power and efficiency - physicstutor.kumarphysicsclasses.com

Power

Power is the rate of doing work or the rate of energy transfer.

Definition

P = W/t

P = power
W = work done or energy transferred
t = time taken
SI unit: watt, W
Dimension: [M L² T^-3]

Fast vs Slow Work

Average Power

Average power is total work divided by total time.

Formula

P_avg = Total Work / Total Time

Useful when power changes during motion but total energy and total time are known.

Solved Examples

A machine does 600 J work in 20 s. Find average power.

Pavg=W/t=600/20=30 W. Exam tip: watt means joule per second.

A lift gains 6000 J energy in 10 s. Find power.

P=6000/10=600 W. Diagram: lift moving upward.

Instantaneous Power

Instantaneous power is the rate of work at a particular instant.

Derivation

P = dW/dt

Since dW = F . ds,

P = F . ds/dtP = F . v

Vector Meaning

Only the component of force along velocity contributes to power.

P = Fv cosθ

Force and Velocity

Horsepower

Horsepower is a practical unit of power used for engines, motors and cars.

Conversion

1 HP = 746 W

Higher horsepower means higher rate of energy transfer, not necessarily higher efficiency.

Comparison Table

Device	Typical Use	Power Unit
Car engine	Acceleration and pulling load	HP or kW
Electric motor	Mechanical output	W or kW
Pump	Lifting water	W or HP

Efficiency

Efficiency compares useful output to total input.

Energy Efficiency

η = Useful Output Energy / Input Energy

Power Efficiency

η = Useful Output Power / Input Power

Percentage

η% = (Output/Input) × 100

Efficiency is always less than or equal to 100% for real machines.

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Applications

Application cards with concept, formula, diagram idea and exam tip.

Electric Motor

Concept: electrical power to mechanical power.

η=Pout/Pin

Exam tip: output is useful shaft power.

Pump

Concept: lifting water.

P=mgh/t

Tip: use mass per second if flow rate is given.

Lift

Concept: raises load.

P=mgv

Tip: constant speed means power against weight.

Car Engine

Concept: force at speed.

P=Fv

Tip: use velocity in m/s.

Generator

Concept: mechanical to electrical power.

η=Electrical output/Mechanical input

Bicycle

Concept: rider power overcomes resistive forces.

P=Fv

Escalator

Concept: raises many people per second.

P=nmg h/t

Crane

Concept: lifting heavy load.

P=W/t=mgh/t

Hydroelectric Plant

Concept: GPE of water to electrical energy.

P=ηρQgh

Thermal Power Plant

Concept: heat energy to electrical output.

η=Pout/Pin

Important Graphs

SVG graph summaries for power, work and efficiency.

Power vs Time

Area under P-t graph gives work/energy.

Power vs Velocity

For constant force, P is proportional to velocity.

Work vs Time

Slope of W-t graph is power.

Efficiency Comparison

Taller bar means higher efficiency.

High-Quality Numericals

Solved numerical bank with question, diagram idea, given, formula, calculation, answer and exam tip.

CBSE: A machine does 1200 J work in 30 s. Find power.

Diagram: machine lifting load. Given W=1200 J, t=30 s. Formula P=W/t. Calculation=40 W. Final answer: 40 W. Exam tip: watt = J/s.

NEET: A force of 50 N moves a body at 4 m/s in same direction. Find power.

P=Fv=50x4=200 W. Tip: use component of force along velocity.

JEE Main: Engine power is 10 kW and speed is 20 m/s. Find driving force.

P=Fv, F=P/v=10000/20=500 N. Tip: kW to W.

JEE Advanced: Power varies as P=3t² from 0 to 4 s. Find work.

W=∫Pdt=∫0^4 3t²dt=t³|0^4=64 J. Tip: area under P-t graph.

IB: Motor input 500 W, output 350 W. Find efficiency.

η=350/500=0.7=70%. Tip: efficiency has no unit.

IGCSE: Lift raises 200 N load through 5 m in 10 s. Power?

Work=200x5=1000 J. Power=1000/10=100 W.

