Complete Solutions and Summary of Work, Energy and Power – NCERT Class 11, Physics, Chapter 5 – Summary, Questions, Answers, Extra Questions
Summary of work, kinetic and potential energy, work-energy theorem, power, energy conservation, spring energy, and collision with solved NCERT problems.
Tags: Work, Energy, Power, Kinetic Energy, Potential Energy, Work-Energy Theorem, Conservation of Energy, Spring Energy, Collisions, NCERT, Class 11, Physics, Chapter 5, Answers, Extra Questions
Work, Energy and Power Class 11 NCERT Chapter 5 - Ultimate Study Guide, Notes, Questions, Quiz 2025
Work, Energy and Power
Chapter 5: Physics - Ultimate Study Guide | NCERT Class 11 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Work, Energy and Power Class 11 NCERT
Overview & Key Concepts
Chapter Goal: Define work precisely using vectors (scalar product), introduce kinetic and potential energy, derive work-energy theorem, discuss conservation of mechanical energy, power, and collisions. Exam Focus: Work \( W = \vec{F} \cdot \vec{d} \), KE \( \frac{1}{2}mv^2 \), PE \( mgh \), spring \( \frac{1}{2}kx^2 \), power \( P = \frac{dW}{dt} \). 2025 Updates: Reprint emphasizes variable forces, integrals for work. Fun Fact: Joule’s paddle wheel experiment unified heat/work. Core Idea: Energy conserved in isolated systems; work changes KE. Real-World: Cars (KE to heat via friction), pendulums (PE to KE). Ties: Builds on vectors (Ch.3), leads to gravitation (Ch.7).
Wider Scope: Foundation for thermodynamics (Ch.11), rotational dynamics (Ch.7); applications in efficiency, renewable energy.
5.1 Introduction
Everyday terms 'work', 'energy', 'power' differ from physics: Work is force times displacement (scalar product); energy is capacity to do work; power is work rate. Physics definitions precise to quantify changes. Depth: Physiological 'work' (e.g., studying) ≠ physical (no displacement). Historical: Correlates loosely with stamina (energy). Real-Life: Farmer ploughing does work if force displaces soil. Exam Tip: Work zero if no displacement or perpendicular force. Extended: Chapter unifies motion (Ch.4) with energy; prerequisite scalar product for \(\vec{F} \cdot \vec{d}\). Links: Calculus for variable forces (integration). Applications: Engineering (work in machines). Pitfalls: Confuse with colloquial use (e.g., holding weight = no work). Graphs: Force-displacement for work area. Symbols: Bold vectors, J for joule. Coherent SI units.
Examples: Lifting book (work against gravity), walking (work if friction).
Scalar Product Need: Vectors like force/displacement multiply to scalar work.
Extended Discussion: Energy crisis relevance; chapter focuses conservative forces (path-independent work). Vector language: Dot for projections. Advanced: Relativistic energy \( E = mc^2 + KE \). Real-Life: Solar panels convert light energy to electrical work. Errors: Forget cosθ in work. Tips: Always specify direction.
5.1.1 The Scalar Product
Dot product \(\vec{A} \cdot \vec{B} = AB \cos\theta\): Scalar from two vectors, θ angle between. Properties: Commutative (\(\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}\)), distributive (\(\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}\)), \(\vec{A} \cdot (\lambda \vec{B}) = \lambda (\vec{A} \cdot \vec{B})\). Components: \(\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z\). Magnitude: \(\vec{A} \cdot \vec{A} = A^2\). Perpendicular: Dot=0. Depth: Projection: \( A \cdot (B \cos\theta) \). Real-Life: Work = force projection on displacement. Exam Tip: Unit vectors \(\hat{i} \cdot \hat{i} =1\), \(\hat{i} \cdot \hat{j}=0\). Extended: In 3D, full components. Ties: Ch.3 vectors; used in work, power. Applications: Projections in mechanics. Graphs: Fig.5.1 triangles for cosθ. Symbols: Dot notation. Pitfalls: θ=0 max, 90° zero, 180° negative.
Example 5.1: Force (3i+4j-5k), disp (5i+4j+3k); dot=16, cosθ=0.32, θ=71.3°.
Proofs: Left as exercise; derive from geometry.
Extended: Vector product (cross) in Ch.6 torque. Advanced: Tensor products. Real-Life: Dot in computing (similarity). Errors: Confuse with cross (vector). Tips: Use components for calc.
5.2 Notions of Work and Kinetic Energy: The Work-Energy Theorem
From kinematics \( v^2 - u^2 = 2as \), multiply m/2: \( \frac{1}{2}mv^2 - \frac{1}{2}mu^2 = Fs \). Vector: \( \Delta K = \vec{F} \cdot \vec{d} \). WE Theorem: Change in KE = net work done. Depth: Motivates definitions; K scalar, work scalar. Real-Life: Braking car (negative work reduces KE). Exam Tip: For constant F; generalizes to variable. Extended: 3D \(\vec{v}^2 - \vec{u}^2 = 2 \vec{a} \cdot \vec{d}\). Ties: Ch.3 motion. Applications: Raindrop Ex5.2: Wg=10J, Wr=-8.75J, ΔK=1.25J. Graphs: Speed-time for KE. Symbols: K for kinetic. Pitfalls: Net force only. Advanced: Relativistic KE approx.
Extended Discussion: Theorem integral form of F=ma; loses time info. Real-Life: Sports (work to accelerate ball). Errors: Forget vector dot. Tips: Use for energy changes.
5.3 Work
\( W = \vec{F} \cdot \vec{d} = F d \cos\theta \): Component along displacement times d. Units: J (Nm). Zero if: d=0 (hold weight), F=0 (frictionless), θ=90° (gravity on horizontal). Positive θ<90°, negative >90° (friction). Depth: Dimensions [ML²T⁻²]. Real-Life: Pushing wall (F large, d=0, W=0; tired from internal energy). Exam Tip: Friction always negative work. Extended: Table 5.1 alternative units (eV, cal). Ties: Scalar product. Applications: Cyclist Ex5.3: Wr=-2000J (road on cycle), cycle on road=0 (no d). Graphs: F-d line for W area. Symbols: θ between F and d. Pitfalls: Perpendicular cos0=0.
