Section 5.5 introduces one of the most powerful extensions of the concept of work — calculating work when force is not constant. In realistic physical systems, forces rarely remain constant. They often depend on displacement, position, configuration, or even velocity. This section develops the calculus-based method required to calculate work in such situations and forms the backbone of advanced mechanics.
At Deeksha Vedantu, we train students to handle both board-level and JEE-advanced level questions from this topic. Mastery of integration and graphical reasoning gives a decisive competitive advantage.
Why Constant Force Formula Is Not Sufficient
Earlier we used:
This works only when force is constant in magnitude and direction.
However, in real systems:
- Spring force changes with displacement.
- Gravitational force changes with distance.
- Electrostatic force varies with separation.
- Many JEE problems define force as a function of x.
Hence we require a general definition.
Elementary Work for Infinitesimal Displacement
Consider a force that varies with position as 
.
For a very small displacement 
, the small amount of work done is:
Total work from 
to 
is obtained by integration:
This is the most general expression for work in one dimension.
Geometrical Interpretation: Area Under F–x Curve
Work done equals the area under the force–displacement graph.
Constant Force
Area = rectangle =
Linearly Increasing Force
If force increases from 0 to F:
Area = triangle =
General Non-Linear Curve
Work equals total area under curve:
If the area lies below the x-axis, work is negative.
Graph interpretation is heavily tested in JEE.
Work Done by a Spring (Hooke's Law Detailed Treatment)
Hooke's Law:
Work done by spring from 0 to x:
Work done by external agent:
This becomes elastic potential energy.
Solved Example 1 (Board Level)
A spring of constant 
is stretched by 
. Find work done.
Work Done by Position-Dependent Force (JEE Level)
Suppose:
Find work done from 0 to 2 m.
This type of polynomial integration is common in JEE Main.
Work Done by Gravitational Force (Advanced Treatment)
Gravitational force at distance r:
Work done from 
to 
:
This directly leads to gravitational potential energy expression.
Advanced JEE Problem (Graph-Based)
A force–displacement graph consists of:
- Rectangle area = 10 J
- Triangle area = 5 J
- Curved negative area = -3 J
Total work:
Students must carefully account for sign of area.
Work Done in Three Dimensions
General vector form:
Total work:
If force has components:
This form is used in advanced JEE Advanced problems.
Conservative vs Non-Conservative Forces (Deeper Insight)
Conservative Force
- Work independent of path
- Potential energy definable
- Work over closed path = 0
Examples:
- Spring force
- Gravitational force
Non-Conservative Force
- Work depends on path
- Mechanical energy not conserved
Example:
- Friction
Understanding this distinction is critical for energy conservation problems.
Advanced Concept: Closed Loop Work
For conservative forces:
This property is frequently used in higher-level mechanics.
High-Level Numerical (JEE Advanced Type)
Force varies as:
Find work from 0 to a.
Such power-function integrations are standard in competitive exams.
Applications in Competitive Exams
Work done by variable force appears in:
- Spring compression and release
- Particle under polynomial force
- Gravitational potential derivation
- Force–displacement graph problems
- Energy conservation derivations
Integration mastery directly improves score.
Common Mistakes to Avoid
- Using
for variable force - Ignoring integration limits
- Forgetting sign of restoring force
- Misreading graph area below axis
- Confusing force–time graph with force–displacement graph
Accuracy in interpretation is essential.
Key Formula Summary
| Concept | Formula |
| Elementary Work | |
| Total Work | |
| Spring Force | |
| Spring Work | |
| Gravitational Work | |
| Vector Form | |
FAQs
Q1. Why can't we use for variable force?
Because force changes during displacement, so simple multiplication does not apply.
Q2. What is the most general formula for work?
Q3. What does the area under the F–x graph represent?
Work done by the force.
Q4. Why is spring work negative when calculated using spring force?
Because spring force is restoring and opposite to displacement.
Q5. Why is this topic important for JEE?
Because advanced mechanics problems frequently define force as a function of position and require integration.
Conclusion
Section 5.5 elevates the concept of work from simple multiplication to a powerful calculus-based tool. The integral form of work enables accurate calculation of energy transfer in realistic physical systems where forces vary continuously.
At Deeksha Vedantu, we ensure students gain conceptual clarity, graphical interpretation skills, and integration mastery so they can confidently solve board-level as well as JEE Advanced-level problems involving variable forces.
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