A-Level: Pump lifts 0.02 m³/s water through 10 m, η=80%. Find input power. ρ=1000, g=10.

Useful power=ρQgh=1000x0.02x10x10=2000 W. Input=2000/0.8=2500 W.

NEET Question Bank

50 high-quality NEET-style MCQs.

1. NEET Exam-style Question: SI unit of power is: A joule B watt C newton D pascal

Answer: B. Power is measured in watts.

2. Formula for average power is: A Wt B W/t C Ft D mv

Answer: B.

3. Instantaneous power equals: A F/v B F.v C mv D mgh

Answer: B.

4. 1 horsepower equals: A 746 W B 100 W C 9.8 W D 1 W

Answer: A.

5. Efficiency is: A input/output B output/input C force/time D work x time

Answer: B.

6. Power dimension is: A ML²T^-2 B ML²T^-3 C MLT^-2 D LT^-1

Answer: B.

7. 100 J work in 5 s gives power: A 10 B 20 C 50 D 500 W

Answer: B.

8. Force 10 N, speed 3 m/s same direction, power: A 3 B 10 C 30 D 300 W

Answer: C.

9. Force perpendicular to velocity gives power: A zero B Fv C infinite D negative only

Answer: A.

10. Useful output 60 J, input 100 J, efficiency: A 40% B 60% C 100% D 160%

Answer: B.

11. Area under P-t graph gives: A force B work C speed D mass

Answer: B.

12. Slope of W-t graph gives: A power B work C displacement D acceleration

Answer: A.

13. If work doubles in same time, power: A half B same C double D four times

Answer: C.

14. If same work is done in half time, power: A half B double C same D zero

Answer: B.

15. A real machine efficiency is usually: A >100% B 1000% C <100% D negative

Answer: C.

16. Power is rate of: A displacement B work C mass D force

Answer: B.

17. P=Fv is valid for: A component along velocity B any perpendicular force C mass only D no motion

Answer: A.

18. 2 HP equals approximately: A 746 W B 1492 W C 373 W D 200 W

Answer: B.

19. 746 W equals: A 1 HP B 2 HP C 0.5 HP D 10 HP

Answer: A.

20. Efficiency has unit: A W B J C no unit D N

Answer: C.

21. Motor input 1000 W, efficiency 80%, output: A 800 W B 80 W C 1250 W D 200 W

Answer: A.

22. Output 500 W, efficiency 50%, input: A 250 W B 500 W C 1000 W D 1500 W

Answer: C.

23. Lift mass m at speed v, useful power: A mgv B mgh C mv² D F/v

Answer: A.

24. Car engine at constant speed against resistive force F has power: A Fv B F/v C v/F D zero

Answer: A.

25. Work 300 J in 15 s: power A 15 B 20 C 30 D 45 W

Answer: B.

26. P-t rectangle height 10 W base 5 s gives work: A 2 B 5 C 50 D 100 J

Answer: C.

27. P-t triangle base 4 s height 20 W gives work: A 20 B 40 C 80 D 100 J

Answer: B.

28. Force opposite velocity gives power: A positive B negative C zero D infinite

Answer: B.

29. Brakes have instantaneous power sign generally: A positive B negative C zero D no unit

Answer: B.

30. Engine accelerating car has power generally: A positive B negative C zero D undefined

Answer: A.

31. 1 watt equals: A 1 J/s B 1 N/s C 1 kg/s D 1 m/s

Answer: A.

32. Kilowatt is: A 10 W B 100 W C 1000 W D 746 W

Answer: C.

33. If P=20 W for 10 s, work: A 2 J B 20 J C 100 J D 200 J

Answer: D.

34. Constant power P over time t gives work: A Pt B P/t C t/P D zero

Answer: A.

35. P=4t from 0 to 5 s, work: A 20 B 50 C 100 D 25 J

Answer: B. Integral 4t dt = 50.

36. η=0.75 as percentage: A 7.5% B 75% C 750% D 0.75%

Answer: B.

37. Useful output cannot exceed input for real machine due to: A losses B no friction C zero work D reference

Answer: A.

