Introduction to Work and Energy
Welcome, students! In our daily lives, we use terms like 'work' and 'energy' very frequently. We talk about 'working hard' on our homework or feeling 'full of energy' after a good meal. But in the world of science, and specifically physics, these terms have very precise and distinct meanings. This chapter, 'Work and Energy' from the NCERT Class 9 Science syllabus, delves into these scientific definitions. It's a foundational chapter that helps us understand how forces cause motion and how this motion relates to energy. We will explore what it means for work to be done, the different forms of energy that make everything happen, and the rate at which this work is performed, which we call power. Understanding these concepts is crucial as they form the basis for much of classical mechanics and have applications in almost every aspect of technology and nature, from launching a rocket to the simple act of lifting a book.
Work: The Scientific Concept
In everyday language, 'work' can refer to any physical or mental effort. Reading a book, thinking about a problem, or even holding a heavy bag without moving are often considered 'work'. However, in physics, the definition is much more specific. Work is done only when a force applied to an object causes that object to move some distance in the direction of the force.
Definition of Work
Scientifically, work done on an object is defined as the product of the magnitude of the force acting on the object and the magnitude of the displacement of the object in the direction of the force.
The formula for work is:
Work (W) = Force (F) × displacement (s)
So, if you push a block with a force 'F' and it moves a distance 's' in the same direction you pushed it, the work you've done on the block is W = Fs.
Conditions for Work Done
From the definition, two essential conditions must be met for work to be done:
- A force must act on the object. If there is no force, there can be no work.
- The object must be displaced. If the object doesn't move, no work is done, no matter how much force is applied. For example, if you push against a solid wall with all your might, you might get tired, but since the wall does not move (s=0), the work done on the wall is zero (W = F × 0 = 0).
Unit of Work
The SI unit of work is the joule (J), named after the English physicist James Prescott Joule.
How do we define one joule?
1 joule is the amount of work done on an object when a force of 1 newton (N) displaces it by 1 meter (m) along the line of action of the force.
So, 1 J = 1 N × 1 m.
Positive, Negative, and Zero Work
Work is a scalar quantity, but it can have a sign (positive, negative, or zero) that gives us important information about the interaction between force and displacement.
Positive Work
Work done is considered positive when the force applied on an object is in the same direction as its displacement. In this case, the angle between the force and displacement is 0°. The force helps the motion.
- Example 1: When you kick a football, the force you apply and the displacement of the ball are in the same direction. You do positive work on the ball.
- Example 2: When a horse pulls a cart, the force exerted by the horse and the movement of the cart are in the same direction. The horse does positive work.
Negative Work
Work done is considered negative when the force applied on an object is in the opposite direction to its displacement. In this case, the angle between the force and displacement is 180°. The force opposes the motion.
- Example 1: When a car is moving, the force of friction from the road and air resistance acts in the opposite direction to the car's motion. Therefore, the work done by friction is negative.
- Example 2: When you lift a bucket of water from a well, the gravitational force is acting downwards, but the displacement is upwards. The work done by the force of gravity is negative. (Note: The work done by you is positive).
Zero Work
Work done is zero in two main situations:
- When there is no displacement (s = 0): As discussed, pushing a wall involves applying a force, but since the wall doesn't move, the work done is zero.
- When the force is perpendicular to the displacement: If the angle between the force and the displacement is 90°, the work done is zero.
- Example 1: A satellite orbiting the Earth. The Earth's gravitational force acts towards the center of the Earth, but the satellite's displacement at any instant is tangential to the orbit. The force and displacement are perpendicular, so the work done by gravity on the satellite is zero.
- Example 2: A coolie carrying a heavy load on his head and walking on a horizontal platform. The force of gravity on the load (and the upward force he applies to support it) is vertical, while his displacement is horizontal. The angle is 90°, so the work done by the coolie against gravity is zero. He gets tired due to the muscular effort required to support the load, but in physics terms, no work is done on the load in the horizontal direction.
Energy: The Capacity to Do Work
If you have to do work, you need energy. Energy is a central concept in all of science. It is the fundamental currency for all interactions and changes in the universe.
What is Energy?
Energy is defined as the capacity or ability to do work. An object that possesses energy can exert a force on another object to do work. When an object does work, it loses energy, and the object on which the work is done gains that energy. In essence, work is the transfer of energy.
