Physics Mechanics - How to Solve: Work, Power & Energy (Work-Energy Theorem, Conservative Forces) – IIT JEE Guide

How to Solve: Work, Power & Energy (Work-Energy Theorem, Conservative Forces) – IIT JEE Guide

Introduction

Mastering the Work-Energy Theorem unlocks 10–15 marks in IIT JEE—enough to push you from a 150 to a 200+ score. Whether it’s a block sliding down a ramp, a spring compressing, or a satellite orbiting Earth, this single concept replaces pages of Newton’s laws with one clean equation. If you can solve work-energy problems, you can solve 90% of JEE mechanics questions faster and with fewer mistakes.

WHAT YOU NEED TO KNOW FIRST

Before diving in, you must already understand: 1. Kinetic Energy (KE) = ½mv² – The energy of motion. 2. Work Done (W) = F·s·cosθ – Force times displacement times the angle between them. 3. Potential Energy (PE) – Energy stored due to position (gravitational, elastic).

If any of these feel shaky, stop now and review them first.

KEY TERMS & FORMULAS

1. Work-Energy Theorem

Formula: W_net = ΔKE = KE_final – KE_initial - W_net = Net work done on the object (Joules, J) - ΔKE = Change in kinetic energy (J) - KE_final, KE_initial = Final and initial kinetic energy (J)

MEMORISE THIS. This is the core equation for this topic.

2. Conservative vs. Non-Conservative Forces

Key Property: - For conservative forces, W = -ΔPE (Work done = Negative change in potential energy). - For non-conservative forces, W ≠ -ΔPE (Energy is lost as heat, sound, etc.).

3. Mechanical Energy Conservation

Formula (Only for Conservative Forces): KE_initial + PE_initial = KE_final + PE_final - MEMORISE THIS. This is only true if no non-conservative forces act (e.g., no friction).

If non-conservative forces act: KE_initial + PE_initial + W_nc = KE_final + PE_final - W_nc = Work done by non-conservative forces (e.g., friction).

4. Power (P)

Formula: P = W / t = F·v (if force and velocity are constant) - P = Power (Watts, W) - W = Work done (J) - t = Time (s) - F = Force (N) - v = Velocity (m/s)

MEMORISE THIS. Often tested in numerical problems (e.g., "Find power of an engine lifting a load").

STEP-BY-STEP METHOD

Step 1: Identify the System & Forces

• Is the system isolated? (No external forces → Mechanical energy conserved.)
• Are there non-conservative forces? (Friction, air resistance → Energy not conserved.)
• List all forces acting on the object.

Step 2: Choose Initial & Final States

• Initial State: Position/velocity at start.
• Final State: Position/velocity at end.
• Define a reference level for PE (usually ground or lowest point).

Step 3: Write Down Known Quantities

• Mass (m), initial velocity (u), final velocity (v), height (h), spring constant (k), displacement (x), etc.

Step 4: Apply the Correct Formula

• No non-conservative forces? → KE_i + PE_i = KE_f + PE_f
• Non-conservative forces present? → KE_i + PE_i + W_nc = KE_f + PE_f
• Only work-energy theorem needed? → W_net = ΔKE

Step 5: Solve for the Unknown

• Plug in values.
• Check units (J for energy, W for power).
• Simplify algebraically before plugging numbers.

Step 6: Verify the Answer

• Does the answer make sense? (e.g., KE should increase if speed increases.)
• Are units correct? (J, m/s, N, etc.)
• Did you account for all forces?

WORKED EXAMPLES

Example 1 – Basic (No Friction)

Problem: A 2 kg block slides from rest down a smooth (frictionless) incline of height 5 m. Find its speed at the bottom.

Solution (Step-by-Step):

1. Identify System & Forces:
2. Only gravity (conservative force) acts.

3. No friction → Mechanical energy conserved.

4. Initial & Final States:

5. Initial: At top (h = 5 m, v = 0).

6. Final: At bottom (h = 0, v = ?).

7. Known Quantities:

8. m = 2 kg
9. h = 5 m
10. g = 9.8 m/s² (use 10 for JEE if not specified)

11. u = 0 m/s

12. Apply Energy Conservation:
    KE_i + PE_i = KE_f + PE_f
    ½mv² (initial) + mgh (initial) = ½mv² (final) + mgh (final)
    0 + (2)(10)(5) = ½(2)v² + 0
    100 = v²
    v = 10 m/s

13. Verify:

14. Speed increases as height decreases → makes sense.
15. Units: m/s → correct.

What we did and why: - Used energy conservation because only gravity (conservative force) acts. - No need for kinematics (faster than using a = g sinθ).

Example 2 – Medium (With Friction)

Problem: A 1 kg block slides down a rough incline (μ = 0.2) of height 3 m and length 5 m. Find its speed at the bottom.

