Work, Energy and Power

Work

1. In physics, work is a measure of energy transfer that occurs when a force is applied to an object, causing it to move a certain distance in the direction of the force. Work is done when a force produces a displacement in the object's position.
2. The amount of work accomplished by a force hinges on two fundamental aspects: the force's strength (magnitude) and the displacement of the object in the direction of that force. This relationship is articulated by the equation:





3. In the International System of Units (SI), work is quantified in Newton metres (Nm), also known as joules (J). A joule is defined as the energy transferred when a one-newton force acts upon an object to move it one metre in the direction of the force.
4. Importantly, for work to occur, the force must induce actual motion. If the object's displacement is zero, then the work done on the object is also zero, regardless of how much force is applied. For instance, pushing a stationary wall may exhaust someone, but since no displacement occurs, the work done on the wall is zero. However, the person expends energy due to muscle exertion.

Work Done Against Gravity

When we lift an object vertically upwards, such as raising a book from a table, we are actually doing work against the force of gravity. This force, exerted by the Earth, pulls objects toward its surface. As we lift the object, we counteract this gravitational force, and the work done in doing so can be calculated using specific formulas.

Calculation of Work Against Gravity

The work done in lifting the object against gravity can be calculated using the formula:

Here, the variables stand for:

W: Work done
m: Mass of the object
g: Acceleration due to gravity
h: Vertical distance through which the object is lifted

Work Done When the Force Acts obliquely to the Direction of Motion

In many cases, the motion of an object is not in the exact direction of the applied force. For instance, when a child pulls a toy car with a string, the car moves horizontally on the ground, but the force applied by the child is at an angle to the ground due to the string's orientation. This means that the whole force is not contributing to the object's movement; only the horizontal component of the force is effective in moving the object.

In such instances, the formula W = F × s cannot be directly used, as the distance moved 's' is not parallel to the applied force 'F'. Instead, the effective force component along the direction of motion needs to be considered. This effective force is represented by 'F cos θ', where 'θ' is the angle between the direction of force and the direction of motion.

Formula for Work Done at an Angle

The formula for calculating work done when a body moves at an angle 'θ' to the direction of force is derived as follows:

Here:

W: Work done
F: Applied force
θ: Angle between force direction and motion direction
s: Distance moved

Work Done When the Force Acts at Right Angles to the Direction of Motion

When a force acts at right angles (90^o) to the direction of motion, no work is done, as the effective force component becomes zero.

Work Done When the Force Acts Opposite to the Direction of Motion

Similarly, if a force acts opposite to the direction of motion (180^o), the work done is negative. In the latter case, the work is calculated as:

Work (W) = -F × s

This means that the force is opposing the motion, leading to negative work.

Positive, Negative, and Zero Work

The concept of positive, negative, and zero work is crucial in understanding how different forces impact the motion of objects.

Positive Work: When a force acts in the same direction as the motion of an object, the work done is considered positive. In this scenario, the force contributes to increasing the speed and energy of the object. For example, if a football is kicked while lying on the ground, the force of the kick acts in the direction of the football's motion. As a result, the work done on the football is positive, causing it to accelerate and gain speed.

Negative Work: Negative work occurs when a force acts in the opposite direction to the motion of an object. In this case, the force acts to slow down or oppose the object's movement. For instance, when a moving football experiences friction with the ground, the force of friction acts against the direction of motion, resulting in negative work. This force reduces the object's speed and kinetic energy.

Zero Work: Zero work is observed when a force acts perpendicular to the direction of motion of an object. In this situation, the force doesn't influence the object's speed, and its energy remains unchanged. An example of zero work is the gravitational force acting on a satellite moving in a circular orbit around the Earth. The gravitational force acts at right angles to the direction of motion, causing no change in the satellite's kinetic energy.

In some cases, multiple forces may be acting on an object. The net work done by these forces will be the sum of their individual work contributions.

Energy

Energy is a fundamental concept that refers to the ability of a system to perform work. Work is done when a force is applied to an object, causing it to move a certain distance. Energy is what enables this process of performing work and bringing about changes in the physical world.