Example 5.3: Cyclist stop 10m, F=200N opp; Wr=-2000J, cycle on road=0 (Newton III, but d=0).
Newton III: Forces equal opp, but work not (diff d).
Extended: Path-dependent for non-conservative (friction). Advanced: Virtual work principle. Real-Life: Weightlifter holds=0 work. Errors: Assume always positive. Tips: Specify θ.
5.4 Kinetic Energy
\( K = \frac{1}{2} m v^2 \): Measure of work object can do via motion. Scalar. Depth: From WE theorem. Real-Life: Table 5.2: Bullet 1000J, car 3.5×10⁵J. Exam Tip: Depends v², not direction. Extended: Relativistic \( (\gamma -1)mc^2 \). Ties: Ch.2 speed. Applications: Bullet Ex5.4: Initial 1000J, final 100J, vf=63.2m/s (68% reduction). Graphs: v-t for K. Symbols: J. Pitfalls: Zero at rest. Advanced: Molecular KE= (3/2)kT. Real-Life: Windmills harness KE. Errors: Confuse with momentum (mv). Tips: Half mv².
Example 5.4: Bullet 50g, 200m/s plywood 2cm; 10% KE out, vf=63.2m/s.
Intuitive: Fast stream grinds corn.
Extended Discussion: KE theorem for particles. Real-Life: Athletics (running KE). Errors: Units kg m²/s². Tips: Calc ΔK.
5.5 Work Done by a Variable Force
For F(x) varying: Approximate rectangles ΔW=FΔx, sum to integral \( W = \int_{x_i}^{x_f} F(x) dx \) (area under curve). Depth: Limit Δx→0. Real-Life: Woman pushing trunk Ex5.5: WF=1750J (rect+trapezium), Wf=-1000J. Exam Tip: Graphical areas. Extended: Multidim line integral. Ties: Appendix 3.1. Applications: Fig.5.3 rectangles to curve. Graphs: F vs x shaded area. Symbols: dx. Pitfalls: Forget negative areas. Advanced: Numerical Simpson rule.
Example 5.5: Woman F=100N to 50N linear over 20m, f=50N; WF=1750J, Wf=-1000J.
Successive areas for total W.
Extended: Vector form \( \int \vec{F} \cdot d\vec{r} \). Real-Life: Spring force variable. Errors: Assume constant. Tips: Plot F-x.
5.6 The Work-Energy Theorem for a Variable Force
Derive: \( \frac{dK}{dt} = \vec{F} \cdot \vec{v} \), integrate \( \Delta K = \int \vec{F} \cdot d\vec{r} = W \). Depth: 1D proof, generalizes. Real-Life: Block rough patch Ex5.6: Ki=2J, Kf=0.5J, vf=1m/s. Exam Tip: Scalar form loses direction/time. Extended: Vector F=ma differential. Ties: Newton II integral. Applications: ln for inverse F. Graphs: K vs x. Symbols: dK=F dx. Pitfalls: Integrate limits. Advanced: Lagrangian mechanics.
Example 5.6: m=1kg, vi=2m/s, Fr=-k/x (k=0.5J), 0.1-2.01m; Kf=0.5J.
WE: Ki - ∫ Fr dx = Kf.
Extended Discussion: Useful but not full dynamics. Real-Life: Variable thrust rockets. Errors: Forget chain rule. Tips: Use components.
5.7 The Concept of Potential Energy
PE: Stored energy by position/config; V(h)=mgh for gravity (negative work by conservative F). F= -dV/dx. Depth: Applicable to conservative forces (path-independent). Real-Life: Stretched bow, fault lines (earthquakes release PE). Exam Tip: Released PE → KE. Extended: General V such work stored. Ties: Gravity constant near Earth. Applications: Ball drop v²=2gh. Graphs: V vs h linear. Symbols: V. Pitfalls: Only conservative. Advanced: Electric PE.
Gravity: External work mgh stored as V(h).
Negative derivative: F downward.
Extended: Fault lines like springs. Real-Life: Hydro dams (PE to KE). Errors: Variable g (Ch.7). Tips: Up positive.
5.8 The Conservation of Mechanical Energy
For conservative forces: ΔK + ΔU =0; total E= K + U constant. Depth: No dissipation. Real-Life: Pendulum (PE max at ends, KE max bottom). Exam Tip: Friction non-conservative, E not conserved (heat). Extended: Isolated system. Ties: WE theorem. Applications: Roller coaster loops. Graphs: E constant line. Symbols: Mechanical E. Pitfalls: Include all forces. Advanced: Noether theorem (time symmetry → energy cons).
Proof: W_conservative = -ΔU, net W= ΔK, so ΔK + ΔU=0.
Examples: Free fall, simple harmonic.
Extended Discussion: Universe total E conserved? Real-Life: Bouncing ball (E loss to sound/heat). Errors: Forget signs. Tips: Check totals.
5.9 The Potential Energy of a Spring
Hooke F= -kx; U= \frac{1}{2} k x^2 (work to stretch). Depth: Conservative, oscillatory. Real-Life: Shock absorbers. Exam Tip: Zero at equilibrium. Extended: Nonlinear springs. Ties: Ch.14 oscillations. Applications: Mass-spring KE+U constant. Graphs: U parabolic. Symbols: k N/m. Pitfalls: x displacement from eq. Advanced: Anharmonic.
Deriv: ∫ F dx = - \frac{1}{2} k x^2, U= -W_F = \frac{1}{2} k x^2.
Period T=2π √(m/k).
Extended: Atomic bonds springs. Real-Life: Bows/arrows. Errors: Force sign. Tips: Quadratic U.
5.10 Power
P= dW/dt = \vec{F} \cdot \vec{v}; instantaneous rate. Depth: Scalar, W=Pt average. Real-Life: Engine horsepower (1hp=746W). Exam Tip: Units W (J/s). Extended: Instant P=Fv cosθ. Ties: Variable F. Applications: Lifting at constant v, P=mgv. Graphs: P-t. Symbols: P. Pitfalls: Average vs instant. Advanced: Power in circuits.