Since energy is measured by the amount of work it can do, the unit of energy is the same as the unit of work: the joule (J).
Forms of Energy
Energy exists in many different forms. The world around us is full of energy transformations. A burning candle converts chemical energy into heat and light energy. A hydroelectric dam converts the potential energy of stored water into kinetic energy and then into electrical energy. Some of the major forms of energy are:
- Mechanical Energy: The sum of kinetic and potential energy in an object.
- Heat Energy: The energy that flows from a hotter object to a colder one.
- Chemical Energy: The energy stored in the bonds of chemical compounds (e.g., in food, batteries, fuel).
- Electrical Energy: The energy of moving electric charges (electrons).
- Light Energy: A form of electromagnetic radiation that is visible to the human eye.
- Nuclear Energy: The energy stored in the nucleus of an atom, released during nuclear reactions like fission and fusion.
In this chapter, we will focus primarily on mechanical energy, which comes in two forms: kinetic energy and potential energy.
Kinetic Energy
Think about a moving cricket ball, a speeding car, or flowing water. All these moving objects can do work. A moving ball can knock over wickets. A speeding car can cause damage in a collision. Flowing water can turn a turbine. The energy possessed by an object due to its motion is called kinetic energy.
The kinetic energy of an object increases with its speed. An object at rest has zero kinetic energy. It also depends on the mass of the object; a heavier object moving at the same speed has more kinetic energy than a lighter one.
Derivation of the Formula for Kinetic Energy (E_k)
Let's consider an object of mass 'm' moving with a uniform velocity 'u'. Let a constant force 'F' act on it, producing an acceleration 'a', and causing it to move a distance 's'. Its velocity changes to 'v'.
- The work done on the object is W = F × s.
- According to Newton's second law of motion, F = ma. So, W = (ma) × s = mas.
- From the third equation of motion, we have v² - u² = 2as. We can rearrange this to get as = (v² - u²)/2.
- Now, substitute this value of 'as' into our work equation: W = m × (v² - u²)/2, which gives W = ½ m(v² - u²).
- This work done on the object has changed its velocity, meaning its kinetic energy has changed. If the object starts from rest, its initial velocity 'u' is 0.
- In this case, the work done is W = ½ mv².
This work done is equal to the kinetic energy gained by the object. Therefore, the formula for kinetic energy is:
E_k = ½ mv²
Where 'm' is the mass and 'v' is the velocity of the object.
Potential Energy
Sometimes, energy is stored in an object not because of its motion, but because of its position or shape. This stored energy is called potential energy. It is the 'potential' for the object to do work.
Consider a stretched rubber band. It has the potential to do work (it can be launched to hit a target). A wound-up toy car spring has potential energy. Water stored at a height in a dam has the potential to do work by flowing down and turning turbines. This is potential energy.
Potential Energy of an Object at a Height
This is the most common type of potential energy we study at this level, often called gravitational potential energy. It is the energy stored in an object because of its vertical position or height above a reference point (usually the ground).
Derivation of the Formula for Potential Energy (E_p)
Let's find the expression for the potential energy of an object of mass 'm' at a height 'h' above the ground.
- To lift the object to this height, we must do work against the force of gravity.
- The minimum force required to lift the object is equal to its weight, which is F = mg (where 'g' is the acceleration due to gravity).
- The object is moved through a vertical distance (displacement) 'h'.
- The work done on the object is W = Force × displacement = (mg) × h = mgh.
This amount of work done is stored in the object as its potential energy. Therefore, the formula for gravitational potential energy is:
E_p = mgh
Important points about potential energy:
- It is a relative quantity. The height 'h' depends on the chosen 'zero level' or reference point. For example, the potential energy of a book on a table can be calculated relative to the tabletop or relative to the floor of the room.
- The path taken to raise the object to the height 'h' does not matter. The work done against gravity is the same whether you lift it straight up or take a longer, winding path.
Law of Conservation of Energy
One of the most fundamental principles in all of science is the Law of Conservation of Energy. It governs all energy transformations, from the smallest atomic interactions to the largest cosmic events.
Statement of the Law
The Law of Conservation of Energy states that energy can neither be created nor destroyed; it can only be changed from one form to another. The total energy of an isolated system remains constant over time.
