Work and Energy

Work and Power

In board examinations, work is defined mathematically as the scalar (dot) product of force ($\vec{F}$) and displacement ($\vec{d}$), expressed as $W = Fd \cos\theta$. Students must remember that work is a scalar quantity and can be positive, negative, or zero depending on the angle $\theta$ between the force and displacement vectors. Zero work is done when force and displacement are perpendicular ($\theta = 90^\circ$). Power ($P$) is another vital concept, defined as the rate of doing work ($P = W/t$ or $P = \vec{F} \cdot \vec{v}$), frequently tested via numerical problems involving pumps, engines, and elevators.

Kinetic and Potential Energy

Energy is the capacity to do work. Kinetic Energy (KE) is the energy possessed by a body due to its motion, given by $KE = \frac{1}{2}mv^2$. The Work-Energy Principle—which states that the work done on a body equals the change in its kinetic energy ($W = \Delta KE$)—is a hot-favorite for short questions and numericals. Potential Energy (PE), on the other hand, is stored energy due to position or configuration. For uniform gravitational fields, gravitational potential energy is expressed as $PE = mgh$. Students must clearly distinguish between conservative forces (like gravity and electrostatic forces, where work done is path-independent and total mechanical energy is conserved) and non-conservative forces (like friction, where mechanical energy is dissipated).

Work Done by Variable Force and Conservation of Energy

When the applied force varies in magnitude or direction (such as the force exerted by a spring, $F = kx$), the total work done is calculated by graphically dividing the force-displacement curve into small rectangular strips and taking the sum of areas, or using integration. This leads directly to the Law of Conservation of Energy, which states that energy can neither be created nor destroyed, though it can transform from one form to another. In conservative systems, the total mechanical energy ($E = KE + PE$) remains constant. Board numericals frequently test this principle using freely falling bodies, simple pendulums, and mass-spring systems.