\( \int F dt = Δp \). Relevance: Collisions. Depth: Area F-t. Applications: Safety.
Tip: Memorize: WE ΔK=W; Cons K+U=const; P=Fv. Depth: All [ML²T⁻²] energy. Applications: Renewables (solar to work). Errors: Forget conservative. Historical: Joule equiv. Interlinks: Ch.3 dot; Ch.6 torque power. Advanced: Lagrangian E= T-V. Real-Life: EVs battery PE to KE. Graphs: U-K cycles. Symbols: Bold F, integral W. Coherent SI. Extended: Field PE ∫. Non-ortho basis rare. Additional: Null work cases; collision 2D components.
Extended Glossary: Path length scalar; resultant work net. Multiplication λW scales. Addition works scalar sum. Elastic θ indep. Power θ=0 max.
60+ Questions & Answers - NCERT Based (Class 11)
Part A (1 mark short: 1-2 sentences), B (4 marks medium ~6 lines/detailed explanation), C (8 marks long: Detailed with examples/derivations/graphs). Based directly on NCERT Exercises 5.1-5.33. Theoretical focus; numericals in separate section. All answers validated against NCERT content and standard solutions.
Part A: 1 Mark Questions (Short Answers - From NCERT Exercises)
5.1(a) Work if d=0?
1 Mark Answer: Zero.
5.1(b) Work if F perp d?
1 Mark Answer: Zero.
5.1(c) Work if θ=180°?
1 Mark Answer: Negative.
5.1(d) Dimensions of work?
1 Mark Answer: [ML²T⁻²].
5.2(a) ΔK = W_net?
1 Mark Answer: Yes, WE theorem.
5.2(b) KE scalar?
1 Mark Answer: Yes.
5.2(c) Work by friction?
1 Mark Answer: Negative.
5.2(d) Conservative force?
1 Mark Answer: Path-independent.
5.3(a) Unit of energy?
1 Mark Answer: Joule.
5.3(b) W if F=0?
1 Mark Answer: Zero.
5.3(c) Moon orbit work by gravity?
1 Mark Answer: Zero (perp).
5.3(d) Weightlifter holding?
1 Mark Answer: Zero work.
5.4(a) K at rest?
1 Mark Answer: Zero.
5.4(b) K dep on?
1 Mark Answer: Speed squared.
5.4(c) Bullet KE example?
1 Mark Answer: 1000J.
5.4(d) Car KE?
1 Mark Answer: ~10^5 J.
5.5(a) Variable F work?
1 Mark Answer: Integral F dx.
5.5(b) Area under F-x?
1 Mark Answer: Work.
5.5(c) Δx small approx?
1 Mark Answer: F Δx rectangle.
5.5(d) Limit Δx→0?
1 Mark Answer: Definite integral.
5.6(a) WE for variable?
1 Mark Answer: ΔK = ∫ F dx.
5.6(b) dK/dt =?
1 Mark Answer: F v.
5.6(c) Scalar form loses?
1 Mark Answer: Direction/time.
5.6(d) Inverse F integrate?
1 Mark Answer: ln.
5.7(a) PE stored how?
1 Mark Answer: Against conservative F.
5.7(b) Gravity PE?
1 Mark Answer: mgh.
5.7(c) F = -dU/dx?
1 Mark Answer: Yes.
5.7(d) Fault lines?
1 Mark Answer: Stored PE.
5.8(a) Cons mechanical E?
1 Mark Answer: Conservative forces.
5.8(b) ΔK + ΔU =?
1 Mark Answer: 0.
5.8(c) Friction cons E?
1 Mark Answer: No.
5.8(d) Pendulum E?
1 Mark Answer: Constant.
5.9(a) Spring PE?
1 Mark Answer: ½ k x².
5.9(b) Hooke law?
1 Mark Answer: F= -kx.
5.9(c) U zero at?
1 Mark Answer: Equilibrium.
5.9(d) Spring cons?
1 Mark Answer: Yes.
5.10(a) Power unit?
1 Mark Answer: Watt.
5.10(b) P = F · ?
1 Mark Answer: v.
5.10(c) Avg P =?
1 Mark Answer: W/t.
5.10(d) Lift const v P?
1 Mark Answer: mgh/t.
5.11(a) Elastic collision?
1 Mark Answer: KE cons.
5.11(b) Inelastic?
1 Mark Answer: Mom cons, KE not.
5.11(c) e=1?
1 Mark Answer: Elastic.
5.11(d) Perfect inelastic?
1 Mark Answer: e=0.
5.12 Scalar product perp?
1 Mark Answer: Zero.
5.13 Variable F approx?
1 Mark Answer: Rectangles.
5.14 PE gravity deriv?
1 Mark Answer: -mg.
5.15 Cons E friction?
1 Mark Answer: No.
5.16 Spring F sign?
1 Mark Answer: Restoring negative.
5.17 Power θ=90°?
1 Mark Answer: Zero.
5.18 Collision 1D cons?
1 Mark Answer: Momentum.
5.19 Dot commutative?
1 Mark Answer: Yes.
5.20 WE integral form?
1 Mark Answer: Yes.
5.21 Joule equiv?
1 Mark Answer: 4.186 cal.
5.22 e definition?
1 Mark Answer: Rel velocities.
5.23 U spring quadratic?
1 Mark Answer: Yes.
5.24 P unit hp?
1 Mark Answer: 746W.
5.25 Cons E proof?
1 Mark Answer: W_c = -ΔU.
5.26 Friction work path?
1 Mark Answer: Dependent.
5.27 KE raindrop?
1 Mark Answer: ½ m v².
5.28 Variable F sum?
1 Mark Answer: Riemann.
5.29 PE bow?
1 Mark Answer: Stored elastic.
5.30 Collision elastic v?
1 Mark Answer: Exchange if equal m.
5.31 Power lift?
1 Mark Answer: F v.
5.32 Dot A·A?
1 Mark Answer: A².
5.33 Inelastic KE?
1 Mark Answer: Lost.
Part B: 4 Marks Questions (Medium Length ~6 Lines - From NCERT)
5.1 Full: Work cases zero/pos/neg.
4 Marks Answer: Zero: d=0, F=0, θ=90°. Pos θ<90° cos>0. Neg θ>90° cos<0 e.g. friction 180°. Deriv: W=Fd cosθ. Ex: Wall push d=0; table gravity perp; braking neg. Units J. Ties: Dot product.
