacement.
In many board-level problems, this simplified form is sufficient.
Work Done by a Variable Force
In real physical systems, force often varies with displacement.
For a small displacement 
:
Total work is obtained by integrating over displacement:
This is the most general expression for work.
This form is heavily tested in JEE where force may depend on position, velocity, or displacement.
Graphical Interpretation of Work
Work done by a force is equal to the area under the force–displacement graph.
Linear Force–Displacement Graph
If force increases linearly with displacement, the graph forms a triangle.
Work equals the triangular area.
Rectangular Graph
If force is constant, work equals rectangular area:
Curved Graph
If the graph is non-linear, work equals area under the curve:
Graph-based problems are common in competitive exams.
Work Done by Gravity
When a body of mass 
moves vertically through height 
:
- During downward motion → work is positive
- During upward motion → work is negative
Gravity is a conservative force, so work depends only on initial and final position.
Work Done by Spring Force
According to Hooke's Law:
The negative sign indicates restoring nature.
Work done in stretching from 0 to x:
This result is important for understanding elastic potential energy.
Work Done by Friction
If frictional force is constant:
Friction always does negative work when opposing motion.
In stopping distance problems, friction removes kinetic energy.
Work Done in Circular Motion
In uniform circular motion:
- Displacement is tangential
- Centripetal force is radial
The angle between them is 
.
Thus, centripetal force changes direction of velocity but not magnitude.
This is a very common MCQ concept.
Work Done by Multiple Forces (Net Work)
If several forces act simultaneously:
Only net work affects kinetic energy.
This concept directly leads to the Work–Energy Theorem.
Units of Work
SI Unit:
CGS Unit:
Work and energy share identical units.
Dimensional Formula of Work
This matches the dimensional formula of energy.
Applications in Competitive Exams
Work concept appears in:
- Inclined plane problems
- Friction and stopping distance problems
- Graph-based work calculations
- Spring compression and release problems
- Circular motion conceptual traps
- Variable force integration questions
Correct sign handling is crucial.
Advanced Insight: Path Dependence
For non-conservative forces like friction, work depends on the path.
For conservative forces like gravity, work depends only on endpoints.
This distinction becomes important in later sections.
Common Mistakes to Avoid
- Ignoring the cosine factor
- Using distance instead of displacement
- Forgetting negative sign of friction
- Assuming all forces do work
- Ignoring that perpendicular force does zero work
Avoiding these mistakes increases exam accuracy.
Comparison: Constant vs Variable Force
| Constant Force | Variable Force |
| |
| Simple multiplication | Requires integration |
| Rectangular graph | Area under curve |
| Board-level application | JEE-level depth |
Key Formula Summary
| Concept | Formula |
| Work (Vector Form) | |
| Work (Magnitude Form) | |
| Variable Force Work | |
| Spring Force | |
| Spring Work | |
| Work by Gravity | |
| Dimensional Formula | |
FAQs
Q1. What is the definition of work in physics?
— Work is the dot product of force and displacement.
Q2. When is work zero?
When force is perpendicular to displacement (
).
Q3. Why does centripetal force do no work?
Because it is always perpendicular to instantaneous displacement.
Q4. How is work calculated for variable force?
Using .
Q5. Can work be negative?
Yes. When force opposes displacement, work is negative and reduces kinetic energy.
Conclusion
Section 5.3 builds the complete mathematical structure of work in physics. Understanding vector form, integration for variable force, graphical interpretation, sign conventions, and physical meaning prepares students for advanced applications in energy, power, and conservation laws.
At Deeksha Vedantu, we ensure students develop deep conceptual clarity and mathematical precision so they can confidently solve both board-level and competitive-level mechanics problems.
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