38. Pump useful power for water flow is related to: A ρQgh B mv C kx D p²/2m

Answer: A.

39. Escalator power increases if number of passengers: A decreases B increases C zero D unrelated

Answer: B.

40. Generator converts mechanical power into: A electrical power B mass C displacement D friction only

Answer: A.

41. Thermal plant efficiency uses useful output as: A electrical energy B wasted heat C smoke D friction

Answer: A.

42. P=Fv cosθ, θ=60, F=10, v=4. Power: A 10 B 20 C 40 D 80 W

Answer: B.

43. θ=90 in P=Fvcosθ gives: A Fv B zero C -Fv D infinite

Answer: B.

44. θ=180 gives power: A positive B negative C zero D maximum positive

Answer: B.

45. A 100 W bulb for 5 s consumes energy: A 20 J B 100 J C 500 J D 1000 J

Answer: C.

46. Energy equals: A P/t B Pt C F/v D η/t

Answer: B for constant power.

47. Average power over complete trip is total work divided by: A total time B initial time C force D velocity only

Answer: A.

48. Instantaneous power may vary even if average power is fixed. A true B false C impossible D no unit

Answer: A.

49. In P=Fv, v must be in: A km/h always B m/s for SI C cm only D no unit

Answer: B.

50. Horsepower is a unit of: A power B work C force D momentum

Answer: A.

JEE Main Question Bank

50 difficult JEE Main questions.

1. A lift raises 500 kg at 2 m/s. Power g=10?

P=mgv=500x10x2=10000 W.

2. Car power 20 kW at 10 m/s. Driving force?

F=P/v=20000/10=2000 N.

3. P=5t² from 0 to 3 s. Work?

W=∫5t²dt=5x27/3=45 J.

4. Force F=20 N at 60° to velocity 5 m/s. Power?

P=Fvcos60=20x5x0.5=50 W.

5. Motor input 2 kW, efficiency 75%. Output?

Pout=0.75x2000=1500 W.

6. Output 3 kW, efficiency 60%. Input?

Pin=3000/0.6=5000 W.

7. Pump lifts 100 kg water per second through 20 m, g=10. Useful power?

P=mgh/t=100x10x20=20000 W.

8. Same pump 80% efficient. Input power?

Pin=20000/0.8=25000 W.

9. Engine does 6x10^5 J work in 1 min. Power?

P=600000/60=10000 W.

10. Convert 5 HP to W.

5x746=3730 W.

11. Convert 1492 W to HP.

1492/746=2 HP.

12. P=Fv, F=kv, find P.

P=kv².

13. Retarding force bv at speed v. Power loss?

P=Fv=bv², negative for object: -bv².

14. Constant power P accelerates mass from rest. KE after time t?

K=Pt.

15. Constant power P, speed after time t from rest?

1/2mv²=Pt, v=√(2Pt/m).

16. Work-time graph W=3t². Power at t=4?

P=dW/dt=6t=24 W.

17. Work-time graph W=10t. Power?

P=10 W.

18. Force vector 3i+4j, velocity 2i. Power?

P=F.v=6 W.

19. Force 3i+4j, velocity 4i-3j. Power?

P=12-12=0.

20. Force 5i, velocity -2i. Power?

P=-10 W.

21. A crane lifts 1000 N load at 0.5 m/s. Power?

P=Fv=1000x0.5=500 W.

22. A 60% efficient crane output 6 kW. Input?

10 kW.

23. 100 W device runs 2 h. Energy in kWh?

0.1 kW x 2 h = 0.2 kWh.

24. 2 kW heater runs 30 min. Energy?

2 kW x 0.5 h=1 kWh.

25. P-t triangle base 10 s height 50 W. Work?

250 J.

26. P-t trapezium sides 20 W, 40 W, width 5 s. Work?

150 J.

27. If velocity doubles at same force, power?

Doubles.

28. If force doubles at same velocity, power?

Doubles.

29. If force and velocity both double, power?

Four times.