This means that whenever energy changes from one form to another, the total amount of energy remains the same. You can't get energy from nothing, and you can't make energy disappear completely.
Conservation of Mechanical Energy
A direct consequence of this law is the conservation of mechanical energy in certain ideal situations. If we consider a system where only gravitational forces are doing work (i.e., we ignore friction and air resistance), then the total mechanical energy of the system remains constant.
Total Mechanical Energy = Kinetic Energy + Potential Energy = Constant
E = E_k + E_p = Constant
Example: A Freely Falling Body
Let's prove this by considering an object of mass 'm' falling freely from a height 'h'.
Position A (At the top, height h):
- The object is stationary, so its initial velocity is 0.
- Kinetic Energy (KE) = ½ m(0)² = 0.
- Potential Energy (PE) = mgh.
- Total Energy = KE + PE = 0 + mgh = mgh.
Position B (During the fall, at height x from the ground):
- The object has fallen a distance of (h - x).
- Let its velocity at this point be v. Using v² = u² + 2as, we get v² = 0 + 2g(h-x) = 2g(h-x).
- Kinetic Energy (KE) = ½ mv² = ½ m[2g(h-x)] = mg(h-x).
- Potential Energy (PE) at height x is mgx.
- Total Energy = KE + PE = mg(h-x) + mgx = mgh - mgx + mgx = mgh.
Position C (Just before hitting the ground, height 0):
- The object has fallen the full distance 'h'.
- Let its final velocity be v_f. Using v_f² = u² + 2as, we get v_f² = 0 + 2gh = 2gh.
- Kinetic Energy (KE) = ½ mv_f² = ½ m(2gh) = mgh.
- Potential Energy (PE) = mg(0) = 0.
- Total Energy = KE + PE = mgh + 0 = mgh.
As you can see, the total mechanical energy at points A, B, and C is the same (mgh). As the object falls, its potential energy decreases, while its kinetic energy increases, but their sum remains constant. This is a perfect illustration of the conservation of energy.
Rate of Doing Work: Power
Imagine two people lifting identical boxes to the same height. One person does it in 10 seconds, while the other takes 30 seconds. Both have done the same amount of work (W = mgh), but they did it at different rates. The concept that describes this rate is 'power'.
Definition of Power
Power is defined as the rate of doing work or the rate of transfer of energy.
If an agent does work 'W' in time 't', then the power is given by the formula:
Power (P) = Work (W) / time (t)
Since work is the transfer of energy, we can also say:
Power (P) = Energy (E) / time (t)
Unit of Power
The SI unit of power is the watt (W), named after the Scottish inventor and engineer James Watt.
1 watt is the power of an agent which does work at the rate of 1 joule per second.
So, 1 W = 1 J/s.
The watt is a relatively small unit, so for many applications, we use larger units:
- Kilowatt (kW): 1 kW = 1000 W
- Megawatt (MW): 1 MW = 1,000,000 W
Another common unit for power, especially for engines, is horsepower (hp), where 1 hp is approximately 746 W.
Commercial Unit of Energy
When we talk about electricity consumption in our homes and industries, the joule is a very small and inconvenient unit. A 100-watt bulb running for just 10 seconds consumes 1000 joules of energy! For this reason, a much larger unit called the kilowatt-hour (kWh) is used for commercial purposes. This is the unit you see on your electricity bill.
What is 1 kWh?
1 kilowatt-hour is the amount of energy consumed when an electrical appliance having a power rating of 1 kilowatt is used for 1 hour.
Relationship between kWh and Joules
Let's convert 1 kWh into joules:
- 1 kWh = 1 kilowatt × 1 hour
- We know that 1 kW = 1000 W and 1 hour = 60 minutes × 60 seconds = 3600 s.
- So, 1 kWh = 1000 W × 3600 s
- Since 1 W = 1 J/s, this becomes: 1 kWh = 1000 J/s × 3600 s
- 1 kWh = 3,600,000 J
- 1 kWh = 3.6 × 10⁶ J
The electricity meter in your home measures energy consumption in these 'units', where 1 unit = 1 kWh.
Important Questions and Answers
Here are some solved problems to help solidify your understanding of the concepts from this chapter.
