Question

State the work energy theorem.

Answer 1

Hint:Here we will first learn about kinetic energy and then derive the work energy theorem. The work-energy theorem is deduced in this section. The work-energy theorem is also known as the theory of work and kinetic energy.

Complete answer:
The work-energy theorem statement says that the overall work performed by all forces acting on a particle equals the difference in the particle's kinetic energy. This can be explained by the function of rotational kinetic energy and torque, this explanation can be applied to rigid bodies.

Let us now know what Kinetic Energy is, and how does it work?
Kinetic energy is the energy which an object has as a result of its motion or movement. It is such an energy of motion, that can be observed as the movement of an object, particle, or group of particles.

Derivation of the Work-Energy Theorem: The work performed by the net force on a particle is proportional to the change in kinetic energy of the particle (KE).
$d=\dfrac{{{v}_{f}}^{2}-{{v}_{i}}^{2}}{2a}$
Consider a situation where the resulting force ‘F' is constant in both direction and magnitude and is parallel to the particle's velocity. The particle is travelling in a straight line with continuous acceleration.

The equation”$F=ma$” (Newton’s second law of motion), gives the relationship between acceleration and net power, the particle's displacement, $d$, can be calculated using the following equation:
${{v}_{f}}^{2}={{v}_{i}}^{2}+2ad$
Obtaining,
$W=\Delta KE=\dfrac{1}{2}m{{v}_{f}}^{2}-\dfrac{1}{2}m{{v}_{i}}^{2}$

The product of the net force's magnitude ($F=ma$) and the particle's displacement is used to measure the net force's function. When the above equations are substituted, we get:
$W=Fd \\
\Rightarrow W=ma\dfrac{{{v}_{f}}^{2}-{{v}_{i}}^{2}}{2a} \\
\Rightarrow W=\dfrac{1}{2}m{{v}_{f}}^{2}-\dfrac{1}{2}m{{v}_{i}}^{2} \\
\Rightarrow W=K{{E}_{f}}-K{{E}_{i}} \\
\therefore W=\Delta KE$

Note:One must know that work done is a scalar quantity. Which means it has only magnitude and no direction. The unit in which the work done is measured is called joules.