1. Work and Energy Relationship: Work and energy are closely connected concepts. When work is done on an object, energy is transferred to that object. Conversely, when work is done by an object, energy is transferred from that object. This connection emphasises the idea that energy is the capacity to accomplish tasks and create changes.
2. Conservation of Energy: The principle of conservation of energy states that energy cannot be created or destroyed; it can only be transferred or converted from one form to another. In a closed system, the total amount of energy remains constant over time. This principle is essential for understanding how energy moves and transforms within different systems.
3. Units of Energy: The standard unit of energy in the International System of Units (SI) is the joule (J). One joule is defined as the amount of energy transferred when a force of one Newton is applied to move an object one metre against a force.
4. Energy Transfer and Transformation: Energy can be transferred from one object to another or transformed from one form to another. For example, when a ball is thrown, the person imparts kinetic energy to the ball. As the ball rises, its kinetic energy decreases while its potential energy increases due to its elevated position.
5. Relationship of Energy with Force and Motion: Energy is deeply intertwined with the concepts of force and motion. Forces can change an object's energy by doing work on it. Likewise, the energy an object possesses can influence its motion, as seen in concepts like kinetic energy and potential energy.

Forms of Energy

Energy manifests in various forms, each playing a distinct role in the physical world. Some of the primary forms of energy include:

1. Kinetic Energy: The energy possessed by a moving object. It depends on the object's mass and velocity.
2. Potential Energy: The energy stored in an object due to its position or state. Gravitational potential energy and elastic potential energy are common examples.
3. Chemical Energy: Energy stored in the bonds between atoms and molecules, released during chemical reactions.
4. Heat Energy: Energy associated with the motion of particles within a substance. It is often transferred between objects as heat.
5. Light Energy: Energy carried by electromagnetic waves, including visible light.
6. Sound Energy: Energy produced by vibrations that travel through a medium as sound waves.
7. Electrical Energy: Energy carried by moving charges, typically in the form of electricity.
8. Nuclear Energy: Energy stored in the nucleus of an atom and released during nuclear reactions.

1. Kinetic Energy

Kinetic energy is a fundamental concept in physics that describes the energy associated with the motion of an object. It's a type of mechanical energy that arises from the movement of an object and its speed. In simpler terms, kinetic energy is the energy that an object possesses due to its motion.

Formula for Kinetic Energy: The kinetic energy (K.E.) of an object with mass 'm' and velocity 'v' is given by the formula:

In terms of momentum (p = mv), Kinetic energy can be quantified as:

Key Points about Kinetic Energy

Motion and Speed: Kinetic energy is only relevant when an object is in motion. An object at rest has no kinetic energy, regardless of its mass.

Dependence of Kinetic Energy on Mass and Speed:

• Kinetic energy is directly proportional to the mass of the object. If the mass doubles, the kinetic energy also doubles, given the same speed.
• Kinetic energy is directly proportional to the square of the speed. If the speed doubles, the kinetic energy increases by a factor of four.

Energy Transfer: When a moving object collides with another object or interacts with it, it can transfer some of its kinetic energy. This energy transfer can lead to changes in the motion or behaviour of the objects involved.

Units: The SI unit of kinetic energy is the joule (J), which is the same unit used for work. The kinetic energy of an object can be measured in joules.

Conservation of Energy: In isolated systems (where no external forces are acting), the total mechanical energy, including both kinetic and potential energy, is conserved. This principle is known as the conservation of mechanical energy.

Work-Energy Theorem: The Work-Energy Theorem is a fundamental concept in physics that relates the work done on an object to its change in kinetic energy. When work is done on an object, it can change the object's kinetic energy.
The Work-Energy Theorem states that the work done on an object is equal to the change in its kinetic energy. Mathematically, this can be written as:

This means that the work done on an object by the applied force causes a change in its kinetic energy. If the work done is positive (force and displacement are in the same direction), the kinetic energy of the object increases. If the work done is negative (force and displacement are in opposite directions), the kinetic energy decreases.

If the object starts from rest (initial velocity, u = 0), the theorem simplifies to:

2. Potential Energy

Potential energy is a form of energy associated with the position or configuration of an object relative to other objects or forces. It is the energy an object possesses due to its location within a gravitational field or due to changes in its shape or configuration. In simpler terms, potential energy is the energy that an object has stored within it, ready to be converted into other forms of energy when certain conditions are met.

Types of Potential Energy

1. Gravitational Potential Energy: This type of potential energy is associated with the position of an object in a gravitational field. The higher an object is positioned above the ground, the greater its gravitational potential energy. The formula for calculating gravitational potential energy is:

   

   Where:

   P.E. is the gravitational potential energy
   m is the mass of the object
   g is the acceleration due to gravity
   h is the height above a reference point
   

2. Elastic Potential Energy: This type of potential energy is associated with objects that can be compressed, stretched, bent, or twisted. When an object is deformed from its natural state, work is done to change its shape, and this work gets stored as elastic potential energy.