Solution (Step-by-Step):

1. Identify System & Forces:
2. Gravity (conservative) + Friction (non-conservative).

3. Energy not conserved → Must account for work done by friction.

4. Initial & Final States:

5. Initial: At top (h = 3 m, v = 0).

6. Final: At bottom (h = 0, v = ?).

7. Known Quantities:

8. m = 1 kg
9. h = 3 m
10. μ = 0.2
11. g = 10 m/s²

12. Incline length (s) = 5 m

13. Find Work Done by Friction (W_nc):

14. Normal force (N) = mg cosθ
15. But θ is unknown → Use s = 5 m (given).
16. Friction force (f) = μN = μmg cosθ
17. But cosθ = adjacent/hypotenuse = 4/5 (from 3-4-5 triangle).
18. So, f = (0.2)(1)(10)(4/5) = 1.6 N

19. Work done by friction (W_nc) = f × s = (1.6)(5) = 8 J (negative, since friction opposes motion).

20. Apply Energy Equation:
    KE_i + PE_i + W_nc = KE_f + PE_f
    0 + (1)(10)(3) – 8 = ½(1)v² + 0
    30 – 8 = ½v²
    22 = ½v²
    v = √44 ≈ 6.63 m/s

21. Verify:

22. Speed is less than 7.75 m/s (frictionless case) → makes sense.
23. Units: m/s → correct.

What we did and why: - Friction is non-conservative → Must include W_nc in energy equation. - Used incline length to find friction work (since θ was not directly given).

Example 3 – Exam-Style (Spring + Friction)

Problem (JEE 2018-Style): A 0.5 kg block is pushed against a spring (k = 200 N/m) compressing it by 0.2 m. The block is released and slides 1 m on a rough surface (μ = 0.1) before stopping. Find the maximum compression of the spring if the block were to return.

Solution (Step-by-Step):

1. Identify System & Forces:
2. Spring force (conservative) + Friction (non-conservative).

3. Energy not conserved → Must account for work done by friction.

4. Initial & Final States:

5. Initial: Spring compressed (x = 0.2 m, v = 0).

6. Final: Block stops after 1 m (v = 0).

7. Known Quantities:

8. m = 0.5 kg
9. k = 200 N/m
10. x_initial = 0.2 m
11. μ = 0.1
12. s = 1 m

13. g = 10 m/s²

14. Find Work Done by Friction (W_nc):

15. Normal force (N) = mg = (0.5)(10) = 5 N
16. Friction force (f) = μN = (0.1)(5) = 0.5 N

17. Work done by friction (W_nc) = f × s = (0.5)(1) = 0.5 J (negative).

18. Apply Energy Equation (First Trip):
    KE_i + PE_i + W_nc = KE_f + PE_f
    0 + ½kx² – 0.5 = 0 + 0
    ½(200)(0.2)² – 0.5 = 0
    4 – 0.5 = 0 → Consistent (block stops).

19. Now, Find Maximum Compression on Return:

20. Block starts from rest at 1 m, moves back, compresses spring.
21. Work done by friction (W_nc) = 0.5 J (same as before, but now opposite direction).

22. Energy Equation (Return Trip):
    KE_i + PE_i + W_nc = KE_f + PE_f
    0 + 0 – 0.5 = 0 + ½kx²
    -0.5 = ½(200)x²
    -0.5 = 100x²
    x² = 0.005 → x = √0.005 ≈ 0.0707 m

23. Verify:

24. Compression is less than initial 0.2 m → makes sense (energy lost to friction).
25. Units: m → correct.

What we did and why: - Two-step problem: First, energy lost to friction. Then, energy remaining for return trip. - Friction work is negative in both directions (opposes motion).

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP (Night Before Exam)

"Listen up—this is your 60-second crash course for Work-Energy in JEE.

1. Work-Energy Theorem: Net work = Change in KE. W_net = ΔKE. That’s it. No forces? No problem.
2. Conservative Forces (gravity, spring): Energy is conserved. KE_i + PE_i = KE_f + PE_f. No friction? Use this.
3. Non-Conservative Forces (friction, air resistance): Energy is not conserved. KE_i + PE_i + W_nc = KE_f + PE_f. W_nc is negative (friction steals energy).
4. Power: P = W/t or P = F·v. Watts = Joules/second.
5. Common Mistakes:
6. Forgetting friction → always check if surface is rough.
7. Wrong sign for W_nc → friction work is negative.
8. Mixing up spring PE (½kx²) and gravitational PE (mgh).
9. Exam Traps:
10. "Smooth" = no friction. "Rough" = friction.
11. If a force depends on path (not just position), it’s non-conservative.
12. Multiple objects? Apply energy to the whole system.

Final Tip: If you see height, speed, or springs, energy methods are faster than Newton’s laws. Now go crush that exam!