Chapter 11: Work and Energy

 

 Work and Energy

Introduction

• Motion, its causes, and gravitation were discussed in previous chapters.
• The concept of work helps in understanding various natural phenomena.
• Energy and power are closely related to work.

Energy in Living Beings

• Living beings need food for energy.
• Energy is required for life processes (breathing, digestion, circulation, etc.).
• Other activities like playing, singing, writing, thinking, running, etc., also need energy.
• Strenuous activities require more energy.

Energy in Animals

• Animals need energy for survival tasks like:
  • Running and jumping
  • Fighting or escaping from enemies
  • Finding food and shelter
• Some animals are used for work, such as:
  • Lifting weights
  • Pulling carts
  • Ploughing fields

Energy in Machines

• Machines need energy to function.
• Examples of machines: Fans, cars, washing machines, computers, elevators, etc.
• Energy sources for machines: Electricity, petrol, diesel, batteries, etc.
• Engines require fuels like petrol and diesel because they convert chemical energy into mechanical energy.

Why Do Living Beings and Machines Need Energy?

• Living beings need energy to perform biological functions and activities.
• Machines need energy to operate and perform tasks efficiently.

Definition of Work

• In everyday life, work refers to any physical or mental effort.
• In science, work is defined differently and depends on two conditions:
  1. A force must be applied.
  2. The object must be displaced in the direction of the applied force.

Analysis of Activities Based on Scientific Work

Activity	Work Done On?	Effect on Object?	Who is Doing Work?
Reading a book	Book	No movement	No one (No work done)
Pushing a rock (not moving)	Rock	No displacement	No one (No work done)
Lifting a box	Box	Moves upwards	Person lifting the box
Walking while holding a bag	Bag	No displacement in force direction	No work done
Climbing stairs	Body	Moves upwards	Person climbing
Playing football	Ball	Ball moves	Player kicks the ball

Examples of Work in Daily Life vs. Science

Activity	Everyday Meaning of Work	Scientific Meaning of Work
Studying for exams	Considered hard work	No work (no displacement of an object)
Pushing a rock (but it doesn’t move)	Hard work, exhausting	No work (no displacement)
Carrying a load while standing still	Feels like work (physical effort)	No work (no displacement)
Climbing stairs or a tree	Considered tiring work	Work is done (displacement against gravity)

Key Takeaways

1. Work is done in science only if an object moves due to an applied force.
2. No displacement = No work (scientifically).
3. Mental effort or exhaustion doesn’t count as work in scientific terms.

 Scientific Conception of Work

Definition of Work in Science

• Work is said to be done when two conditions are satisfied:
  1. A force is applied on an object.
  2. The object is displaced in the direction of the applied force.
• If any of these conditions are not met, no work is done in the scientific sense.

Examples of Work Done

Situation	Force Applied?	Displacement?	Work Done?
Pushing a pebble (it moves)	✅ Yes	✅ Yes	✅ Yes (Work is done)
Pulling a trolley (it moves)	✅ Yes	✅ Yes	✅ Yes (Work is done)
Lifting a book upwards	✅ Yes	✅ Yes	✅ Yes (Work is done)
Pushing a wall (it doesn’t move)	✅ Yes	❌ No	❌ No work done
Holding a heavy bag while standing	✅ Yes	❌ No	❌ No work done
Bullock pulling a cart (cart moves)	✅ Yes	✅ Yes	✅ Yes (Work is done)

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Situations Involving Work (Activity 11.2)

Think about daily life activities and check whether work is done:

1. Situations where force is applied, but no displacement:
   • Trying to push a parked car.
   • Holding a suitcase without moving.
   • Pressing against a wall.
     (No work done in these cases.)
2. Situations where displacement occurs, but no external force is applied:
   • A ball rolling on a frictionless surface.
   • 
   • A spaceship moving in space without any thrust.
     (Work is not being done after the initial force is applied.)

