Work Power And Energy Notes

Introduction

In the realm of physics, the concepts of work, power, and energy are fundamental to understanding the interactions and transformations that occur in physical systems. These concepts are interconnected and form the basis for analyzing various phenomena in mechanics and other areas of physics. This study note will delve into each of these concepts, providing a comprehensive understanding tailored for JEE Advanced Physics.

Work

Definition of Work

Work is defined as the transfer of energy that occurs when a force is applied to an object causing a displacement. The mathematical expression for work is:

$$ W = \vec{F} \cdot \vec{d} $$

where:

• $W$ is the work done,
• $\vec{F}$ is the force applied,
• $\vec{d}$ is the displacement of the object,
• $\cdot$ denotes the dot product.

Conditions for Work

For work to be done:

1. A force must be applied.
2. There must be displacement.
3. The force must have a component in the direction of the displacement.

Work Done by a Constant Force

If the force is constant and the displacement is in a straight line, the work done can be simplified to:

$$ W = F \cdot d \cdot \cos(\theta) $$

where $\theta$ is the angle between the force vector and the displacement vector.

Work Done by a Variable Force

When the force varies with position, the work done is given by the integral:

$$ W = \int_{x_i}^{x_f} F(x) , dx $$

Power

Definition of Power

Power is the rate at which work is done or energy is transferred. It is defined as:

$$ P = \frac{dW}{dt} $$

where:

• $P$ is the power,
• $dW$ is the infinitesimal work done,
• $dt$ is the infinitesimal time interval.

Average and Instantaneous Power

• Average Power over a time interval $\Delta t$ is: $$ P_{avg} = \frac{W}{\Delta t} $$
• Instantaneous Power is: $$ P = \vec{F} \cdot \vec{v} $$

where $\vec{v}$ is the instantaneous velocity.

Energy

Definition of Energy

Energy is the capacity to do work. It exists in various forms such as kinetic energy, potential energy, thermal energy, etc.

Kinetic Energy

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

$$ K.E. = \frac{1}{2} mv^2 $$

where:

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

Potential Energy

Potential energy is the energy possessed by an object due to its position or configuration. The most common forms are gravitational potential energy and elastic potential energy.

Gravitational Potential Energy

For an object of mass $m$ at a height $h$ in a uniform gravitational field:

$$ U = mgh $$

Elastic Potential Energy

For a spring with spring constant $k$ compressed or stretched by a distance $x$:

$$ U = \frac{1}{2} k x^2 $$

Conservation of Energy

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

$$ E_{total} = K.E. + P.E. = \text{constant} $$

Work-Energy Theorem

The work-energy theorem links work and kinetic energy. It states that the work done by the net force on an object is equal to the change in its kinetic energy:

$$ W_{net} = \Delta K.E. = \frac{1}{2} mv_f^2 - \frac{1}{2} mv_i^2 $$

Summary

Understanding work, power, and energy is crucial for analyzing mechanical systems in physics. These concepts are interrelated and provide a framework for solving a wide range of problems in mechanics. By mastering these principles, you will be well-equipped to tackle questions in JEE Advanced Physics.