Power in Physics The Rate of Doing Work

1. Introduction

In everyday life, when we say someone is “powerful,” we usually mean strong or influential. In physics, however, power has a very specific meaning:

Power is the rate at which work is done or energy is transferred.

This concept is crucial for understanding how quickly machines, engines, and even our own bodies can perform tasks.

For example:

• Two people may both do the same amount of work (say, climbing a set of stairs). But if one does it faster, that person has greater power output.
• Similarly, a car and a truck may both reach the same speed, but the one that accelerates faster delivers more power.

In this article, we’ll explore:

• The definition and mathematical formula of power
• Units of power (watt, horsepower, etc.)
• Average vs instantaneous power
• Relation between force, velocity, and power
• Applications in daily life and technology
• Examples and solved problems
• Misconceptions about power
• Advanced concepts like mechanical power, electrical power, and human power

By the end, you’ll have a deep and practical understanding of power in physics, not just as a formula, but as a bridge between work, energy, and time.

---

2. What is Power? – Basic Definition

In physics: P=WtP = \frac{W}{t}P=tW​

where:

• PPP = Power
• WWW = Work done
• ttt = Time taken

So, power measures how fast work is being done.

Example:

If you lift a 20 kg box to a height of 2 m:

• Work done = mgh=20×9.8×2=392Jmgh = 20 \times 9.8 \times 2 = 392 Jmgh=20×9.8×2=392J.
• If you do it in 2 seconds: P=3922=196 WP = \frac{392}{2} = 196 \, WP=2392​=196W
• If you do it in 10 seconds: P=39210=39.2 WP = \frac{392}{10} = 39.2 \, WP=10392​=39.2W

👉 Same work, different power because of time difference.

---

3. Units of Power

• SI unit: Watt (W) 1 W=1 J/s1 \, W = 1 \, J/s1W=1J/s
• Practical units:
  • Kilowatt (kW): 1 kW=1000 W1 \, kW = 1000 \, W1kW=1000W
  • Megawatt (MW): 1 MW=106 W1 \, MW = 10^6 \, W1MW=106W
  • Horsepower (hp): 1 hp≈746 W1 \, hp ≈ 746 \, W1hp≈746W

👉 Engines and motors are often rated in horsepower or kilowatts.

---

4. Average Power vs Instantaneous Power

4.1 Average Power

Pavg=WtP_{avg} = \frac{W}{t}Pavg​=tW​

It gives the overall rate of work in a time interval.

4.2 Instantaneous Power

At any instant: P=dWdtP = \frac{dW}{dt}P=dtdW​

Since dW=F⋅dxdW = F \cdot dxdW=F⋅dx: P=F⋅vP = F \cdot vP=F⋅v

👉 Instantaneous power depends on the force and velocity at that moment.

---

5. Relation Between Power, Force, and Velocity

We saw that: P=F⋅vP = F \cdot vP=F⋅v

• If the force is constant and in the direction of velocity:

P=FvP = F vP=Fv

• If angle θ\thetaθ exists between force and velocity:

P=Fvcos⁡θP = F v \cos \thetaP=Fvcosθ

This is very useful in mechanics and engineering.

---

6. Examples from Daily Life

1. Climbing Stairs: Two people of same weight climbing stairs; faster one develops more power.
2. Vehicles: A sports car accelerates quickly → high power engine.
3. Lifting Weights: Bodybuilders lift same weight but faster lifts mean more power.
4. Appliances: A 1000 W microwave works faster than a 600 W one.
5. Hydropower Plants: Convert water’s potential energy into electrical power.

---

7. Power in Machines

Power rating of a machine tells how fast it can do work.

Efficiency and Power

Not all power supplied is useful. Machines waste energy as heat, sound, or friction. Efficiency=Useful Power OutputPower Input×100%\text{Efficiency} = \frac{\text{Useful Power Output}}{\text{Power Input}} \times 100\%Efficiency=Power InputUseful Power Output​×100%

Example:

• Motor input = 1000 W
• Useful output = 800 W
• Efficiency = 80%

---

8. Human Power Output

Humans also have power limits.

• Walking = 100–150 W
• Cycling = 200–400 W (average)
• Sprinting = 1000 W (short bursts)

👉 This is why machines are needed for heavy, fast, or continuous work.

---

9. Electrical Power

In electricity, power is energy transferred per unit time: P=VIP = VIP=VI

where VVV = potential difference, III = current.