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Work Done by a Constant Force

• Formula: $$W = F \times s$$ where,
  • $W$ = Work done (Joule, J)
  • $F$ = Force applied (Newton, N)
  • $s$ = Displacement (meter, m)
• Work is a scalar quantity (has magnitude but no direction).

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SI Unit of Work

• The unit of work is Newton meter (N·m) or Joule (J).
• 1 Joule (J) = Work done when 1 Newton (N) force moves an object 1 meter (m) in the direction of the force. $$1J = 1N \times 1m$$

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Positive and Negative Work

• Positive Work:

  • When force and displacement are in the same direction.
  • Example:
    • Pulling a toy car parallel to the ground.
    • Lifting an object upwards (force exerted by hands).
    • A horse pulling a cart forward.

• Negative Work:

  • When force and displacement are in opposite directions.
  • Example:
    • A retarding force (friction) slowing down an object.
    • Gravity acting on an object being lifted.
    • A person pushing a box up a slope while friction resists it.
  • Work done is calculated as: $$W = - F \times s$$

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Activity 11.4 – Lifting an Object Up

• Forces Acting on the Object:
  • Applied force (by hand): Positive work (object moves upwards).
  • Gravitational force (pulling down): Negative work (opposes displacement).
• Conclusion:
  • The force applied by hand does positive work.
  • The force of gravity does negative work.

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Key Takeaways

✅ Work is done when force causes displacement.
✅ No displacement = No work (scientifically).
✅ Work can be positive (force & displacement same direction) or negative (opposite direction).
✅ SI unit of work = Joule (J).

 Energy

Definition of Energy

• Energy is the capacity to do work.
• Any object that possesses energy can exert a force on another object and transfer energy to it.
• The object that loses energy does work, and the object that gains energy receives work.

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Sources of Energy 

Natural Sources:

• The Sun 🌞 – The largest natural energy source.
• Nuclear Energy – Released from atomic nuclei.
• Geothermal Energy – Heat from the Earth's interior.
• Tidal Energy – Energy from ocean tides.

Other Energy Sources:

Source	Primary Origin	Relation to Sun
Solar Energy	Sun	Direct (sunlight)
Wind Energy	Sun	Indirect (heats air, creating wind)
Hydroelectric Energy	Sun	Indirect (water cycle driven by Sun)
Fossil Fuels	Ancient Sunlight	Indirect (stored solar energy in plants/animals)
Biomass	Plants	Indirect (photosynthesis from Sun)
Nuclear Energy	Atoms	Not related to the Sun
Geothermal Energy	Earth's core heat	Not related to the Sun
Tidal Energy	Moon’s gravity	Not related to the Sun

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Activity 11.5: Energy Discussions

• List other sources of energy such as wind, hydro, biomass, etc.
• Discuss which sources are due to the Sun (e.g., solar, wind, hydro).
• Identify sources NOT dependent on the Sun (e.g., nuclear, geothermal, tidal).

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How Does an Object Possess Energy?

• An object has energy if it has the ability to do work.
• Energy is transferred when one object does work on another.
• Examples:
  1. A fast-moving cricket ball transfers energy to the wicket.
  2. A raised hammer has energy to drive a nail.
  3. A wound-up toy car moves using stored energy.
  4. A pressed balloon stores potential energy, which can burst if compressed hard.

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Unit of Energy

• Energy is measured in Joules (J), the same unit as work.
• 1 Joule (J) = Energy required to do 1 Joule of work.
• Larger unit: Kilojoule (kJ) = 1000 Joules (J).

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Key Takeaways

✅ Energy is the capacity to do work.
✅ Work transfers energy from one object to another.
✅ Most energy sources are derived from the Sun.
✅ Some energy sources (nuclear, geothermal, tidal) are NOT from the Sun.
✅ The SI unit of energy is the Joule (J), with larger units in kilojoules (kJ).