Key Points about Potential Energy

1. Dependence on Position: Gravitational potential energy depends on the height above a reference point. The higher an object is positioned, the greater its potential energy.
2. Work-Energy Principle: Potential energy is related to the concept of work. When work is done against a force (e.g., gravity) to change the position or shape of an object, potential energy is stored.
3. Conversion of Energy: Potential energy can be converted into other forms of energy. For example, when an object falls, its gravitational potential energy is converted into kinetic energy as it gains speed.
4. Mechanical Energy Conservation: In a closed system without non-conservative forces (like friction), the total mechanical energy (sum of kinetic and potential energies) remains constant. This principle is known as the conservation of mechanical energy.
5. Energy Transformations: Potential energy can be transformed into kinetic energy and vice versa. For example, when a pendulum swings, its potential energy is highest at the top of the swing, while its kinetic energy is highest at the bottom.
6. Units: The SI unit of potential energy is the joule (J), which is the same as other forms of energy.

Power

Power is the rate at which work is done or energy is transferred or converted. It measures how quickly energy is used or produced. Mathematically, power is defined as the amount of work done (or energy transferred) per unit of time. It is a measure of how quickly a task can be completed or how fast energy is being used or produced. Power is a scalar quantity; it has magnitude but no direction.

Formula for Power: The formula for calculating power is given by:

Where:

P is the power,
W is the work done or energy transferred
t is the time taken to do the work or transfer the energy.

Units of Power: The SI unit of power is the watt (W), named after James Watt, the inventor and engineer. One watt is equal to one joule of work done (or energy transferred) per second. In other words, it's the power required to do one joule of work in one second. Mathematically:

1 watt = 1 joule/second

Larger units of power are also commonly used:

• Kilowatt (kW): 1 kW = 1000 W
• Megawatt (MW): 1 MW = 1000 kW = 10⁶ W
• Horsepower (hp): Commonly used for engines and motors. 1 hp is approximately equal to 746 W.

Relation Between Power and Energy: Power and energy are related concepts. Power is the rate of energy transfer or conversion. Energy is the total amount of work done or transferred. The relationship between power, energy, and time is given by:

Where:

E is the energy (work done or transferred)
P is the power
t is the time.

Commercial Unit of Energy: The commercial unit of energy used for billing purposes is the kilowatt-hour (kWh). One kilowatt-hour is the amount of energy consumed when using a device with a power rating of one kilowatt for one hour. It's equivalent to 3.6 million joules.

1 kWh = 3.6 × 10⁶ J

This conversion allows us to relate energy usage in everyday terms to the standard SI unit of energy, the joule.

Transformation of Energy

Energy transformation refers to the process where energy changes from one form to another. This happens constantly in various systems and processes. Some examples of energy transformation are:

Law of Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed; it can only be transferred or converted from one form to another. In a closed system, the total amount of energy remains constant over time. This law is a fundamental principle of physics and is applicable to all types of energy, including mechanical, thermal, chemical, electromagnetic, and more.

Examples of Conservation of Energy

1. Free Fall of an Object: When an object falls freely under the influence of gravity, its potential energy is converted into kinetic energy as it gains speed. The total mechanical energy (sum of potential and kinetic energy) remains constant as long as air resistance is negligible.
2. Simple Pendulum: A swinging pendulum is a classic example of the conservation of mechanical energy. As it swings back and forth, its potential energy is converted into kinetic energy and vice versa. Friction and air resistance gradually dampen the motion by converting mechanical energy into heat.
3. Mechanical Work: When we perform mechanical work, the energy we put into a system is converted into various forms of energy depending on the situation. For example, when we lift an object, we transfer energy to it in the form of potential energy.
4. Electric Appliances: When electrical energy is consumed by devices like light bulbs or heaters, it is converted into light energy or heat energy. The total energy consumed remains the same, even though the initial electrical energy is transformed.
5. Chemical Reactions: In chemical reactions, the energy stored in the bonds of reactants is converted into energy stored in the bonds of products. The total energy before and after the reaction remains unchanged.
6. Thermal Energy: When hot and cold objects are in contact, heat energy flows from the hotter object to the colder one until thermal equilibrium is reached. While the temperature changes, the total energy remains constant.

Implications of the Law

The law of conservation of energy has profound implications in various scientific and practical contexts:

1. Energy Efficiency: In any process or device, the goal is to minimise energy losses and maximise useful energy output. This is crucial for energy-efficient technologies and sustainable practices.
2. Renewable Energy: The transformation of energy from renewable sources like solar, wind, and hydroelectric power adheres to the law of conservation of energy. These sources harness existing energy and convert it into usable forms.
3. Environmental Impact: Conservation of energy is relevant to environmental issues. For instance, the conversion of energy in vehicles contributes to emissions and pollution, emphasising the need for cleaner energy sources.