Question 1: A force of 7 N acts on an object. The displacement is, say, 8 m in the direction of the force. Let us take it that the force acts on the object through the displacement. What is the work done in this case?
Answer:
Given:
- Force (F) = 7 N
- Displacement (s) = 8 m
The force and displacement are in the same direction.
The formula for work done is W = F × s.
Substituting the given values:
W = 7 N × 8 m = 56 N m
Since 1 N m = 1 joule (J),
Work done (W) = 56 J.
Question 2: An object of mass 15 kg is moving with a uniform velocity of 4 m/s. What is the kinetic energy possessed by the object?
Answer:
Given:
- Mass of the object (m) = 15 kg
- Velocity of the object (v) = 4 m/s
The formula for kinetic energy is E_k = ½ mv².
Substituting the given values:
E_k = ½ × 15 kg × (4 m/s)²
E_k = ½ × 15 × 16 J
E_k = 15 × 8 J
Kinetic Energy (E_k) = 120 J.
Question 3: Find the energy in kWh consumed in 10 hours by four devices of power 500 W each.
Answer:
First, calculate the total power of the four devices.
- Power of one device = 500 W
- Total power (P) = 4 × 500 W = 2000 W
To calculate energy in kWh, we need power in kW. Convert watts to kilowatts:
P = 2000 W = 2000 / 1000 kW = 2 kW
Now, calculate the energy consumed.
- Time (t) = 10 hours
- Energy (E) = Power (P) × time (t)
E = 2 kW × 10 h = 20 kWh
The total energy consumed by the four devices is 20 kWh (or 20 units).
Question 4: An object of mass 40 kg is raised to a height of 5 m above the ground. What is its potential energy? If the object is allowed to fall, find its kinetic energy when it is half-way down. (Take g = 10 m/s²)
Answer:
Part 1: Potential Energy at the top
Given:
- Mass (m) = 40 kg
- Height (h) = 5 m
- Acceleration due to gravity (g) = 10 m/s²
The formula for potential energy is E_p = mgh.
E_p = 40 kg × 10 m/s² × 5 m = 2000 J
The potential energy of the object at a height of 5 m is 2000 J.
Part 2: Kinetic Energy half-way down
By the law of conservation of energy, the total mechanical energy remains constant (assuming no air resistance). The total energy at the top is 2000 J (all potential).
Half-way down, the height of the object is h' = 5 m / 2 = 2.5 m.
Let's calculate the potential energy at this new height:
E_p' = mgh' = 40 kg × 10 m/s² × 2.5 m = 1000 J
Now, use the conservation of energy principle:
Total Energy = Potential Energy at half-way + Kinetic Energy at half-way
2000 J = 1000 J + E_k'
E_k' = 2000 J - 1000 J = 1000 J
The kinetic energy of the object when it is half-way down is 1000 J.
Question 5: Define 1 watt of power.
Answer:
One watt is the power of an agent or a machine that does work at the rate of 1 joule per second. If a device consumes or converts energy at a rate of 1 joule every second, its power is said to be 1 watt.
1 watt = 1 joule / 1 second (1 W = 1 J/s).
Chapter Summary
Here is a quick summary of the key concepts, formulas, and units we have covered in this comprehensive guide to Work and Energy.
- Work Done: Work is done when a force causes displacement. Formula: W = F × s. The SI unit is the joule (J).
- Types of Work: Work can be positive (force and displacement in the same direction), negative (force and displacement in opposite directions), or zero (no displacement or force perpendicular to displacement).
- Energy: The capacity to do work. The SI unit is also the joule (J).
- Kinetic Energy (E_k): Energy possessed by an object due to its motion. Formula: E_k = ½ mv².
- Potential Energy (E_p): Energy stored in an object due to its position or configuration. For an object at a height 'h', the gravitational potential energy is given by: E_p = mgh.
- Law of Conservation of Energy: Energy can neither be created nor destroyed, only transformed from one form to another. For a system under gravitational force alone, the total mechanical energy (KE + PE) is conserved.
- Power: The rate at which work is done or energy is transferred. Formula: P = W / t. The SI unit is the watt (W).
- Commercial Unit of Energy: The kilowatt-hour (kWh), commonly called a 'unit'. It is the energy used by a 1 kW device in 1 hour. 1 kWh = 3.6 × 10⁶ J.