Forms of Energy

The world provides energy in various forms, which can be classified as:

Type of Energy	Description	Examples
Mechanical Energy	Energy due to motion (kinetic) or position (potential).	Moving car, stretched rubber band, falling apple.
Heat Energy	Energy due to temperature difference.	Sun’s heat, boiling water, burning wood.
Chemical Energy	Stored energy in substances, released during reactions.	Food, fuels (petrol, coal), batteries.
Electrical Energy	Energy from electric current flow.	Lightning, power from batteries, household electricity.
Light Energy	Energy that travels in waves, visible to us.	Sunlight, bulbs, lasers.

Understanding Energy Forms

• How do we recognize a form of energy?
  • It has the ability to do work (move objects, produce heat, generate electricity).
  • It can be converted into other energy forms (e.g., electrical energy to light energy in a bulb)

James Prescott Joule: Key Points

• Full Name: James Prescott Joule

• Nationality: British

• Profession: Physicist

• Notable Contributions:

  1. Research in Electricity & Thermodynamics: Joule made significant contributions to understanding electricity and heat.
  2. Heating Effect of Electric Current: He formulated a law explaining how electric current produces heat.
  3. Law of Conservation of Energy: Joule experimentally verified this fundamental law, showing that energy cannot be created or destroyed, only transformed.
  4. Mechanical Equivalent of Heat: He discovered that mechanical energy can be converted into heat energy, leading to the concept of the mechanical equivalent of heat.

• Legacy: The unit of energy and work is named the joule (J) in his honor.

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Kinetic Energy

Definition:

• Kinetic Energy (KE) is the energy possessed by an object due to its motion.
• It depends on the mass of the object and its velocity.
• The faster an object moves, the more kinetic energy it has.

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Key Concepts:

• A moving object can do work due to its motion.
• The kinetic energy of an object increases with its speed.
• Examples of objects with kinetic energy include:
  • A falling coconut
  • A speeding car
  • A flying aircraft
  • Flowing water, blowing wind, and a running athlete

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Activity 11.6

• Objective: Observe how the depth of the depression increases with the height from which the ball is dropped.
• Analysis: The ball creates a deeper dent with increasing height due to the greater kinetic energy acquired as it gains speed. The faster the ball moves, the more energy it transfers to the sand.

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Activity 11.7

• Objective: A trolley moves forward and hits a block. The block moves, and work is done on the block by the trolley.
• Analysis: The moving trolley has kinetic energy and does work on the block by transferring energy to it. The energy of the moving trolley depends on its mass and speed.

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Derivation of Kinetic Energy Formula

1. Work Done on the Object:

   • Consider an object of mass (m) moving with an initial velocity u.
   • The object is displaced by a distance s, and a force (F) acts on the object in the direction of displacement.

   The work done (W) is given by:

   $$W = F \times s \quad \text{(Equation 11.1)}$$

2. Equations of Motion:

   • The relationship between the initial velocity u, final velocity v, acceleration a, and displacement s is given by:
   $$v^2 - u^2 = 2as \quad \text{(Equation 8.7)}$$
   • Rearranging for s:
   $$s = \frac{v^2 - u^2}{2a} \quad \text{(Equation 11.2)}$$

3. Force and Acceleration:

   • From Newton's second law of motion, the force F acting on the object is:
   $$F = ma \quad \text{(Equation 9.4)}$$

4. Substitute for Acceleration (a):

   • From Equation (3) and (4) , substitute s in the work equation (Equation 1):
   $$W = F \times s = ma \times \frac{v^2 - u^2}{2a}$$
   • Simplifying:
   $$W = \frac{m}{2} (v^2 - u^2) \quad \text{(Equation 11.3)}$$

5. Final Work Done if Object Starts from Rest:

   • If the object starts from rest, then u = 0:
   $$W = \frac{1}{2} m v^2 \quad \text{(Equation 11.4)}$$

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Conclusion:

• The kinetic energy (KE) of an object is given by:

$$KE = \frac{1}{2} m v^2$$

where:

• m is the mass of the object,
• v is the velocity of the object.

