Section 5.8 introduces one of the most elegant and powerful laws in mechanics — the Conservation of Mechanical Energy. After understanding kinetic energy, potential energy, and the Work–Energy Theorem, we now combine them to establish a universal principle that simplifies complex motion problems.
At Deeksha Vedantu, we train students to recognize when conservation can be applied instead of using force equations. In competitive exams like JEE Main and JEE Advanced, energy methods often reduce multi-step force problems into a single equation.
What Is Mechanical Energy?
Mechanical energy is the sum of kinetic energy and potential energy of a system.
Mathematically:
Where:
may be gravitational or elastic potential energy.
Thus,
Mechanical energy represents the total usable energy in a conservative system.
Statement of Conservation of Mechanical Energy
If only conservative forces act on a system, the total mechanical energy remains constant.
Mathematically:
Or,
This principle directly follows from combining the Work–Energy Theorem and the definition of potential energy.
Mathematical Derivation
From Work–Energy Theorem:
For conservative forces:
Thus:
Rearranging:
Therefore:
This proves conservation of mechanical energy.
Conditions for Conservation
Mechanical energy is conserved only when:
- Only conservative forces act.
- No friction or air resistance.
- No external non-conservative work is done.
If non-conservative forces act:
Mechanical energy changes by that amount.
Understanding these conditions is crucial in JEE conceptual questions.
Application 1: Free Fall Motion
For a body dropped from height 
:
Initial energy:
At height 
:
Thus,
This shows velocity depends only on vertical drop, not path.
Application 2: Vertical Circular Motion (JEE Classic)
Consider a mass moving in a vertical circle of radius 
.
Energy at bottom:
Energy at top:
Using conservation:
Minimum speed condition at top:
Thus required bottom speed:
This derivation appears frequently in JEE Advanced.
Application 3: Escape Velocity (Gravitational Field)
Initial energy at Earth's surface:
At infinity:
Thus,
Escape velocity:
This entire derivation is purely energy-based.
Multi-Step Mixed Energy Problem 1 (Incline + Spring)
A 2 kg block slides down a frictionless incline of height 3 m and compresses a spring of constant 
.
Step 1: Initial energy
Step 2: At maximum compression, KE = 0
This avoids resolving forces along inclines.
Multi-Step Mixed Energy Problem 2 (Incline + Friction + Spring)
Now suppose the friction coefficient 
on incline length 5 m.
Work by friction:
Assume 
.
Effective energy reaching spring:
Now:
This shows how non-conservative work modifies conservation equations.
Multi-Step Mixed Energy Problem 3 (Loop-the-Loop + Height)
A particle slides from height into a vertical loop of radius .
Condition to complete loop:
At top:
Using conservation from height h to top:
This classic JEE problem demonstrates multi-stage energy reasoning.
Energy Graph Interpretation
Plotting KE and U vs position:
- Total energy is horizontal.
- Turning points occur where
. - Motion allowed only where
.
.This graphical approach is extremely powerful in advanced mechanics.
Advanced JEE Problem (Variable Potential)
Given:
Total energy = 18 J.
Maximum displacement:
Velocity at 
:
Energy method simplifies such problems dramatically.
When Mechanical Energy Is Not Conserved
If friction acts:
Mechanical energy converts to thermal energy.
However, the total energy of the universe remains conserved.
This distinction is tested in conceptual questions.
Comparison: Work–Energy vs Conservation
| Work–Energy Theorem | Conservation of Mechanical Energy |
| Always valid | Valid only for conservative forces |
| |
Students must clearly distinguish between these two principles.
Common Mistakes to Avoid
- Applying conservation when friction is present without accounting work.
- Ignoring gravitational potential reference.
- Forgetting energy at turning points.
- Mixing escape velocity derivation with kinematics.
Precision ensures accuracy.
Key Formula Summary
| Concept | Formula |
| Mechanical Energy | |
| KE |  |
| PE (near Earth) |  |
| Elastic PE |  |
| Conservation | |
| Escape Velocity | |
FAQs
Q1. When is mechanical energy conserved?
When only conservative forces act.
Q2. What happens if friction is present?
Mechanical energy decreases by amount of work done by friction.
Q3. Why is conservation useful in JEE?
It reduces complex force problems into simple energy equations.
Q4. What is the turning point in the energy graph?
Point where 
and motion reverses.
Q5. Is total energy always conserved?
Yes, but mechanical energy may convert into other forms.
Conclusion
Section 5.8 establishes the powerful law of Conservation of Mechanical Energy. When only conservative forces act, the sum of kinetic and potential energies remains constant.
From free fall and circular motion to escape velocity and spring systems, this principle simplifies multi-step problems and forms the backbone of advanced mechanics.
At Deeksha Vedantu, we ensure students master both conceptual understanding and advanced applications so they can confidently solve board-level and JEE Advanced-level problems using energy methods.
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