Introduction
In the study of physics, the concepts of work, energy, and power are fundamental and interrelated. These concepts not only help us understand various physical phenomena but also form the basis for solving many problems in mechanics. This study note will cover the definitions, formulas, and applications of work, energy, and power, as well as provide examples and tips to help you master these topics for the NEET exam.
Work
Definition of Work
Work is done when a force acts on an object and causes it to move. The mathematical definition of work is given by:
$$ W = \vec{F} \cdot \vec{d} = Fd \cos \theta $$
where:
- $W$ is the work done
- $\vec{F}$ is the force applied
- $\vec{d}$ is the displacement of the object
- $\theta$ is the angle between the force and the displacement
Units of Work
The SI unit of work is the joule (J), where $1 , \text{J} = 1 , \text{N} \cdot \text{m}$.
Types of Work
- Positive Work: When the force and displacement are in the same direction ($0^\circ \leq \theta
< 90^\circ$). 2. Negative Work: When the force and displacement are in opposite directions ($90^\circ < \theta \leq 180^\circ$). 3. Zero Work: When the force is perpendicular to the displacement ($\theta = 90^\circ$).
ExampleExample Calculation: A force of $10 , \text{N}$ is applied to push a box $5 , \text{m}$ along the floor. The force is applied at an angle of $30^\circ$ to the horizontal. Calculate the work done by the force.
$$ W = Fd \cos \theta = 10 , \text{N} \times 5 , \text{m} \times \cos 30^\circ = 10 \times 5 \times \frac{\sqrt{3}}{2} = 25\sqrt{3} , \text{J} $$
TipAlways resolve the force into components parallel and perpendicular to the displacement to simplify the calculation of work.
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 the velocity of the object
Potential Energy
Potential energy is the energy possessed by an object due to its position or configuration. The most common form is gravitational potential energy, given by:
$$ P.E. = mgh $$
where:
- $m$ is the mass of the object
- $g$ is the acceleration due to gravity
- $h$ is the height above the reference point
Example Calculation: Calculate the kinetic energy of a $2 , \text{kg}$ object moving at a velocity of $3 , \text{m/s}$.
$$ K.E. = \frac{1}{2} mv^2 = \frac{1}{2} \times 2 , \text{kg} \times (3 , \text{m/s})^2 = 9 , \text{J} $$
NoteEnergy is a scalar quantity and does not have a direction.
Power
Definition of Power
Power is the rate at which work is done or energy is transferred. It is given by:
$$ P = \frac{W}{t} $$
where:
- $P$ is the power
- $W$ is the work done
- $t$ is the time taken
Units of Power
The SI unit of power is the watt (W), where $1 , \text{W} = 1 , \text{J/s}$.
Average and Instantaneous Power
- Average Power: The total work done divided by the total time taken.
- Instantaneous Power: The power at a specific moment, given by $P = \vec{F} \cdot \vec{v}$.
Example Calculation: A machine does $500 , \text{J}$ of work in $10 , \text{s}$. Calculate its power.
$$ P = \frac{W}{t} = \frac{500 , \text{J}}{10 , \text{s}} = 50 , \text{W} $$
Common MistakeA common mistake is to confuse power with energy. Remember, power is the rate of doing work, while energy is the capacity to do work.
Work-Energy Theorem
Statement of the Theorem
The work done by the net force on an object is equal to the change in its kinetic energy:
$$ W_{\text{net}} = \Delta K.E. = \frac{1}{2} mv_f^2 - \frac{1}{2} mv_i^2 $$
where:
- $v_f$ is the final velocity
- $v_i$ is the initial velocity
Example Calculation: A car of mass $1000 , \text{kg}$ accelerates from $10 , \text{m/s}$ to $20 , \text{m/s}$. Calculate the work done by the net force.
$$ W_{\text{net}} = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 = \frac{1}{2} \times 1000 , \text{kg} \times (20 , \text{m/s})^2 - \frac{1}{2} \times 1000 , \text{kg} \times (10 , \text{m/s})^2 $$ $$ W_{\text{net}} = 1000 \times 200 - 1000 \times 50 = 200000 - 50000 = 150000 , \text{J} $$
Conservation of Energy
Principle of Conservation of Energy
The total energy of an isolated system remains constant. Energy can neither be created nor destroyed but can only be transformed from one form to another.
Mechanical Energy Conservation
In the absence of non-conservative forces (like friction), the total mechanical energy (sum of kinetic and potential energy) of a system remains constant:
$$ K.E.{\text{initial}} + P.E.{\text{initial}} = K.E.{\text{final}} + P.E.{\text{final}} $$
ExampleExample Calculation: A ball of mass $2 , \text{kg}$ is dropped from a height of $10 , \text{m}$. Calculate its speed just before hitting the ground (neglect air resistance).
Initial potential energy:
$$ P.E._{\text{initial}} = mgh = 2 , \text{kg} \times 9.8 , \text{m/s}^2 \times 10 , \text{m} = 196 , \text{J} $$
Since the ball is initially at rest, $K.E._{\text{initial}} = 0$. Using energy conservation:
$$ 196 , \text{J} = \frac{1}{2} mv^2 $$ $$ 196 = \frac{1}{2} \times 2 \times v^2 $$ $$ v^2 = 196 $$ $$ v = \sqrt{196} = 14 , \text{m/s} $$
Conclusion
Understanding the concepts of work, energy, and power is crucial for solving many problems in physics. These concepts are interconnected and form the foundation for more advanced topics. By mastering these basics, you will be well-prepared to tackle questions on the NEET exam.
TipPractice solving problems from previous NEET exams to familiarize yourself with the types of questions asked and to improve your problem-solving speed and accuracy.