This shows that kinetic energy is directly proportional to both the mass and the square of the velocity.

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Potential Energy

Definition:

• Potential Energy (PE) is the energy possessed by an object due to its position or configuration.
• This energy is stored in the object and can be released to do work.
• Common forms of potential energy include gravitational potential energy and elastic potential energy.

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Activities:

1. Activity 11.8 (Rubber Band):

   • Process: Stretch a rubber band and release it.
   • Observation: The rubber band acquires energy when stretched and tends to regain its original length, demonstrating that energy is stored in the stretched position.
   • Conclusion: The stored energy in the rubber band is elastic potential energy.

2. Activity 11.9 (Slinky):

   • Process: Stretch and release a slinky.
   • Observation: The slinky acquires energy when stretched, which can also happen when it is compressed.
   • Conclusion: The energy stored in the stretched or compressed slinky is elastic potential energy.

3. Activity 11.10 (Toy Car):

   • Process: Wind a toy car using its key and release it.
   • Observation: The car moves due to the energy stored by winding the key.
   • Conclusion: The energy stored in the wound spring is elastic potential energy.

4. Activity 11.12 (Bow and Arrow):

   • 
     Process: Stretch the string of a bow and release the arrow.
   • Observation: The potential energy stored in the stretched string is converted into the kinetic energy of the arrow.
   • Conclusion: The energy stored in the bow is elastic potential energy.

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Gravitational Potential Energy:

• Definition: The gravitational potential energy of an object at a height is the energy it possesses due to its position relative to the Earth's surface.

• When an object is raised to a height, work is done against gravity. This work is stored as potential energy in the object.

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Derivation of Gravitational Potential Energy Formula:

1. Work Done Against Gravity:

   • Consider an object of mass (m).
   • The object is raised to a height h above the ground.
   • The force required to raise the object is equal to its weight, which is mg (where g is the acceleration due to gravity).

   The work done to raise the object is:

   $$\text{Work done (W)} = \text{Force} \times \text{Displacement}$$

   Since the force required is equal to the weight (mg) and the displacement is the height (h), we have:

   $$W = mg \times h$$

2. Energy Gained by the Object:

   • The object gains energy equal to the work done on it.
   • Therefore, the energy gained is:
   $$E_{\text{p}} = mg \times h$$

   where Eₚ is the potential energy of the object.

3. Final Formula:
   The gravitational potential energy (Eₚ) is given by:

   $$E_{\text{p}} = mgh \quad \text{(Equation 11.7)}$$

   where:

   • m = mass of the object,
   • g = acceleration due to gravity,
   • h = height from the ground.

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Key Points:

• Potential Energy Depends on Height: The potential energy of an object depends on its height above the reference point (ground level).
• Reference Level: The potential energy of an object is relative to the reference level or the chosen ground level.
• Path Independence: The work done by gravity depends only on the vertical displacement (height difference), not the path taken.

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More to Know:

• If an object is raised from position A to position B, the work done against gravity is the same regardless of the path taken, as long as the vertical displacement (h) is the same.

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Conclusion:

• Potential Energy (Ep) is the energy stored in an object due to its position or configuration.
• The gravitational potential energy of an object at a height is given by:

$$E_{\text{p}} = mgh$$

This explains how energy is stored and can be later converted into kinetic energy or other forms of energy.

Law of Conservation of Energy

Definition:

• Law of Conservation of Energy states that energy cannot be created or destroyed. It can only be converted from one form to another. The total energy of an isolated system remains constant during any transformation.

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Explanation:

1. Transformation of Energy:

   • Whenever energy changes from one form to another (e.g., from potential energy to kinetic energy), the total amount of energy remains the same.
   • For example, when an object falls, its potential energy (due to height) gets converted into kinetic energy (due to motion).