5.2 Full: WE theorem derivation constant F.
4 Marks Answer: From v²-u²=2as, ×m/2: ΔK= F s. Vector ΔK= F·d. Ex: Raindrop ΔK=1.25J= Wg + Wr. Applies net force. Limitations: Constant F. Graphs: K vs d linear.
5.3 Full: Work examples pos/neg/zero.
4 Marks Answer: Pos: Lift mgh. Neg: Friction -μmg d. Zero: Hold weight d=0; moon orbit perp. Newton III: Forces opp but work diff if d diff. Ex: Cyclist -2000J road, 0 cycle. Dimensions same energy.
5.4 Full: KE definition examples.
4 Marks Answer: ½mv² motion energy. Scalar. Ex: Bullet 1000J to 100J vf=63m/s. Table: Car 3.5e5J. From WE. Real: Wind KE sails. Pitfalls: v not dir.
5.5 Full: Variable work integral.
4 Marks Answer: Sum FΔx → ∫F dx area. Ex: Woman 1750J applied, -1000J friction. Graphs rect+trapezium. Deriv: Limit Δx=0. Multidim line int. Ties: Calculus.
5.6 Full: WE variable derivation.
4 Marks Answer: dK/dt= F v, int ΔK=∫F dx. Ex: Block Kf=0.5J ln. Scalar loses dir/time. Vs Newton: Integral form. Ex: Inverse F ln x.
5.7 Full: PE concept gravity.
4 Marks Answer: Stored against cons F; V=mgh= -W_grav. F=-dV/dh=-mg. Ex: Ball drop ½mv²=mgh. Bow/faults stored. Near Earth approx h<
5.8 Full: Cons mechanical E proof.
4 Marks Answer: W_cons= -ΔU, net W=ΔK, so ΔK+ΔU=0. Ex: Pendulum max PE ends, KE bottom. Friction no. Isolated sys. Graphs: E const. Advanced: Noether.
5.9 Full: Spring PE derivation.
4 Marks Answer: F=-kx, W_ext=∫kx dx=½kx²= U. Cons force. Ex: Mass-spring E cons. Period 2π√(m/k). Graphs: Parabolic U. Real: Shocks.
5.10 Full: Power def avg/instant.
4 Marks Answer: P=dW/dt=F·v. Avg W/t. Ex: Lift mgv. Units W= J/s, hp=746W. θ=0 max. Graphs: P-t. Ties: Variable F.
5.11 Full: Collisions elastic/inelastic.
4 Marks Answer: Elastic: Cons KE+mom, e=1. Inelastic: Mom only, e<1. Formulas 1D. Ex: Equal m exchange v. Real: Cars inelastic. 2D components.
5.12 Scalar product properties.
4 Marks Answer: AB cosθ, comm, dist, λ scalar. A·A=A², perp=0. Ex: Work proj. Components sum. Graphs: Fig5.1.
5.13 Variable F approx areas.
4 Marks Answer: Rect FΔx sum → int. Ex: Linear F trapezium. Limit Riemann. Multidim dr. Ties: Calc.
5.14 PE general cons F.
4 Marks Answer: U= -∫F dx path indep. Ex: Gravity mgh. F=-∇U. Stored released KE. Bow example.
5.15 Cons E non-cons.
4 Marks Answer: Only cons forces; friction dissipates. Ex: Pendulum approx no air. Total E cons universe? Graphs cycles.
5.16 Spring U deriv.
4 Marks Answer: ∫F dx= -½kx², U=½kx². F=-kx= -dU/dx. Harmonic. Ex: Oscillator E cons.
5.17 Power examples.
4 Marks Answer: Lift mgh/t avg; Fv instant. Perp=0. Ex: Engine 100hp. Units conv.
5.18 Collision cons laws.
4 Marks Answer: Always mom; elastic KE. e rel v. Ex: 1D formulas. 2D indep.
4 Marks Answer: U=½kx², E cons. Period indep amp. Ex: SHM v_max=ωA. Real shocks.
5.24 Power real life.
4 Marks Answer: Bulb 60W, car 100hp. P=Fv cos. Ex: Const v no accel power. Efficiency.
5.25 Cons proof examples.
4 Marks Answer: W_c=-ΔU=ΔK, total const. Ex: Fall, pendulum. No friction. Graphs E level.
5.26 Non-cons work.
4 Marks Answer: Path dep, e.g. friction -μmg d. Dissipates heat. No U. Ex: Braking.
5.27 KE notions.
4 Marks Answer: ½mv² from WE. Ex: Raindrop, bullet. Scalar v². Real stream grind.
5.28 Variable graphs.
4 Marks Answer: Area ∫F dx. Rect sum limit. Ex: Linear trap. Multidim.
5.29 PE examples.
4 Marks Answer: Gravity mgh, bow elastic. Released KE. F=-dU. Earthquakes.
5.30 Elastic formulas.
4 Marks Answer: v1f= (m1-m2)/(m1+m2) u1 + ... Equal m exchange. Cons KE+mom. Deriv rel v.
5.31 Instant P.
4 Marks Answer: F v cosθ. Along v. Ex: Perp 0. Deriv dW/dt= F dr/dt.
5.32 Dot components.
4 Marks Answer: Sum A_i B_i. Unit i·i=1, i·j=0. Ex: 3D calc. Proj.
5.33 Inelastic loss.
4 Marks Answer: KE to heat/deform. v_com= (m1u1+m2u2)/(m1+m2). Max loss stick. Ex: Clay.
Part C: 8 Marks Questions (Detailed Long Answers - From NCERT)
5.1 Detailed: Work definition properties examples.
8 Marks Answer: W= F d cosθ= F·d scalar. Deriv: Proj F along d. Pos <90°, neg >90° friction, zero d=0/F=0/90° moon. Ex: Wall push d=0 tired internal; cyclist -2000J neg stop, road 0 no d Newton III unequal work. Units J= Nm, table alt eV cal. Graphs F-d area. Ties: Dot Ch3. Advanced: Line int. Pitfalls: Colloquial confuse. Real: Machines efficiency. Errors: Forget cos. Tips: Always θ specify, net for WE.
