6.4 Power

When we talk about work and energy, we often care about how quickly the work is done or how fast the energy is converted. For example, two cars might both travel the same distance, but the one that does it in less time is more powerful. Power is the rate at which work is done or energy is converted. It is a fundamental concept in physics and engineering, helping us compare the performance of machines, engines, and even human bodies.

Definition

Power is the time rate at which work is done or the rate at which energy is consumed or converted. It is a scalar quantity.

• P is power.
• W is work done.
• E is energy converted or consumed.
• t is the time taken.

The SI unit of power is the watt (W), named after James Watt, a Scottish physicist who invented the first steam engine.

1 W = 1 J/s One watt is equal to one joule of work done per second.

Main Idea

The main idea is that power measures how fast work is done or energy is transformed. A more powerful machine can do the same amount of work in less time, or do more work in the same amount of time. Power is not about the total amount of work or energy, but about the rate at which it is used. This distinction is crucial in choosing the right machine or engine for a task, from a light bulb to a car engine.

Core Concept

The Formula for Power The text states: "Power is the time rate at which work is done or the rate at which energy is consumed. Mathematically P = Wt = Et Like work power is a scalar quantity. The SI unit of power is watt (W), in honour of James Watt, a Scottish physicist who invented the first steam engine."

The Unit of Power The SI unit of power is the watt (W). 1 watt = 1 joule / 1 second or in symbols, 1 W = 1 J/s

Larger Units of Power For practical purposes, a larger unit is often used, the horsepower (hp). One horsepower is defined as 550 ft·lb/s, which equals 746 W. 1 hp = 746 W

The Kilowatt-Hour (kWh) The text states: "The electricity bills that we pay are measured in terms of this unit i.e 1 unit = 1 kWh." 1 kWh = (10³ W)(3600 s) = 3.6 × 10⁶ J The kilowatt-hour is a unit of energy, not power. It is the energy consumed by a 1000 W device running for one hour.

Example from Text The text provides an example (Example 6.5): "How powerful will be a Tesla Car's engine if it does 3 × 10⁷ J of work in 1 minute? What will be its power in hp?" - Given: Work W = 3 × 10⁷ J, Time t = 1 min = 60 s - Solution: P = 3 × 10⁷60 = 5 × 10⁵ W = 500 kW - In hp: P = 5 × 10⁵746 ≈ 670 hp

Detailed Explanation

Power is a fundamental concept in physics and engineering. It measures how quickly energy is transferred or work is done.

Power vs. Work and Energy Work and energy are measured in joules (J). Power is measured in watts (W), which is joules per second (J/s). A machine that does 1000 J of work in 1 second has a power of 1000 W (1 kW). A machine that does the same 1000 J of work in 10 seconds has a power of only 100 W. The total work is the same, but the power is different because the time is different.

The Watt The watt is a small unit of power. A typical LED light bulb uses about 10 W, a laptop uses about 50 W, and a microwave oven uses about 1000 W (1 kW). A car engine might have a power of 100 kW (about 134 hp), and a large power plant might have a power of 1000 MW (1 GW).

Horsepower Horsepower is a non-SI unit of power that is still commonly used for engines and motors. It was originally defined by James Watt to compare the power of steam engines to the power of horses. One horsepower is equal to 746 W.

The Kilowatt-Hour (kWh) The kilowatt-hour is a unit of energy commonly used for electricity billing. It is the energy consumed by a device with a power of 1 kW running for 1 hour. 1 kWh = 1000 W × 3600 s = 3.6 × 10⁶ J For example, a 100 W light bulb left on for 10 hours consumes 1 kWh of energy.

• Mechanical Power: The rate at which work is done by a machine (e.g., a car engine, a motor).
• Electrical Power: The rate at which electrical energy is consumed or produced (e.g., a light bulb, a power plant).
• Human Power: The rate at which the human body does work (e.g., a person lifting weights, running).

