Understanding Power and Energy Units

This question asks us to match common units and quantities related to power and energy. Let's break down each item in List I and find its correct match in List II.

List I contains:

1. W (Watt)
2. kW (kilowatt)
3. 1 kW.h (kilowatt-hour)
4. 1 HP (Horsepower)

List II contains potential matches:

1. Nm
2. $3.6 \times 10^6$ J
3. 1000 W
4. 746 W

Analyzing the Matches

Let's examine each item from List I:

1. W (Watt):

• The Watt (W) is the standard unit of power in the International System of Units (SI).
• Power is defined as the rate at which work is done or energy is transferred.
• One Watt is equal to one joule per second ($1 \text{ W} = 1 \text{ J/s}$).
• Work done is force times displacement, and the SI unit for work is the Joule (J). Force is measured in Newtons (N), and displacement in meters (m). So, work is measured in N.m (Newton-meter). Thus, $1 \text{ J} = 1 \text{ N.m}$.
• Therefore, $1 \text{ W} = 1 \text{ N.m/s}$.
• List II contains 'Nm'. While N.m is a unit of work or energy (Joule), Watt is a unit of power (rate of energy transfer). Let's look at other options first to be sure about the best match provided. Sometimes related units are listed if direct ones aren't available.

2. kW (kilowatt):

• The kilowatt (kW) is a multiple of the Watt.
• The prefix 'kilo' means 1000.
• So, $1 \text{ kW} = 1000 \text{ W}$.
• Looking at List II, option C is '1000 W'. This is a direct match.

3. 1 kW.h (kilowatt-hour):

• The kilowatt-hour (kW.h) is a unit of energy.
• Energy is Power multiplied by time ($E = P \times t$).
• 1 kW.h is the energy consumed or transferred by a power of 1 kilowatt operating for 1 hour.
• Let's convert this to the standard SI unit of energy, the Joule (J).
• $1 \text{ kW} = 1000 \text{ W}$
• $1 \text{ hour} = 60 \text{ minutes} \times 60 \text{ seconds} = 3600 \text{ seconds}$
• $1 \text{ kW.h} = (1 \text{ kW}) \times (1 \text{ h}) = (1000 \text{ W}) \times (3600 \text{ s})$
• $1 \text{ kW.h} = 3,600,000 \text{ W.s}$
• Since 1 W = 1 J/s, then 1 W.s = 1 J.
• So, $1 \text{ kW.h} = 3,600,000 \text{ J} = 3.6 \times 10^6 \text{ J}$.
• Looking at List II, option B is '$3.6 \times 10^6$ J'. This is a direct match.

4. 1 HP (Horsepower):

• Horsepower (HP) is an older, non-SI unit of power.
• It is still used in some fields, such as automotive and motor industries.
• There are different definitions of horsepower, but the most common one (mechanical horsepower) is approximately equal to 745.7 watts.
• For practical purposes, it is often rounded to 746 W.
• Looking at List II, option D is '746 W'. This is a direct match.

Now let's revisit item 1, W (Watt). The remaining option in List II is 'Nm'. As discussed, Nm is a unit of work or energy (Joule), not power (Watt). However, if we consider the relationship $1 \text{ W} = 1 \text{ J/s}$ and $1 \text{ J} = 1 \text{ Nm}$, then $1 \text{ W} = 1 \text{ Nm/s}$. While Nm itself is not a Watt, it is directly related to the definition of work/energy which forms the basis of power calculation. Given the other perfect matches, it seems there might be a slight imprecision in option A, but it's the only remaining option for Watt. Perhaps it's intended to link the unit of power to its fundamental components involving Force and Distance, even though time is missing in 'Nm'. Let's proceed with the other confirmed matches and see if this pattern holds up.

Based on our analysis:

• 1. W matches with A. Nm (This is the least precise match, but the only remaining one).
• 2. kW matches with C. 1000 W.
• 3. 1 kW.h matches with B. $3.6 \times 10^6$ J.
• 4. 1 HP matches with D. 746 W.

Summary Table

The matching pattern is: 1-A, 2-C, 3-B, 4-D.

Conclusion

By understanding the definitions and conversions of different units of power and energy, we can correctly match the items from List I to List II. The relationship between W and Nm is less direct than the others, but within the given options, it is the most plausible pairing, likely intended to relate power (J/s) to work (Nm).

Revision Table: Key Units and Conversions

Additional Information on Power and Energy Units

Understanding the relationship between power and energy is crucial in physics. Power is the rate at which energy is used or transferred. The standard SI unit for energy is the Joule (J), and for power is the Watt (W).

• Joule (J): Defined as the work done when a force of one Newton moves an object one meter in the direction of the force. It is also the energy transferred when a power of one Watt is applied for one second.
• Watt (W): Defined as one Joule per second (1 W = 1 J/s). It measures how quickly energy is being used or converted.
• Kilowatt (kW): A common unit for measuring power in everyday life, such as the power rating of appliances or motors. 1 kW = 1000 W.
• Kilowatt-hour (kW.h): A unit of energy, often used for billing residential electricity consumption. It represents the total energy consumed by a device drawing 1 kW of power for one hour. It's a unit of energy, not power.
• Horsepower (HP): Historically significant unit of power, originally related to the power output of a horse. Still used for engines and motors.
• Newton-meter (Nm): This is the unit for Torque, but it is also dimensionally equivalent to the Joule (J) and is used as a unit for work or energy. However, when describing work or energy, it's usually called a Joule. Torque and work are different concepts even though they share the same units in terms of dimensions.

It is important to distinguish between units of power (rate of energy) and units of energy (total amount of energy). W, kW, and HP are units of power. J and kW.h are units of energy.