5.2 Detailed: WE theorem constant/variable derivation applications.
8 Marks Answer: Constant: v²-u²=2as ×m/2 ΔK=Fs= F·d. Variable: dK/dt=Fv, int ΔK=∫F·dr. Ex: Raindrop Wg=10J grav pos, Wr=-8.75J resist neg, ΔK=1.25J balance. Block inverse ln Kf=0.5J. Scalar loses dir/time vs F=ma vector instant. Useful energy changes. Graphs K-d. Ties: Newton integral. Advanced: Lagrangian. Real: Brakes calc dist. Errors: Net only. Tips: Components 2D.
5.3 Detailed: Work cases derivations graphs.
8 Marks Answer: W= F cosθ d. Deriv: Dot proj. Zero: Hold 150kg 30s d=0; frictionless block F_h=0; gravity horiz θ=90° cos0; moon circular radial-tang perp. Pos lift θ=0 mgh; neg friction 180° -f d heat. Ex: Cyclist road -2000J WE stop, cycle road 0 d=0 III forces opp work diff. Dimensions energy. Graphs F-d triangle area. Ties: Units table. Advanced: Virtual work. Real: Sports neg friction slow. Errors: Perp not zero. Tips: Sign conv.
5.4 Detailed: KE notions derivation examples table.
8 Marks Answer: K=½mv² from WE left side. Scalar v² dep. Deriv: Integrate ½m v dv= F dx. Ex: Bullet 50g 200m/s 1000J plywood 10% out vf=63m/s 68% red not 90% v not linear. Table car 3.5e5J, bullet 1e3J, stone 10J. Real: Stream KE grind, ships wind. Graphs v² line K. Ties: Ch2 speed. Advanced: Rel γmc². Pitfalls: Dir indep. Real: Athletics sprint. Errors: Units. Tips: ΔK calc.
5.5 Detailed: Variable work Riemann integral graphs.
8 Marks Answer: Constant rare; varying F(x) ΔW=FΔx rect sum W=∑FΔx → lim Δx=0 ∫F dx area curve. Ex: Woman trunk 100N-50N linear 20m f=50N WF=1000 rect +750 trap=1750J, Wf=-50*20=-1000J neg side. Fig5.3 rect to curve. Multidim ∫F·dr. Ties: Appendix calc. Advanced: Numerical trap. Real: Muscle fatigue var F. Graphs shaded areas. Errors: Neg areas. Tips: Plot first.
5.6 Detailed: WE variable proof limitations Ex.
8 Marks Answer: 1D dK/dt= m v dv/dt= v F= F dx/dt, dK=F dx int Ki to Kf ∫xi^xf F dx= ΔK. Ex: Block m=1kg vi=2 Fr=-0.5/x 0.1-2.01m ∫=0.5 ln20=1.5, Kf=2-1.5=0.5J vf=1m/s. Scalar no dir/time vs F=ma. Integral over int loses temporal. Useful variety probs. Graphs K-x. Ties: Chain rule. Advanced: Hamilton. Real: Thrust var. Errors: Limits. Tips: Subst u=dx.
5.7 Detailed: PE concept derivation gravity Ex.
8 Marks Answer: Stored by config/pos; appl to cons F work stored released KE. Gravity near Earth g const h<
5.8 Detailed: Cons mechanical E proof applications graphs.
8 Marks Answer: Cons forces W_c= -ΔU, net W=ΔK (WE), ΔK + ΔU=0 E=K+U const isolated. Proof: Path indep closed loop W=0=ΔK no ΔU. Ex: Pendulum PE max ends KE=0, bottom KE max PE min total const; roller no friction loops. Friction non-cons dissipates heat total mech dec. Graphs: E horizontal line, K+U curve. Ties: Isolated. Advanced: Noether time sym. Real: Orbits approx cons. Errors: Include all. Tips: Calc totals check.
5.9 Detailed: Spring PE deriv cons Ex.
8 Marks Answer: Hooke F=-kx cons, W_ext to stretch= ∫0^x kx dx= ½ k x²= U(x) zero eq. F=-dU/dx=-kx restor. Deriv: W_F= -∫F dx= -½kx², U= -W_F. Ex: Mass-spring pull x compress release osc E=½kx² max= ½ m v_max² cons. Period T=2π√(m/k) amp indep. Graphs: U parabolic min 0. Real: Shocks absorb, atoms bonds. Ties: Ch14 SHM. Advanced: Nonlinear k(x). Errors: x from eq. Tips: Quadratic U v_max=√(k/m) A.
5.10 Detailed: Power def deriv avg/instant Ex units.
8 Marks Answer: Rate dW/dt; instant P= F·v= F v cosθ along v. Deriv: W=∫F·dr=∫F v dt, dW/dt=Fv. Avg P= W/Δt total/time. Ex: Crane lift const v P=mg v; engine 100hp=74.6kW. Perp θ=90° P=0 no work rate. Units W=J/s, hp=746W hist horse lift 550ft lb/min. Graphs: P-t peaks. Ties: Var F. Advanced: Circ power torque ω. Real: Bulbs 60W light/heat. Errors: Avg vs inst. Tips: Cosθ include.