30. A car climbs slope gaining height 100 m in 50 s, mass 1000 kg. Useful power g=10?

P=mgh/t=1000x10x100/50=20000 W.

31. Efficiency 25%, input 800 J. Useful output?

200 J.

32. Efficiency 40%, useful output 200 J. Input?

500 J.

33. Motor wastes 200 W and output 800 W. Efficiency?

Input=1000 W, η=80%.

34. Device input 500 W, efficiency 70%. Wasted power?

Output=350 W, wasted=150 W.

35. P=2t+3 from 0 to 5. Work?

W=[t²+3t]0^5=25+15=40 J.

36. P=12 constant, work in 8 s?

96 J.

37. P=Fvcosθ, θ=120, F=10, v=2.

P=10x2x(-1/2)=-10 W.

38. A lift moves down at constant speed while motor supports it. Sign of motor power on lift?

Negative if upward motor force and downward velocity.

39. A cyclist overcomes 40 N resistance at 8 m/s. Power?

320 W.

40. Generator mechanical input 5 kW, electrical output 4 kW. Efficiency?

80%.

41. Hydroelectric plant η=0.9, ρQgh=100 MW. Output?

90 MW.

42. Output 90 MW, η=0.9. Water useful input?

100 MW.

43. Power required to lift mass m through height h in time t.

P=mgh/t.

44. Same lifting with efficiency η. Input power?

Pin=mgh/(ηt).

45. Instantaneous power if F=10 N, v=0 at instant.

P=0.

46. P=dK/dt for net power. If K=5t², net power at t=3?

P=10t=30 W.

47. Work done by 4 kW engine in 15 s.

W=4000x15=60000 J.

48. Power needed for 3000 J in 2 min.

P=3000/120=25 W.

49. P=Fv, constant power means force varies as?

F=P/v, inversely with velocity.

50. At high speed for constant power engine, available driving force?

Decreases as F=P/v.

JEE Advanced Question Bank

50 difficult JEE Advanced questions on P=Fv, variable power, efficiency and energy transfer.

1. Net power P0 constant from rest. Find v(t).

P0=dK/dt=d(1/2mv²)/dt. Thus 1/2mv²=P0t, v=√(2P0t/m).

2. Constant power P0. Find acceleration as function of v.

P=Fv=mav, so a=P0/(mv).

3. Resistive force bv, speed v. Power dissipated?

P_loss=bv².

4. Resistive force cv², speed v. Power dissipated?

P_loss=cv³.

5. Engine constant power P against resistance bv. Terminal speed?

At terminal, P=bv², so v=√(P/b).

6. Engine constant power P against resistance cv². Terminal speed?

P=cv³, so v=(P/c)^(1/3).

7. Force F=αx and velocity v=βx. Instantaneous power?

P=Fv=αβx².

8. Work if power P=kt³ from 0 to T.

W=∫0^T kt³dt=kT⁴/4.

9. Kinetic energy K=at³. Net power?

P=dK/dt=3at².

10. Velocity vector perpendicular to force vector. Power?

Zero because P=F.v.

11. Force is central radial and velocity tangential. Power?

Zero instantaneously.

12. Force is -bv and velocity v. Power by force?

P=F.v=-bv².

13. Variable efficiency η(t), input power Pin constant. Useful energy 0 to T?

Eout=∫0^T η(t)Pin dt.

14. η=η0 t/T, Pin=P0, 0 to T. Useful energy?

E=P0η0∫0^T t/T dt=P0η0T/2.

15. Pump flow rate Q, height h, efficiency η. Input power?

Pin=ρQgh/η.

16. Lift mass m at acceleration a and velocity v upward. Instantaneous power?

Force=m(g+a), P=m(g+a)v.

17. Lift descending with upward tension T and downward velocity v. Power by tension?

P=T v cos180=-Tv.

18. A crane lifts variable mass flow λ kg/s through height h. Useful power?

P=λgh.

19. A particle has position x=t² and force F=3t. Power?

v=2t, P=Fv=6t².

20. Work from previous from 0 to T?

W=∫0^T 6t²dt=2T³.