Intermediate Level

In physics, power is the rate of doing work or transferring energy: P = Wt. It is a scalar quantity measured in watts (W), where 1 W = 1 J/s. Power is distinct from work and energy—work and energy are amounts (measured in joules), while power is a rate (measured in watts).

For machines, power is often expressed in horsepower (hp), where 1 hp = 746 W. This unit is still used for car engines and other motors. For electricity, the kilowatt-hour (kWh) is commonly used as a unit of energy (not power). It is the energy consumed by a 1 kW device in 1 hour, equal to 3.6 × 10⁶ J.

Power is crucial in engineering because it determines how quickly a machine can perform a task. A more powerful engine can accelerate a car faster or climb a hill more easily. However, greater power usually comes with greater energy consumption and cost.

Important Points

• - Power: The rate of doing work or transferring energy.
• - Formula: P = Wt = Et.
• - Unit: Watt (W). 1 W = 1 J/s.
• - Scalar quantity.
• - Horsepower: 1 hp = 746 W.
• - Kilowatt-hour (kWh): A unit of energy, not power. 1 kWh = 3.6 × 10⁶ J.
• - Power depends on both work and time.

Remember

• - Power = Work / Time.
• - Unit: Watt (W).
• - 1 W = 1 J/s.
• - 1 hp = 746 W.
• - 1 kWh = 3.6 × 10⁶ J (energy, not power).
• - More power = faster work.

Common Mistakes

• - Confusing power with energy: Power is a rate (joules per second). Energy is an amount (joules). A 100 W bulb uses 100 J every second, but the total energy it uses depends on how long it is on.
• - Forgetting that kWh is a unit of energy: The kilowatt-hour is a unit of energy, not power. It is the energy consumed by a 1 kW device in 1 hour.
• - Thinking more power always means more work: A more powerful machine can do more work in the same time, or the same work in less time. The total work depends on both power and time.

Quick Revision

Power is the rate at which work is done or energy is transferred. It is calculated as work divided by time (P = Wt) and is measured in watts (W), where 1 W = 1 J/s. Power is a scalar quantity. Other units include horsepower (1 hp = 746 W). The kilowatt-hour (kWh) is a unit of energy, not power, equal to 3.6 × 10⁶ J.

60-Second Revision

• Formula: P = Wt.
• Unit: Watt (W). 1 W = 1 J/s.
• 1 hp = 746 W.
• 1 kWh = 3.6 × 10⁶ J (energy).
• Scalar quantity.
• More power = faster work.

Frequently Asked Questions (FAQs)

• Q1. What is power?
• Answer: Power is the time rate at which work is done or the rate at which energy is consumed or converted. It is given by P = Wt.

• Q2. What is the SI unit of power?
• Answer: The SI unit of power is the watt (W). One watt is equal to one joule per second (1 W = 1 J/s).

• Q3. Is power a scalar or a vector?
• Answer: Power is a scalar quantity. It has magnitude but no direction.

• Q4. What is horsepower?
• Answer: Horsepower (hp) is a non-SI unit of power commonly used for engines and motors. One horsepower is equal to 746 watts (1 hp = 746 W).

Q5. What is a kilowatt-hour (kWh)? Answer: A kilowatt-hour is a unit of energy, not power. It is the energy consumed by a device with a power of 1 kW running for 1 hour. 1 kWh = 3.6 × 10⁶ J.

Q6. What is the difference between power and energy? Answer: Energy is the capacity to do work (measured in joules). Power is the rate at which work is done or energy is transferred (measured in watts). Energy is an amount; power is a rate.

Q7. How is power related to work and time? Answer: Power is directly proportional to work and inversely proportional to time: P = Wt. More work done in the same time means more power. The same work done in less time means more power.

• Q8. A machine does 1000 J of work in 5 seconds. What is its power?
• Answer: P = Wt = 10005 = 200 W.