2. Example (Free Fall of an Object):

   • At the start:
     • The object has potential energy equal to mgh (where m is mass and h is the height).
     • The object has zero kinetic energy because its initial velocity is zero.
   • As the object falls:
     • The potential energy decreases while the kinetic energy increases.
     • At any point, the sum of kinetic energy (½mv²) and potential energy (mgh) remains constant.
   • At the ground:
     • The potential energy becomes zero because the height h is zero.
     • The kinetic energy is maximum because the velocity is the highest.
   • The total mechanical energy is constant throughout the fall:
   $$\text{Potential Energy} + \text{Kinetic Energy} = \text{Constant}$$
   • This can be written as:
   $$mgh + \frac{1}{2}mv^2 = \text{Constant} \quad \text{(Equation 11.7)}$$

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Derivation of Sum of Kinetic Energy (K.E.) and Potential Energy (P.E.) is Constant for a Falling Ball

Consider a ball of mass m dropped from a height h. The total mechanical energy of the ball at any point during its fall is the sum of its kinetic energy (K.E.) and potential energy (P.E.). We will show that the sum of these two energies remains constant during the fall, as per the Law of Conservation of Energy.

Step 1: Initial Energy at the Height $h$

When the ball is at the height h from the ground and is initially at rest, its kinetic energy (K.E.) is zero because its velocity is zero. Its potential energy (P.E.) is given by:

$$\text{P.E.} = mgh$$

where:

• m is the mass of the ball,
• g is the acceleration due to gravity (approximately 10 m/s²),
• h is the height from which the ball is dropped.

So, the total mechanical energy at the start is:

$$\text{Total Energy} = \text{K.E.} + \text{P.E.} = 0 + mgh = mgh$$

Step 2: Energy During the Fall

As the ball falls, its height decreases, and its potential energy (P.E.) decreases, but its kinetic energy (K.E.) increases because it gains speed.

Let the ball fall to a height h' from the ground, where h' < h. At this point:

• The potential energy is given by: $$\text{P.E.} = mgh'$$
• The kinetic energy is given by: $$\text{K.E.} = \frac{1}{2}mv^2$$ where v is the velocity of the ball at height h'.

By the work-energy principle, the work done by gravity on the ball during the fall is equal to the change in its kinetic energy. The work done by gravity is the force due to gravity (mg) acting over a displacement of h - h'. This work is the change in potential energy, which is transformed into kinetic energy.

Thus, the change in potential energy is:

$$\Delta \text{P.E.} = mgh - mgh' = mg(h - h')$$

This change in potential energy is converted into kinetic energy. Therefore, the increase in kinetic energy is:

$$\Delta \text{K.E.} = mg(h - h')$$

Step 3: Final Energy Just Before Hitting the Ground

When the ball reaches the ground (height h' = 0), all its potential energy is converted into kinetic energy. At this point:

• The potential energy is zero: $$\text{P.E.} = 0$$
• The kinetic energy is at its maximum, given by: $$\text{K.E.} = \frac{1}{2}mv^2$$

The velocity v of the ball just before it hits the ground can be found using the equation of motion:

$$v^2 = u^2 + 2gh$$

Since the ball is dropped from rest, u = 0. So:

$$v^2 = 2gh$$

Thus, the kinetic energy at this point is:

$$\text{K.E.} = \frac{1}{2}m(2gh) = mgh$$

Step 4: Conclusion

At all points during the fall, the total mechanical energy (sum of kinetic energy and potential energy) remains constant. This is a direct consequence of the Law of Conservation of Energy, which states that energy cannot be created or destroyed, only transformed from one form to another.

Thus, the total energy at any point during the fall is:

$$\text{Total Energy} = \text{K.E.} + \text{P.E.} = \text{constant}$$

This proves that the sum of kinetic energy and potential energy is constant during the free fall of the ball.