21. Force vector (2t)i+3j, velocity 4i+t j. Power?

P=8t+3t=11t.

22. Work 0 to T for previous.

W=∫0^T 11t dt=11T²/2.

23. Constant power P moves body along rough horizontal with μ. Equation for K?

dK/dt=P-μmgv, with v=√(2K/m).

24. Maximum speed with constant power P against constant resistance R?

At terminal P=Rv, so v=P/R.

25. Power needed to move at constant speed v up incline angle θ without friction.

P=mgv sinθ.

26. Same with friction coefficient μ.

P=mgv(sinθ+μcosθ).

27. Power required to maintain circular motion by radial force?

Zero for radial centripetal force because perpendicular to velocity.

28. Tangential force Ft in circular motion speed v. Power?

P=Ftv.

29. Motor delivers torque τ at angular speed ω. Power?

P=τω.

30. Rotating motor 100 N m at 20 rad/s. Power?

2000 W.

31. Efficiency with rotational output τω and electrical input VI.

η=τω/(VI).

32. Generator output VI, mechanical input τω. Efficiency?

η=VI/(τω).

33. Force F=F0 cosωt, velocity v0 cosωt. Average power over cycle?

Average of F0v0 cos²ωt is F0v0/2.

34. Force F0 sinωt, velocity v0 cosωt. Average power?

Average of sinωt cosωt over cycle is zero.

35. P-t graph below axis means?

Negative work or energy removed from system.

36. Net power is dK/dt. If speed constant, net power?

Zero, though individual powers may be non-zero.

37. Engine power positive and friction power negative at constant speed. Relation?

They cancel: P_engine + P_friction = 0.

38. Useful power to raise water at mass rate ṁ.

P=ṁgh.

39. If useful power fixed, time to do work W?

t=W/P.

40. Machine has input energy Ein and loss Eloss. Efficiency?

η=(Ein-Eloss)/Ein.

41. If losses are 20% of input, efficiency?

80%.

42. Power in terms of momentum and force?

P=F.v=F.p/m for constant mass.

43. If p changes under constant force F, instantaneous power?

P=F p/m along same direction.

44. Power needed for vertical climb speed v with drag D.

P=(mg+D)v.

45. Power produced by gravity on falling mass at speed v.

P=mgv if falling downward.

46. Power by gravity on upward moving mass speed v.

P=-mgv.

47. Average power over interval equals?

Total work over total time.

48. Instantaneous power equals limit of?

lim Δt→0 ΔW/Δt.

49. If P=av² and v=bt, work from 0 to T?

P=ab²t², W=ab²T³/3.

50. If P=Fv and force is fixed, P-v graph slope?

Slope is F.

IB / IGCSE / A-Level Questions

Separate question sets with answers and explanations.

IB Questions - 25

IB 1. Define power.

Rate of doing work or energy transfer.

IB 2. SI unit of power.

Watt.

IB 3. Formula for average power.

Pavg=W/t.

IB 4. Formula for instantaneous power.

P=dW/dt=F.v.

IB 5. Define efficiency.

Useful output divided by input.

IB 6. 500 J in 10 s power?

50 W.

IB 7. Force 20 N, speed 3 m/s power?

60 W if parallel.

IB 8. What is 1 HP?

746 W.

IB 9. Efficiency 0.8 as percent.

80%.

IB 10. Area under P-t graph.

Work or energy transferred.

IB 11. Slope of W-t graph.

Power.

IB 12. Power by perpendicular force.

Zero.

IB 13. Power by friction sign.

Usually negative.

IB 14. Motor input 200 W output 150 W efficiency.

75%.

IB 15. Output 100 W efficiency 50% input.

200 W.

IB 16. Dimension of power.

ML²T^-3.

IB 17. Why efficiency less than 100%?

Energy losses such as heat and sound.

IB 18. Lift power at constant speed.

P=mgv.

IB 19. Pump useful power.

ρQgh.

IB 20. P=4t, work 0 to 2.

8 J.

IB 21. Work if power 30 W for 4 s.

