Work Done — Constant and Variable Force

• Work by constant force: W = F·d·cosθ — scalar product of force and displacement.
• Work by variable force: integrate dW = F(x) dx; area under F-x graph equals work done.
• Work-energy theorem: W_net = ΔKE = ½mv² − ½mv₀² — net work equals change in kinetic energy.
• Conditions for zero work, negative work, and maximum positive work.
• Application to inclined planes, spring forces, and friction in NCERT and JEE problems.

Level 1 — Foundations

Verbal: Work is done on an object when a force causes displacement in a component of its own direction.

Formula: W = F d cosθ, where θ is the angle between force vector and displacement vector.

Zero work cases: Force ⊥ displacement (θ = 90°, e.g., centripetal force on circular path); or displacement = 0 (holding heavy bag stationary).

Negative work: θ > 90°; friction does negative work on a sliding block.

Units: Joule (J) = N·m = kg·m²·s⁻².

Kinetic energy: KE = ½mv² — energy possessed due to motion.

Level 2 — JEE / NEET depth

Work-energy theorem derivation: Using v² = v₀² + 2as → mas = ½mv² − ½mv₀² → F·d = ΔKE (for constant force along d).

Variable force: dW = F(x) dx, so W = ∫[x₁ to x₂] F(x) dx.

Spring work: F_spring = −kx, so W by spring = −½kx² (agent does +½kx² while compressing).

F-x graph: area under F(x) vs x curve between limits = work done by that force.

Work done against gravity: W = mgh (for height h, independent of path — gravity is conservative).

Conservative vs non-conservative: Gravity and spring are conservative (work path-independent). Friction is non-conservative.

Superposition: W_net = ΣW_each force = ΔKE (only kinetic energy change — crucial for multi-force problems).

Block pulled on rough surface

A 5 kg block is pulled 4 m along a horizontal surface by a force
F = 20 N at θ = 37° above horizontal. μ_k = 0.2. Find net work done.

Step 1 — Work by applied force:
W_F = F d cosθ = 20 × 4 × cos37° = 20 × 4 × 0.8 = 64 J

Step 2 — Normal force:
N = mg − F sinθ = 5×10 − 20×0.6 = 50 − 12 = 38 N

Step 3 — Friction force:
f_k = μ_k N = 0.2 × 38 = 7.6 N

Step 4 — Work by friction:
W_f = −f_k × d = −7.6 × 4 = −30.4 J (negative, opposes motion)

Step 5 — Work by gravity and normal = 0 (⊥ to displacement).

Step 6 — Net work = 64 − 30.4 = 33.6 J

Step 7 — Verify: ΔKE = 33.6 J → final v = √(2×33.6/5) ≈ 3.66 m/s

Spring compression

A spring with k = 500 N/m is compressed by x = 0.1 m from natural length.
Find work done by the agent compressing the spring.

Step 1 — Spring force is variable: F(x) = kx (magnitude, opposing compression).

Step 2 — Work by agent (in direction of compression):
W = ∫[0 to 0.1] kx dx = ½k x² = ½ × 500 × 0.01 = 2.5 J

Step 3 — Work done by spring on agent = −2.5 J (spring pushes back).

Step 4 — This stored energy = elastic PE = 2.5 J (released when spring relaxes).

• A force of 30 N acts at 60° to displacement of 5 m — find work done.
• Under what conditions is work done by gravity zero?
• A 2 kg block accelerates from 3 m/s to 7 m/s. What is the net work done?
• Sketch the F-x graph for a spring; what does the area represent?
• Stretch: A variable force F = 3x² acts from x = 0 to x = 2 m. Find work done.

NCERT Chapter 5 link: Work-energy theorem connects dynamics (forces) to energetics (motion change). Always check sign of work before applying the theorem — negative work reduces kinetic energy.

Exam connections: JEE frequently tests: work done on inclined plane with friction; spring work (agent vs spring sign); variable force integration; identifying zero-work forces (centripetal, normal). Work-energy theorem shortcut avoids solving differential equations of motion.

Study strategy: Draw a free-body diagram, identify ALL forces, compute each W = F·d·cosθ, then sum for W_net = ΔKE. For variable force, look for recognizable integrals (linear → triangle area, quadratic → cubic formula).

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.