Mechanical Energy:

• The sum of kinetic energy (K.E.) and potential energy (P.E.) is called the total mechanical energy.
• As the object falls, the decrease in potential energy is equal to the increase in kinetic energy.
• Mechanical Energy is conserved during the free fall of an object (ignoring air resistance).

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Example Problem:

An object of mass 20 kg is dropped from a height of 4 m. We need to compute the potential energy (P.E.), kinetic energy (K.E.), and the total energy (P.E. + K.E.) at various heights during its fall.

1. Formula for Potential Energy: $$P.E. = mgh$$
2. Formula for Kinetic Energy: $$K.E. = \frac{1}{2}mv^2$$
3. Total Mechanical Energy: $$\text{Total Energy} = P.E. + K.E.$$

Given:

• m = 20 kg (mass of object)
• h = 4 m (initial height)
• g = 10 m/s² (acceleration due to gravity)

Now, we will fill in the table with calculations:

Here's the code you can use to create a table in Blogger using HTML: ```html

Height (m)	Potential Energy (P.E.) (J)	Kinetic Energy (K.E.) (J)	Total Energy (P.E. + K.E.) (J)
4	800 J	0 J (object is stationary initially)	800 J (Constant)
3	600 J	200 J (Energy is converted to K.E.)	800 J
2	400 J	400 J	800 J
1	200 J	600 J	800 J
Just above the ground	0 J	800 J	800 J

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Conclusion:

• Total Energy remains constant throughout the fall.
• Potential Energy is converted into Kinetic Energy as the object falls.
• The law of conservation of energy is proven here, as the sum of potential and kinetic energy remains constant throughout the fall.
• Nature's Energy Transformation: If energy couldn't be transformed, life would not be possible because all processes in nature (e.g., photosynthesis, digestion, movement) depend on the conversion of one form of energy to another.

Rate of Doing Work

• Definition of Power:
  Power is the rate at which work is done or energy is transferred. It measures how fast or slow work is done. Power is defined as:

  $$\text{Power} = \frac{\text{Work}}{\text{Time}} = \frac{W}{t}$$

• Unit of Power:
  The unit of power is watt (W).

  • 1 watt = 1 joule/second (1 W = 1 J/s)
  • 1 kilowatt (kW) = 1000 watts = 1000 J/s

• Explanation through Activity:
  Two children (A and B) weigh the same and climb a rope to a height of 8 m:

  • Child A takes 15 seconds to reach the height, and Child B takes 20 seconds.
  • Both do the same work (same height and weight), but A does the work faster (in less time).
  • Power is determined by how quickly the work is done, so A is more powerful than B.

• Power in Machines:

  • Machines like vehicles are classified based on their power because the rate at which they perform work or transfer energy determines their efficiency and performance. A more powerful machine completes tasks faster.

• Average Power:
  If the power of an agent changes over time, average power is the total energy consumed divided by the total time taken.

  $$\text{Average Power} = \frac{\text{Total Energy Consumed}}{\text{Total Time Taken}}$$

This formula gives an overall measurement of how efficiently an agent uses energy over a period.

Commercial Unit of Energy

• Kilowatt Hour (kWh):
  The unit joule is too small for large quantities of energy, so we use kilowatt hour (kWh) as a larger, more convenient unit.

• What is 1 kWh?

  • 1 kWh is the energy used by a machine consuming energy at the rate of 1000 J/s (1 kW) continuously for one hour.
  • To calculate: $$1 \, \text{kWh} = 1 \, \text{kW} \times 1 \, \text{hour} = 1000 \, \text{W} \times 3600 \, \text{seconds} = 3,600,000 \, \text{J} = 3.6 \times 10^6 \, \text{J}$$

• Usage of kWh:

  • Energy consumption in households, industries, and commercial establishments is commonly expressed in kilowatt hours.
  • For example, electrical energy used during a month is often expressed in units, where 1 unit = 1 kWh.