120 J.

IB 22. Is horsepower SI?

No, watt is SI.

IB 23. Engine at constant speed against resistance R.

P=Rv.

IB 24. Generator efficiency formula.

Electrical output/mechanical input.

IB 25. Net power relation to KE.

Pnet=dK/dt.

IGCSE Questions - 25

IGCSE 1. Power means?

Rate of doing work.

IGCSE 2. Unit of power?

Watt.

IGCSE 3. Formula power.

Power=work/time.

IGCSE 4. 100 J in 5 s.

20 W.

IGCSE 5. 60 W for 10 s energy?

600 J.

IGCSE 6. Efficiency formula.

Useful output/input.

IGCSE 7. Efficiency as percent.

(Useful output/input)x100.

IGCSE 8. Input 100 J output 70 J.

70%.

IGCSE 9. Unit watt equals?

J/s.

IGCSE 10. Work done by 200 W in 3 s.

600 J.

IGCSE 11. Machine output less than input because?

Energy losses.

IGCSE 12. Lift weight 400 N at 2 m/s.

800 W.

IGCSE 13. Pump lifts water, useful energy becomes?

Gravitational potential energy.

IGCSE 14. Horsepower is unit of?

Power.

IGCSE 15. 1 HP is about?

746 W.

IGCSE 16. Faster machine doing same work has?

Greater power.

IGCSE 17. Same power for longer time transfers?

More energy.

IGCSE 18. 1 kW equals?

1000 W.

IGCSE 19. 2 kW for 1 hour energy?

2 kWh.

IGCSE 20. Useful output 30 J input 60 J.

50%.

IGCSE 21. Efficiency can be 120%?

No.

IGCSE 22. A motor changes electrical to?

Mechanical energy.

IGCSE 23. Generator changes mechanical to?

Electrical energy.

IGCSE 24. Crane useful work is?

Lifting load against gravity.

IGCSE 25. Power is scalar?

Yes.

A-Level Questions - 25

A-Level 1. Derive P=F.v.

P=dW/dt=F.ds/dt=F.v.

A-Level 2. F=3i+4j, v=5i. Power?

15 W.

A-Level 3. F=3i+4j, v=4i-3j.

0 W.

A-Level 4. Net power equals?

dK/dt.

A-Level 5. Constant power acceleration from rest.

v=√(2Pt/m).

A-Level 6. Force from constant power.

F=P/v.

A-Level 7. Pump input with efficiency η.

Pin=ρQgh/η.

A-Level 8. Lift accelerating upward power.

P=m(g+a)v.

A-Level 9. Work from P-t graph.

Area under graph.

A-Level 10. Power from W-t graph.

Gradient.

A-Level 11. P=t², work 0 to 3.

9 J.

A-Level 12. P=6t, work 0 to 2.

12 J.

A-Level 13. Torque power relation.

P=τω.

A-Level 14. Torque 50 Nm, ω=10.

500 W.

A-Level 15. Electrical input power.

P=VI.

A-Level 16. Efficiency motor.

Mechanical output/electrical input.

A-Level 17. Resistive power for drag kv².

kv³.

A-Level 18. Constant speed engine power.

Equals resistive force times speed.

A-Level 19. Up slope power no friction.

mgv sinθ.

A-Level 20. Up slope with friction.

mgv(sinθ+μcosθ).

A-Level 21. Gravity power falling.

mgv.

A-Level 22. Gravity power rising.

-mgv.

A-Level 23. Average power formula.

Total work/total time.

A-Level 24. Instantaneous power formula.

dW/dt.

A-Level 25. Efficiency with losses L from input E.

η=(E-L)/E.

Assertion Reason

30 assertion-reason questions.

1. Assertion: Power is rate of work. Reason: P=W/t for average power.

Both true; reason explains.

2. Assertion: Watt is SI unit of power. Reason: 1 W = 1 J/s.

Both true.

3. Assertion: P=Fv always scalar. Reason: It is dot product F.v.

Both true.

4. Assertion: Perpendicular force does no power. Reason: cos90=0.

Both true.