Multiple Choice Questions (MCQs)

• Q1. Power is defined as:
• A. Work × time
• B. Work / time
• C. Time / work
• D. Work + time

• Correct Answer: B
• Explanation: The text defines power as the time rate at which work is done, P = Wt.

• Q2. The SI unit of power is the:
• A. Joule
• B. Newton
• C. Watt
• D. Pascal

• Correct Answer: C
• Explanation: The SI unit of power is the watt (W).

• Q3. One watt is equal to:
• A. 1 J × 1 s
• B. 1 J / 1 s
• C. 1 N × 1 m
• D. 1 N / 1 m

• Correct Answer: B
• Explanation: 1 W = 1 J/s.

• Q4. Power is a:
• A. Vector quantity
• B. Scalar quantity
• C. Base quantity
• D. Derived quantity

• Correct Answer: B
• Explanation: The text states that power is a scalar quantity.

• Q5. One horsepower is equal to:
• A. 100 W
• B. 550 W
• C. 746 W
• D. 1000 W

• Correct Answer: C
• Explanation: The text states that 1 hp = 746 W.

• Q6. The kilowatt-hour (kWh) is a unit of:
• A. Power
• B. Energy
• C. Force
• D. Time

• Correct Answer: B
• Explanation: The kWh is a unit of energy, equal to 3.6 × 10⁶ J. It is the energy consumed by a 1 kW device in 1 hour.

• Q7. A machine does 5000 J of work in 10 seconds. Its power is:
• A. 50 W
• B. 500 W
• C. 5000 W
• D. 50,000 W

• Correct Answer: B
• Explanation: P = Wt = 500010 = 500 W.

• Q8. A 100 W light bulb is left on for 10 hours. The energy consumed is:
• A. 1 kWh
• B. 10 kWh
• C. 100 kWh
• D. 1000 kWh

• Correct Answer: A
• Explanation: Energy = Power × Time = 100 W × 10 h = 1000 Wh = 1 kWh.

• Q9. Which of the following has the greatest power?
• A. A machine that does 1000 J of work in 10 s
• B. A machine that does 500 J of work in 5 s
• C. A machine that does 2000 J of work in 20 s
• D. A machine that does 100 J of work in 1 s

• A: 100010 = 100 W
• B: 5005 = 100 W
• C: 200020 = 100 W
• D: 1001 = 100 W

• Q10. The power of a car engine is 100 kW. In horsepower, this is approximately:
• A. 100 hp
• B. 134 hp
• C. 746 hp
• D. 1000 hp

• Correct Answer: B
• Explanation: 100 kW = 100,000 W. 100,000746 ≈ 134 hp.

Frequently Asked Questions (FAQs)

Q1. What is the difference between power and energy? Answer: Energy is the capacity to do work (measured in joules). Power is the rate at which work is done or energy is transferred (measured in watts). Energy is an amount; power is a rate.

Q2. Why is the kilowatt-hour (kWh) used for electricity bills? Answer: The kWh is a convenient unit for measuring large amounts of electrical energy. A typical household uses several kWh of electricity per day, so the kWh is a practical unit for billing.

Q3. How is power related to force and velocity? Answer: Power can also be expressed as the product of force and velocity: P = F × v. This is useful for analyzing the power of a moving object (e.g., a car engine).

Q4. What is the power of a human? Answer: The power of a human depends on the activity. A person walking might use about 100 W. A person running or cycling vigorously might use 500-1000 W. An athlete in a sprint might briefly produce over 2000 W.

Q5. What is the difference between average power and instantaneous power? Answer: Average power is the total work done divided by the total time. Instantaneous power is the power at a specific instant in time (e.g., the power of a car engine at a given moment).

Q6. How does the power of a machine affect its performance? Answer: A more powerful machine can do more work in the same time, or the same work in less time. For example, a more powerful car engine can accelerate the car faster or climb a hill more easily.

• Q7. What is the power of the Sun?
• Answer: The Sun radiates energy at a rate of about 3.8 × 10²⁶ W. This is the power output of the Sun.

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