Work, Energy and PowerMind Map

Work is done when a force produces displacement of its point of application. For a constant force, work is W = Fs cosθ, where θ is the angle between force and displacement. Work is positive when force helps motion, negative when force opposes motion and zero when force is perpendicular to displacement or displacement is zero. For variable force, work is found by adding small works and equals area under the force-displacement graph. When multiple forces act, net work is the sum of work done by all forces or work done by the resultant force. The work-energy theorem states that net work done on a particle equals change in kinetic energy.

Kinetic energy is the energy possessed by a body due to motion. For translational motion, K = 1/2 mv², so it depends on mass and square of speed. It is also related to momentum by K = p²/2m. Work done by net force changes kinetic energy. Potential energy is energy stored because of position or configuration. Near Earth’s surface, gravitational potential energy is U = mgh relative to a chosen reference level. More generally, gravitational potential energy is negative and depends on separation from Earth or another mass. Mechanical energy is the sum of kinetic and potential energies. Many problems involve conversion between kinetic and potential energy.

A variable force changes in magnitude or direction during motion. Its work cannot generally be calculated by simple W = Fs cosθ; instead, the displacement is divided into small parts and work is integrated. Graphically, work done by a force along a line is the area under the force-displacement graph. Conservative forces are special forces for which work depends only on initial and final positions, not on path. Gravity and spring force are common examples. For conservative forces, work done equals negative change in potential energy. Non-conservative forces, such as friction and air resistance, depend on path and usually convert mechanical energy into heat.

The law of conservation of energy states that energy can neither be created nor destroyed; it only transforms from one form to another. In mechanics, if only conservative forces such as gravity or spring force do work, total mechanical energy remains constant. Thus K + U at one point equals K + U at another point. This principle solves free fall, smooth inclined plane, pendulum and roller coaster problems without directly using acceleration equations. In free fall, gravitational potential energy converts into kinetic energy. On a smooth incline, loss of height decides speed, not the path length. In a pendulum, energy exchanges between kinetic and potential forms during swing.

A spring stores energy when stretched or compressed. For an ideal spring, Hooke’s law gives F = -kx, where k is the spring constant and x is displacement from natural length. The elastic potential energy stored is U = 1/2 kx². This energy can convert into kinetic energy when the spring is released. Springs connected in series become softer, while springs in parallel become stiffer. Power measures how fast work is done or energy is transferred. Average power is total work divided by total time, while instantaneous power is the rate at a particular instant and can be written as P = F · v. Efficiency measures useful output compared with input.

A collision is a short-duration interaction between bodies during which internal forces are very large compared with external forces. Therefore, total linear momentum of the system is conserved if external impulse is negligible. Kinetic energy may or may not be conserved. In an elastic collision, both momentum and kinetic energy are conserved. In an inelastic collision, momentum is conserved but kinetic energy decreases, usually converting into heat, sound or deformation. In a perfectly inelastic collision, bodies stick together. The coefficient of restitution measures how elastic a collision is and is the ratio of relative speed of separation to relative speed of approach. One-dimensional collisions are common NEET problems.