IIT TOUGH PROBLEMS – SET 5 (PART VII)
Work, Energy & Power

This is the final and hardest problem set in Work–Energy–Power. These problems test energy philosophy, inequalities, hidden work, and examiner psychology. If you are comfortable here, you are IIT-ready.

Problem 1: Energy Inequality + Reachability

Question:
A particle of mass m moves in a one-dimensional potential U(x) = ax⁴ − bx² (a, b > 0). For what range of total energy E is the motion bounded? Identify stable equilibrium positions.

Key analysis:

• Turning points from E = U(x)
• Bounded motion when E < maximum of U(x)

Equilibrium:

dU/dx = 0 → x = 0, ±√(b/2a)

IIT Insight: Motion limits are decided by energy landscape, not force magnitude.

Problem 2: Energy + Non-Inertial Frame

Question:
A block slides on a smooth wedge which is accelerating horizontally with acceleration a. Discuss how energy analysis changes in the wedge frame.

Advanced reasoning:

• Pseudo force appears
• Pseudo force may do work

Conclusion:

Mechanical energy is NOT conserved in non-inertial frame.

Rank-deciding idea: Energy depends on reference frame when pseudo forces exist.

Problem 3: Energy + Time-Independent Proof

Question:
Prove that if force depends only on position, the work done between two points is independent of time.

Proof idea:

• Work = ∫F(x)dx
• No explicit time term appears

Conceptual depth: Explains why energy methods eliminate time.

Problem 4: Energy + Partial Constraints

Question:
A particle is constrained to move on a curve. Show that the constraint force does no work if it is always perpendicular to velocity.

Reasoning:

Work = F · ds = 0 when F ⟂ ds

Hidden idea: Constraints can guide motion without changing energy.

Problem 5: Energy + Maximum Work Principle

Question:
A force of fixed magnitude F acts on a particle. What orientation of force produces maximum work over a given displacement?

Solution:

Maximum work when force is parallel to displacement.

IIT Angle: Uses vector projection, not calculus.

Problem 6: Energy + Catastrophic Loss

Question:
A moving particle suddenly loses all its kinetic energy. Describe the physical situations where this can occur.

Examples:

• Perfectly inelastic collision
• Sudden capture by a potential well

Concept: Energy conversion, not disappearance.

Problem 7: Energy + Extremum Condition

Question:
Show that at a point of stable equilibrium, the potential energy has a minimum.

Mathematical condition:

dU/dx = 0 and d²U/dx² > 0

IIT Standard: Links calculus directly with physical stability.

Problem 8: Energy + No-Work Forces

Question:
Give examples of forces that can never change the kinetic energy of a particle.

Answer:

• Centripetal force
• Ideal constraint forces

Deep idea: Energy change requires force component along velocity.

Final Self-Test (After Set-5)

• You think in energy landscapes
• You identify hidden work instantly
• You are calm with inequalities
• You trust physics over algebra

“If Set-5 feels logical, you are IIT-Advanced ready.”

Work, Energy & Power – Complete Physics Library

This is the MASTER LIBRARY PAGE for the complete chapter Work, Energy and Power, prepared for Intermediate, IIT-JEE (Main & Advanced), NEET and competitive exams.

All concepts are explained from basic to IIT level, including theory, derivations, numerical problems, objective questions, previous year questions, tough IIT problems, tricks and cautions.

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📚 Complete Lesson Index (Part 1 – Part 39)

1. Part 1 – Introduction to Work
2. Part 2 – Types of Work
3. Part 3 – Variable Force & Graphs
4. Part 4 – Kinetic Energy
5. Part 5 – Work–Energy Theorem
6. Part 6 – Potential Energy
7. Part 7 – Conservative Forces
8. Part 8 – Mechanical Energy
9. Part 9 – Power
10. Part 10 – Power Applications
11. Part 11
12. Part 12
13. Part 13
14. Part 14
15. Part 15
16. Part 16
17. Part 17
18. Part 18
19. Part 19
20. Part 20
21. Part 21
22. Part 22
23. Part 23
24. Part 24
25. Part 25
26. Part 26
27. Part 27
28. Part 28
29. Part 29
30. Part 30
31. Part 31
32. Part 32
33. Part 33
34. Part 34
35. Part 35
36. Part 36
37. Part 37
38. Part 38
39. Part 39 – Final IIT Tough Problems & Solutions

Prepared by: Shaktimatha Learning 🌱

Strong Concepts • Smart Practice • Exam Success