5.11 Detailed: Collisions types formulas 1D/2D e.
8 Marks Answer: Impulse ∫F dt=Δp cons mom always. Elastic cons KE+mom e=1 rel v sep=approach; inelastic mom only KE loss deform/heat e<1; perfect stick e=0 max loss. 1D elastic v1f=(m1-m2)u1/(m1+m2)+2m2 u2/(m1+m2) equal m exchange; inelastic v=(m1u1+m2u2)/(m1+m2). 2D components indep. Deriv: Cons + e= -(v2-v1)/(u2-u1). Ex: Billiard elastic, car crash inelastic. Graphs: v-t impulse. Real: Safety airbags inelastic slow. Errors: 2D forget comp. Tips: e calc rel.
5.13 Detailed: Variable work approx to int Ex graphs.
8 Marks Answer: F var Δx small F approx const ΔW=FΔx rect, sum W=∑ ΔW → lim Δx=0 ∫F dx def int area. Ex: Trunk linear F 100-50N 20m rect 1000 + trap 750=1750J f rect neg -1000J. Fig5.3 rect stack curve. Multidim ∫F·dr path. Ties: Riemann calc. Advanced: Monte Carlo. Real: Var thrust. Graphs: Shaded pos/neg. Errors: Sign areas. Tips: Trap (a+b)/2 h.
5.14 Detailed: PE cons F deriv Ex bow.
8 Marks Answer: Cons F path indep closed W=0 def U= -W_F path. General U(x)= -∫F dx. Gravity g const h<
5.15 Detailed: Cons E proof pend Ex graphs.
8 Marks Answer: Cons F W_c= -ΔU (def), net W=ΔK (WE), ΔK= -ΔU total const no non-cons. Proof: Closed path ∮F·dr=0=ΔK=0 no ΔU. Ex: Pend bob max h PE=mgh ends v=0, bottom KE= mgh PE=0 total mgh; roller loops if v_top²>5gr. Friction air/sound mech dec total cons heat. Isolated no ext. Graphs: E horiz, K sin U cos. Ties: Noether. Real: Swings approx. Errors: Air drag. Tips: Total calc.
5.16 Detailed: Spring PE deriv F U graphs Ex.
8 Marks Answer: Hooke linear F=-kx restor cons. W_ext stretch= ∫0^x k s ds= ½ k x² (s var)= U(x) zero unstretched. Deriv: W_spring= ∫F dx= -½ k x² (F=-ks), U= -W_F= ½ k x². F= -dU/dx= -k x. Ex: Block attach pull x release osc max U= ½ k A²= max KE= ½ m v_m² v_m= A √(k/m) total cons. T=2π √(m/k). Graphs: U parab min0, total E horiz. Real: Suspensions, molecules. Ties: SHM Ch14. Advanced: Damp k. Errors: x eq=0. Tips: Int limits.
5.17 Detailed: Power deriv Ex units conv graphs.
8 Marks Answer: Work rate P= dW/dt; W=∫F·dr=∫F·(v dt) dW= F·v dt P=F·v cosθ proj v. Avg P= ΔW/Δt total. Ex: Elevator m=1000kg h=10m t=20s P_avg= mgh/t=500W; const v up P= mg v. Perp F v=0 no rate. Units W=J/s= Nm/s, 1hp=746W horse lift 75kg 1m/s. Graphs: P-t var F peaks. Ties: Var W int. Advanced: Mech P= τ ω rot. Real: Fans 50W, cars 150hp. Errors: Avg total. Tips: Dot v.
5.18 Detailed: Collisions cons deriv formulas Ex 1D/2D.
8 Marks Answer: Short time large F impulse J=∫F dt= Δp cons mom always isolated. Elastic cons KE+mom reversible e=1 -(v2f-v1f)/(u2-u1); inelastic mom KE→other e<1; perfect stick e=0 v_f= (m1 u1 + m2 u2)/(m1+m2) max ΔKE. Deriv elastic: Mom m1 u1 + m2 u2= m1 v1 + m2 v2; rel v cons u1-u2= v2-v1 (e=1); solve v1= (m1-m2)u1/(sum) + 2 m2 u2/sum equal exchange. 2D comp indep. Ex: Balls equal m elastic exchange v; cars inelastic common v. Graphs: p cons line, KE elastic same. Real: Airbags slow inelastic. Errors: 2D comp. Tips: e rel.
5.19 Detailed: Dot deriv prop Ex applications.
8 Marks Answer: Scalar A·B= A (B cosθ) proj B on A= sum A_i B_i comp. Deriv: Geom triangle cos; algebra (i·i=1 etc ortho basis). Prop: Comm swap; dist A·B + A·C= A·(B+C); λ A·(λB)=λ A·B. A·A= A² |A|; perp cos90=0 dot0; parallel ±AB. Ex: Ex5.1 F=3i+4j-5k d=5i+4j+3k dot=15+16-15=16 cos=16/√50 √50=0.32 θ=71°. Graphs: Fig5.1 proj lines. Ties: Work P. Advanced: Gram-Schmidt ortho. Real: ML dot sim. Errors: θ between. Tips: Comp 3D.
5.20 Detailed: WE var proof Ex limitations.
8 Marks Answer: Var F dK= m v dv= F dx (v=d x/dt) int Ki^Kf= ∫xi^xf F dx ΔK=W. Proof: Chain dv/dt= (dv/dx)(dx/dt)= v dv/dx= F/m, m v dv= F dx. Ex: Block rough Fr=-k/x k=0.5 ∫0.1^2.01 k/x dx= k ln(20.1)=1.5 ΔK=-1.5 Kf=2-1.5=0.5J. Limitations: Scalar no dir vec F=ma; int over path loses time explicit Newton. Useful non-time probs. Graphs: K-x curve. Ties: Diff eq. Advanced: Energy methods. Real: Var accel. Errors: Int dir. Tips: Sub ln.
5.21 Detailed: Units alt table Ex conv.
8 Marks Answer: Work/energy [ML²T⁻²] SI J=1N·m=1kg m²/s². Table: 1eV=1.6e-19J electron volt; 1cal=4.18J heat; 1kWh=3.6e6J elec; 1kW h=3.6e6J. Ex: Bullet KE=1000J= 1000/4.18≈239 cal; raindrop 1.25J=1.25/1.6e-19≈7.8e18 eV. Hist Joule paddle heat=mech. Ties: Thermo equiv. Graphs: Unit scales. Advanced: erg=1e-7J CGS. Real: Food cal=4.18 kJ. Errors: Prefixes. Tips: Conv factors.
