Work, Energy And Power

Work

Definition: Work is defined as the transfer of energy when a force causes an object to move in the direction of the force. Mathematically, work ($W$) is given by: $W = \text{Force} \times \text{Distance} \times \cos(\theta)$ where:

• Force ($F$) is the applied force.
• Distance ($d$) is the distance moved in the direction of the force.
• $\theta$ is the angle between the force and the direction of movement.

Work Done by a Constant Horizontal Force

When a force is applied horizontally and causes an object to move horizontally, the angle $\theta$ between the force and the direction of movement is zero. Hence, the work done ($W$) is: $W = \text{Force} \times \text{Distance} \times \cos(0^\circ)$ Since $\cos(0^\circ) = 1$, $W = \text{Force} \times \text{Distance}$ The work done is simply the product of the force applied and the distance moved in the direction of the force.

Work Done by a Constant Force Acting Obliquely

When a force is applied at an angle to the direction of movement, the work done is given by: $W = \text{Force} \times \text{Distance} \times \cos(\theta)$ where $\theta$ is the angle between the force and the direction of movement. This formula accounts for only the component of the force that acts in the direction of movement.

Types of Work Done

1. Positive Work: When the force and displacement are in the same direction, work is positive. For example, lifting an object vertically involves positive work.

2. Negative Work: When the force and displacement are in opposite directions, work is negative. For example, lowering an object involves negative work, as the force applied (downwards) is opposite to the displacement (upwards).

3. Zero Work: When the force is perpendicular to the direction of displacement or when there is no displacement, the work done is zero. For example, carrying an object horizontally while walking does not involve work in the direction of movement (if the object is lifted and held at a constant height).

Work Done Against Gravity

When an object is lifted against the force of gravity, the work done is: $W = mgh$ where:

• m is the mass of the object,
• g is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$ on Earth),
• h is the height to which the object is lifted.

This work is equal to the increase in the object’s gravitational potential energy.

Understanding these concepts helps in analyzing different scenarios involving forces and motion, and how energy is transferred or transformed in various physical processes.

Energy

Definition: Energy is the ability to do work or cause change. It exists in various forms and can be converted from one form to another.

Kinetic Energy

Definition: Kinetic energy is the energy possessed by an object due to its motion.

Formula: 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 its velocity.

Work-Energy Theorem

Definition: The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy.

Formula: If $W$ is the work done on an object and $KE_i$ and $KE_f$ are its initial and final kinetic energies, respectively, then: $W = KE_f - KE_i$

Potential Energy

Definition: Potential energy is the energy possessed by an object due to its position or configuration.

Types of Potential Energy:

1. Gravitational Potential Energy: Energy due to an object's height above the ground.
2. Elastic Potential Energy: Energy stored in elastic materials when they are stretched or compressed (e.g., in springs).

Potential Energy of a Body at Some Height:

Formula: The gravitational potential energy ($PE$) of an object at height $h$ above the ground is given by: $PE = mgh$ where:

• m is the mass of the object,
• g is the acceleration due to gravity (approximately $9.8 \, \text{m/s}^2$ on Earth),
• h is the height above the reference point.

Forms of Energy

Energy exists in various forms, including:

• Kinetic Energy: Energy of motion.
• Potential Energy: Stored energy due to position or configuration.
• Thermal Energy: Energy due to temperature.
• Chemical Energy: Energy stored in chemical bonds.
• Electrical Energy: Energy from electric currents.
• Nuclear Energy: Energy from nuclear reactions.

Transformation of Energy

Energy can be transformed from one form to another. For example:

• Mechanical Energy to Thermal Energy: Friction converts mechanical energy into heat.
• Chemical Energy to Kinetic Energy: Burning fuel releases chemical energy that powers vehicles.

Law of Conservation of Energy

Definition: The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. The total energy in an isolated system remains constant.

Law of Conservation of Energy of a Freely Falling Body

For a freely falling body, the potential energy decreases while the kinetic energy increases. The total mechanical energy (sum of kinetic and potential energy) remains constant if air resistance is neglected.

Formula: If $PE$ is converted into $KE$, then: $mgh = \frac{1}{2} mv^2$ where:

• h is the height from which the object falls,
• v is the velocity of the object just before impact.

Rate of Doing Work (Power)

Definition: Power is the rate at which work is done or energy is transferred.

Formula: Power ($P$) is given by: $P = \frac{\text{Work}}{\text{Time}}$ or $P = \frac{\text{Energy}}{\text{Time}}$

Unit: The unit of power is the watt (W), where one watt equals one joule per second.

Commercial Unit of Energy

Definition: In the commercial context, energy is often measured in kilowatt-hours (kWh), which is used to quantify electrical energy consumption.

Conversion: 1 kilowatt-hour is equal to 3.6 million joules (1 kWh = 3.6 × 10^6 J).

Understanding these concepts provides a comprehensive view of how energy functions in different scenarios, how it is transferred, and how it is conserved in physical processes.