5. Assertion: Efficiency can exceed 100%. Reason: Output may exceed input.

Both false for real machines.

6. Assertion: 1 HP = 746 W. Reason: HP is a power unit.

Both true.

7. Assertion: Average and instantaneous power are always same. Reason: Power can vary.

Assertion false, reason true.

8. Assertion: Area under P-t graph gives work. Reason: W=∫Pdt.

Both true.

9. Assertion: Slope of W-t graph is power. Reason: P=dW/dt.

Both true.

10. Assertion: Braking power is negative. Reason: braking force opposes velocity.

Both true.

11. Assertion: Engine power is positive during acceleration. Reason: force has component along velocity.

Both true.

12. Assertion: Power dimension is ML²T^-3. Reason: power is energy/time.

Both true.

13. Assertion: Efficiency has no unit. Reason: it is a ratio.

Both true.

14. Assertion: P=Fv can be used with km/h directly in SI. Reason: SI velocity is m/s.

Assertion false, reason true.

15. Assertion: Constant power engine has force decreasing with speed. Reason: F=P/v.

Both true.

16. Assertion: Net power equals dK/dt. Reason: net work changes kinetic energy.

Both true.

17. Assertion: A motor is 100% efficient in practice. Reason: there are no losses.

Both false.

18. Assertion: Pump power depends on flow rate. Reason: more water per second requires more energy per second.

Both true.

19. Assertion: Lift power at constant speed is mgv. Reason: force equals mg.

Both true.

20. Assertion: Horsepower measures work. Reason: horsepower is old unit.

Assertion false; horsepower measures power.

21. Assertion: Work can be found from constant power as Pt. Reason: W=∫Pdt.

Both true.

22. Assertion: Negative power means energy removed. Reason: force has opposite component to velocity.

Both true.

23. Assertion: Generator efficiency output/input. Reason: useful output is electrical power.

Both true.

24. Assertion: Thermal plant has losses. Reason: heat engines are not perfectly efficient.

Both true.

25. Assertion: Power is scalar. Reason: F.v is scalar product.

Both true.

26. Assertion: More power always means more work. Reason: time may differ.

Assertion false without time; reason points out issue.

27. Assertion: Same work in less time means more power. Reason: P=W/t.

Both true.

28. Assertion: P=Fv for constant force only. Reason: instantaneous formula uses instantaneous F and v.

Assertion false, reason true.

29. Assertion: Efficiency percentage is η×100. Reason: η is fractional ratio.

Both true.

30. Assertion: Useful output excludes wasted heat. Reason: efficiency counts useful transfer only.

Both true.

Case Study Questions

Case studies on lift, motor, pump, car engine and generator.

Case 1: Lift raises 400 kg at 1.5 m/s. Find useful power. g=10.

P=mgv=400x10x1.5=6000 W. If efficiency is 75%, input power=8000 W.

Case 2: Motor input 2 kW, output shaft power 1.6 kW.

Efficiency=1.6/2=0.8=80%. Wasted power=0.4 kW.

Case 3: Pump lifts 50 kg water per second through 12 m.

Useful power=ṁgh=50x10x12=6000 W. With 60% efficiency, input=10000 W.

Case 4: Car engine provides 30 kW at 15 m/s.

Driving force=P/v=30000/15=2000 N. At same power, force decreases if speed increases.

Case 5: Generator mechanical input 8 kW and electrical output 6 kW.

Efficiency=6/8=75%. Losses=2 kW as heat, sound and internal resistance.

Common Student Mistakes

Avoid these errors in Power and Efficiency questions.

Confusing Work And Power

Work is energy transfer; power is rate of transfer.

Average vs Instantaneous

Use average power for total work/time and instantaneous power for a particular instant.

Ignoring Efficiency Losses

Input power is larger than useful output power for real machines.

Horsepower Unit Error

Use 1 HP = 746 W, not 1 HP = 1000 W.

Wrong Use Of P = Fv

Use component of force along velocity: P=Fvcosθ.

Wrong Units

Convert km/h to m/s and kW to W before calculation.