5.22 Detailed: e def deriv collisions types.
8 Marks Answer: Coeff restit e= rel v sep / rel v app= -(v2f - v1f)/(u2 - u1). Deriv: From impulse def, elastic e=1 rel cons, inelastic <1 loss. Types: e=1 perfect elastic KE+mom cons; 0
5.23 Detailed: Spring U F deriv SHM ties.
8 Marks Answer: Linear spring F=-k x restor prop deform cons. U= work store= ∫0^x k s ds= ½ k x² (s dummy). Deriv: W_s= ∫ F dx= ∫ -k x dx= -½ k x², U= - W_F (F cons)= ½ k x² zero x=0 eq. F= - dU/dx= -k x. Ex: Block m pull A release SHM max U= ½ k A² bottom= max KE= ½ m (ω A)^2 ω=√(k/m) total E cons amp indep. T=2π/ω. Graphs: U parab, pos sin, total const. Ties: Ch14 osc. Real: Clocks, bridges damp. Advanced: Var k anhar. Errors: x signed. Tips: Energy ½ k A²= ½ m v_m².
5.24 Detailed: Power real Ex deriv units hp.
8 Marks Answer: Energy rate P= dE/dt= dW/dt; mech P= F · v= F v cosθ proj v dir. Deriv: W(t)= ∫ F · dr= ∫ F · v dt, diff P= F · v. Avg ΔW/Δt finite. Ex: Man lift 50kg h=2m t=4s P_avg= m g h /t= 245W; const v up P= m g v= 122W if v=0.5m/s. θ=90° F perp v P=0. Units W= J/s= V A elec; 1hp=746W=550 ft lb/s horse hist. Graphs: P-t engine var. Ties: Rot P=τ ω Ch7. Advanced: Eff= P_out/P_in. Real: Bike 200W human, solar panel kW. Errors: Avg total W. Tips: Cos include.
5.25 Detailed: Cons E deriv proof pend roller Ex.
8 Marks Answer: Mech E= K + U cons if only cons forces no dissip. Deriv: Cons F def W_c path indep closed ∮ dr · F=0; gen W_c= - ΔU (U= -∫ F dr); net W= W_c (no non-cons)= ΔK (WE thm); ΔK= -ΔU total ΔE=0. Proof: Isolated sys no ext work. Ex: Simple pend small amp approx cons max PE= m g l (1-cosθ) ends v=0, bottom KE= m g l (1-cosθ) PE=0; roller coaster hill PE→KE loop if top v²> g r (cent). Friction mech→heat total cons. Graphs: E line, K+U osc. Ties: Isolated. Advanced: Symmetry cons. Real: Sat orbits. Errors: Air. Tips: Δ totals zero.
5.26 Detailed: Non-cons work path dep Ex friction.
8 Marks Answer: Non-cons F work path dep closed ∮ F·dr ≠0 dissipates mech other forms heat/sound. No def U. Ex: Friction f=μ N W_f= - f d cos180= -μ m g d always neg indep θ but path total d longer more loss; block rough patch ∫ f dx= -f Δx. Cons gravity W= -m g Δh height only path indep. Deriv: Hysteresis loop area loss. Graphs: F-d cycle area work. Ties: Total E cons universe. Advanced: Viscous drag. Real: Brakes heat, tires wear. Errors: Assume cons. Tips: Calc total path for f.
5.27 Detailed: KE intuitive deriv table Ex.
8 Marks Answer: KE= work motion can do; intuitive fast flow grind corn wind sails. Deriv: WE thm left ½ m (v² - u²)= W right. Scalar ½ m v² zero rest v² dep. Ex: Raindrop fall h=1km v=50m/s ΔK=1.25J from rest; bullet 50g 200m/s 1000J plywood 2cm 10% KE out vf=√(2*100/0.05)=63m/s red 68% v√KE. Table: Car 70kg 25m/s 2e4J wait 3.5e5? 70*625/2=2e4 no car mass 1000kg 25m/s 3e5J; bullet 1e3; stone 10m 14m/s 1J; rain term vel 9m/s 0.04J; air mol 500m/s 3e-21J. Graphs: K-v parab. Ties: Thermo mol KE. Real: Tsunamis KE. Errors: Linear v. Tips: ½ forget.
5.28 Detailed: Variable F graphs Riemann Ex woman.
8 Marks Answer: Real F var; small Δx F approx const ΔW= F Δx rect area, total W= ∑ rect → lim n→∞ Δx→0 Riemann sum= def ∫ F(x) dx curve area. Ex: Woman trunk rough 10m F=100N then linear to 50N next 10m total 20m f=50N const opp; plot F pos applied neg f; WF= rect 100*10=1000 + trap (100+50)/2 *10=750 total 1750J; Wf= rect -50*20=-1000J neg axis. Fig5.3 stack rect curve. Multidim line int path. Ties: Calc approx. Advanced: Gauss quad. Real: Muscle var. Graphs: Linear slope trap. Errors: Avg f. Tips: Area calc trap (a+b)h/2.
5.29 Detailed: PE stored released Ex gravity bow.
8 Marks Answer: PE capacity action stored config/pos; appl cons F work against stored on release KE. Def U= - W_F (F cons path indep). Gravity near surf g=const h<
5.30 Detailed: Elastic collision deriv formulas equal m Ex.
8 Marks Answer: Collision short Δt large F impulse J= ∫ F dt ≈ F_avg Δt= Δp cons mom isolated m1 u1 + m2 u2= m1 v1 + m2 v2. Elastic add cons KE ½ m1 u1² + ½ m2 u2²= ½ m1 v1² + ½ m2 v2² + e=1 rel v cons u1 - u2= v2 - v1 (approach sep). Solve sys: v1f= (m1 - m2) u1 / (m1 + m2) + 2 m2 u2 / sum; v2f= (m2 - m1) u2 / sum + 2 m1 u1 / sum. Equal m1=m2 v1f= u2, v2f= u1 exchange v dir. Deriv: From rel + mom. Ex: Billiard equal mass head-on strike stop shooter, target takes v. 2D comp along normal e, tang undeform. Graphs: v before after. Real: Nucleus scatter. Errors: Rel sign. Tips: Solve linear.
