5.6 The Work-Energy Theorem for a Variable Force – Class 11 Physics

Section 5.6 establishes one of the most powerful and universal results in mechanics — the Work–Energy Theorem remains valid even when the force acting on a particle varies continuously. In earlier sections, we derived the theorem under the assumption of constant force and constant acceleration. However, real-world physical systems rarely involve constant forces. Instead, forces depend on position, configuration, or even velocity.

This section proves mathematically that irrespective of how complicated the force variation may be, the net work done on a particle is always equal to the change in its kinetic energy.

At Deeksha Vedantu, we ensure that students understand not only the mathematical derivation but also the deep conceptual meaning of this theorem, because it forms the foundation for conservation laws, gravitational potential derivation, spring systems, and advanced mechanics used in JEE and NEET.

Recap: Work–Energy Theorem for Constant Force

Previously, we derived:

Or equivalently,

This derivation used constant acceleration from Newton's Second Law combined with kinematics.

Now we remove the assumption of constant force and develop a general proof using calculus.

General Derivation for a Variable Force (Using Calculus)

Start from Newton's Second Law in its most general form:

Multiply both sides by velocity :

Since velocity is related to displacement as:

We substitute:

Multiplying both sides by :

The left side represents elementary work:

Thus,

Now integrate both sides between initial and final states:

Evaluating the integral:

Therefore,

This proves that the Work–Energy Theorem holds even when force varies continuously with position or time.

This result is independent of the nature of force — constant, variable, conservative, or non-conservative.

Deep Physical Interpretation

The Work–Energy Theorem tells us that:

• Work is the mechanism by which kinetic energy changes.
• Forces do not directly change velocity; they perform work.
• The net work done by all forces determines the final kinetic energy.

Important conclusions:

• Positive net work → → speed increases.
• Negative net work → → speed decreases.
• Zero net work → → constant speed.

Thus, the theorem connects force, motion, and energy in a single powerful equation.

Application 1: Particle Under Position-Dependent Force

Suppose force varies as:

Work done from to :

Using Work–Energy Theorem:

Thus,

This avoids solving differential equations directly.

Application 2: Spring–Block System (Detailed Insight)

For spring force:

Work done by spring from to :

Using Work–Energy Theorem:

If block starts with velocity :

This equation is used to calculate maximum compression in spring-block systems.

Application 3: Variable Gravitational Force (Orbital Motion Foundation)

Gravitational force at distance :

Work done from to :

Applying Work–Energy Theorem:

This forms the basis of escape velocity and orbital energy derivations in higher classes.

Advanced Solved Example 1 (JEE Advanced Level)

Force varies as:

Mass of particle = 1 kg, starts from rest at x = 0.

Find velocity at x = 2 m.

Work done:

Using theorem:

This type of integration-based velocity calculation is common in JEE Advanced.

Advanced Solved Example 2 (Graph-Based Multi-Region)

Suppose area under F–x graph is:

• Region 1: +15 J
• Region 2: −5 J
• Region 3: +10 J

Total net work:

If initial KE is 5 J:

Final velocity for mass 2 kg:

This demonstrates how graphical interpretation links directly with kinetic energy change.

Why Work–Energy Theorem Is More General Than Energy Conservation

Work–Energy Theorem always holds because it is derived directly from Newton's Laws.

However, conservation of mechanical energy holds only when:

• No non-conservative forces act.
• Only conservative forces are present.

If friction acts:

Mechanical energy decreases, but the Work–Energy Theorem remains valid.

Thus, the Work–Energy Theorem is a more universal principle.

Mathematical Insight: Independence from Path of Integration

In one dimension:

In three dimensions:

The theorem remains valid regardless of path taken.

This universality makes it extremely powerful in advanced mechanics.

Competitive Exam Insights

• Many JEE Advanced problems define force as polynomial in x.
• Graph-based area problems test sign interpretation.
• Variable gravitational force appears in orbital motion.
• Spring compression questions combine integration + theorem.
• Problems avoid solving differential equations by using an energy approach.

Mastery of this section significantly reduces solution time.

Common Mistakes to Avoid

• The thinking theorem applies only to constant force.
• Forgetting integration limits.
• Ignoring signs of work when the area lies below the axis.
• Confusing energy conservation with Work–Energy Theorem.
• Applying conservation when non-conservative forces act.

Clarity in these concepts improves accuracy.

Key Formula Summary

Concept	Formula
Differential Work
Integral Form
Work–Energy Theorem
Change in KE
Vector Form

FAQs

Q1. Does the Work–Energy Theorem apply when force varies continuously?

Yes. It is derived using calculus and holds universally.

Q2. Why is this theorem more general than conservation of energy?

Because it applies even when non-conservative forces are present.

Q3. What mathematical concept is essential for understanding this section?

Integration.

Q4. How are graph problems solved using this theorem?

The area under the F–x graph gives net work, which equals change in kinetic energy.

Q5. Why is this topic important for JEE Advanced?

Because many advanced problems define force as a function of position and require applying integration with the Work–Energy Theorem.

Conclusion

Section 5.6 establishes the Work–Energy Theorem as a universal law valid for all types of forces, whether constant or variable. By deriving it rigorously using calculus, we see that energy methods are fundamentally rooted in Newton's Laws.

At Deeksha Vedantu, we ensure students gain both mathematical depth and conceptual clarity so they can confidently apply this theorem to board exams, JEE Main, and JEE Advanced level problems. Mastery of this section forms a strong bridge toward conservation of mechanical energy and advanced mechanics topics.