• If resistance RRR is present:

P=I2R=V2RP = I^2 R = \frac{V^2}{R}P=I2R=RV2​

Electrical appliances are rated by power (e.g., 60 W bulb, 1500 W heater).

---

10. Mechanical Power

• Linear Motion:

P=FvP = FvP=Fv

• Rotational Motion:

P=τωP = \tau \omegaP=τω

where τ\tauτ = torque, ω\omegaω = angular velocity.

👉 Engines, turbines, and motors use rotational power equations.

---

11. Power and Work–Energy Theorem

Since work done = change in kinetic energy: P=dWdt=ddt(12mv2)P = \frac{dW}{dt} = \frac{d}{dt}\left(\frac{1}{2} mv^2\right)P=dtdW​=dtd​(21​mv2)

So power is directly linked with rate of change of kinetic energy.

---

12. Solved Examples

Example 1: Horsepower of an Engine

A car engine does 150 kJ work in 10 s. Find power in hp. P=15000010=15000WP = \frac{150000}{10} = 15000 WP=10150000​=15000W

Convert: 15000746≈20.1 hp\frac{15000}{746} ≈ 20.1 \, hp74615000​≈20.1hp

---

Example 2: Stair Climbing

A 60 kg student runs up 5 m high stairs in 4 seconds. Find power. W=mgh=60×9.8×5=2940JW = mgh = 60 \times 9.8 \times 5 = 2940 JW=mgh=60×9.8×5=2940J P=29404=735WP = \frac{2940}{4} = 735 WP=42940​=735W

---

Example 3: Rotational Power

A motor provides torque of 50 Nm at 20 rad/s. P=τω=50×20=1000WP = \tau \omega = 50 \times 20 = 1000 WP=τω=50×20=1000W

---

Example 4: Electrical Power

An electric heater operates on 220 V, current = 5 A. Find power. P=VI=220×5=1100WP = VI = 220 \times 5 = 1100 WP=VI=220×5=1100W

---

13. Graphical Understanding

• Power vs Time graph: shows how power varies during motion.
• Work vs Time slope: represents power.
• Force–velocity product: helps visualize instantaneous power.

---

14. Misconceptions about Power

❌ “Power means strength.” → Not exactly; it means speed of doing work.
❌ “More power always means more efficiency.” → No, efficiency depends on useful work, not just power rating.
❌ “If work is zero, power must be zero.” → Not true; instantaneous power can exist even if net work over time is zero (like in oscillations).

---

15. Applications of Power Concept

1. Automobiles: Engine power determines acceleration and towing capacity.
2. Construction: Cranes rated in kW or hp.
3. Power Plants: Generate MW of power for cities.
4. Sports Science: Athletes’ power measured for performance.
5. Technology: Computers, LEDs, and electronics rated by power consumption.

---

16. Advanced Power Concepts

Relativistic Power

At near-light speeds: P=dEdtP = \frac{dE}{dt}P=dtdE​

where EEE = relativistic energy.

Power Transmission

In electricity: P=VIcos⁡ϕP = VI \cos \phiP=VIcosϕ

where cos⁡ϕ\cos \phicosϕ = power factor.

👉 High-voltage transmission reduces power loss.

---

17. Practice Problems

1. A pump delivers 200 L of water per minute to height of 10 m. Find power required.
2. A cyclist develops 250 W power. How much work in 1 hour?
3. A 100 W bulb works for 5 hours. Find energy consumed in kWh.
4. A motor rotates at 1500 rpm with torque 10 Nm. Find power in kW.
5. A truck engine does 500 kJ work in 25 s. Find power in hp.

---

18. Power and Energy Conservation

• Power tells how fast energy changes form.
• Energy may convert (chemical → mechanical → electrical), but total remains conserved.
• Power analysis is key in designing efficient systems.

---

19. Summary of Key Formulas

1. Average Power:

P=WtP = \frac{W}{t}P=tW​

2. Instantaneous Power:

P=dWdt=FvP = \frac{dW}{dt} = F vP=dtdW​=Fv

3. Rotational Power:

P=τωP = \tau \omegaP=τω

4. Electrical Power:

P=VI=I2R=V2RP = VI = I^2 R = \frac{V^2}{R}P=VI=I2R=RV2​

5. Efficiency:

η=PoutPin×100%\eta = \frac{P_{out}}{P_{in}} \times 100\%η=Pin​Pout​​×100%