5.31 Detailed: Instant power deriv Ex lift perp.
8 Marks Answer: Power rate energy transfer P= dE/dt= dW/dt mech. Instant P= d(∫ F·dr)/dt= F · (dr/dt)= F · v cosθ proj F on v or v on F. Deriv: Chain W(t)= ∫_0^t F · v(t') dt' diff P= F · v. Avg finite ΔW/Δt= total W / time. Ex: Worker lift m=50kg h=3m t=5s const accel? Avg P= m g h / t ≈ 294W; const v= h/t=0.6m/s P_inst= m g v= 294W. Perp ex: Satellite orbit gravity F radial v tang θ=90° cos0 P=0 no work. Graphs: P-t var v peaks. Ties: Rot analog. Advanced: Eff P_out/in. Real: Human 100W sustain. Errors: Avg inst confuse. Tips: Dot essential.
5.32 Detailed: Dot components deriv unit Ex 3D.
8 Marks Answer: Dot scalar mult vec to scalar A·B= |A| |B| cosθ angle between= proj A on B × |B|. Deriv: Geom cos law triangle; algebra ortho basis i j k unit perp i·i=1 i·j=0 expand (A_x i + A_y j + A_z k) · (B_x i + ...)= A_x B_x + A_y B_y + A_z B_z comp sum. Prop: Comm dot sym; dist linear; λ scalar. A·A= A²= x²+y²+z² mag sq; perp θ=90° cos0=0 dot0; parallel θ=0/180 ±AB. Ex: 3D F=3i+4j-5k |F|=√(9+16+25)=√50 d=5i+4j+3k |d|=√50 dot=15+16-15=16 cos=16/50=0.32 θ=cos^{-1}0.32≈71°. Graphs: Fig5.1 proj dashed. Ties: Work cos. Advanced: Ortho proj. Real: Signal corr. Errors: θ obtuse neg. Tips: Comp calc easy.
5.33 Detailed: Inelastic KE loss deriv perfect Ex cars.
8 Marks Answer: Inelastic collision cons mom isolated but KE not→ internal heat deform sound. Impulse J=Δp cons m1 u1 + m2 u2= m1 v1 + m2 v2. Partial e<1 rel v sep= e approach; perfect e=0 v1f= v2f= v= (m1 u1 + m2 u2)/(m1 + m2) stick max ΔKE= ½ μ (u1 - u2)^2 μ red mass loss. Deriv: From e=0 rel v_f=0. Ex: Two cars m=1000kg u1=10m/s u2=0 crash v=5m/s initial KE=½*1000*100=5e4J final ½*2000*25=2.5e4J loss 2.5e4J heat. Graphs: KE before after drop. Real: Clay drop stick e=0; crashes airbags slow inelastic min injury. Errors: Assume elastic. Tips: ΔKE= initial - final pos loss.
Tip: Include diagrams/eqs in long; practice exercises.
Key Concepts - In-Depth Exploration
Core ideas with derivations, examples, common pitfalls, interlinks (4+ pages sim). Emphasize theorems, conservation, variable forces.
Scalar Product
Deriv: Geom proj AB cosθ= comp sum. Pitfall: θ>90 neg. Ex: Work F·d. Interlink: Ch3 vectors.
Work Definition
Deriv: F cosθ d proj. Pitfall: Zero cases forget. Ex: Friction neg. Interlink: Energy dim.
WE Theorem Constant
Deriv: Kin ×m/2 ΔK=Fs. Pitfall: Net F. Ex: Raindrop balance. Interlink: Ch3 kin.
Advanced: Rel E=γmc². Pitfalls: Cons only mech; int signs. Interlinks: Ch3 dot work; Ch11 thermo total E. Real: Renewables cons solar→elec. Depth: Var g PE -GMm/r. Examples: Ex5.2 resist; Ex5.6 ln. Graphs: U-K pend sin. Calculus: dK=Fv dt. Errors: Units mix. Tips: Cons check Δ=0; plot F-x area.
Extended: Conceptual: First law thermo ΔU=Q-W. Math: Hamilton E=T+V. Applications: Game phys energy. Common: Forget net W. Principles: Energy forms interconv cons total. Advanced: Field theory PE. Vector fields preview.
Extended: Error %ΔK ≈ %W. Pitfalls: Signs U. Real: Softw energy calc.
Applications & Real-Life Relevance
Automobiles: Brakes WE neg ΔK heat; engines power hp.
Renewables: Hydro PE→KE turbine work; wind KE blades.
Sports: Golf elastic collision club ball; running KE.
Engineering: Springs dampers U absorb; cons eff machines.
Earthquakes: Fault PE release seismic KE.
Tip: Relate energy crisis. Depth: EVs battery PE chem→elec KE. Climate: Solar cons.
Global: Power grids W. Extended: Fusion E cons.
30 Solved Numerical Problems - Step by Step from NCERT & Variations
Based on NCERT exercises (5.7,5.10,5.16,5.20,5.24,5.29) and chapter examples/variations for work, energy, power, collisions. g=9.8 m/s² or 10. Step-by-step with eqs.
1. NCERT 5.7: Forces 50N θ=60°, W if d=5m?
Step 1: For work, but resultant R=√(50²+50²+2*50*50*cos60°)=50√3 N, but W=R d cos0 if along= 50√3 *5 ≈433N m=433J.
Wall push 0W tired internal; UCM gravity 0W perp; cons only mech; var int ln inverse.
Mnemonics: Work "Force Dot Displacement" (FDD). WE "Delta K Equals W Net" (DKEWN). Cons "K Plus U Constant" (KPUC). Power "Force Dot Velocity" (FDV). Spring "Half K X Squared" (HKXS). Collisions "Momentum Always Elastic Maybe" (MAEM).
Tip: g=10 approx; practice 5 probs daily; units J consistent.
