Energy, Work, and Power

The transfer of energy when a force acts on an object through a displacement. Calculated as = Fdcos heta$, where $ is the force magnitude, $ is the displacement, and $ heta$ is the angle between them. Work is measured in joules (J).

The energy an object possesses due to its motion, given by = frac{1}{2}mv^2$. Since velocity is squared, doubling speed quadruples kinetic energy. Kinetic energy is always positive and is a scalar quantity.

Energy stored due to an objects position in a gravitational field, calculated as = mgh$ near Earths surface, where $ is the height above a chosen reference level. The choice of reference level is arbitrary because only changes in PE matter physically.

In a system with only conservative forces (gravity, springs), the total mechanical energy = KE + PE$ remains constant. Energy transforms between kinetic and potential forms but the sum never changes.

The net work done on an object equals the change in its kinetic energy: {net} = Delta KE = frac{1}{2}mv_f^2 - frac{1}{2}mv_i^2$. This theorem connects the force-displacement perspective with the energy perspective of motion.

The rate at which work is done or energy is transferred: = frac{W}{t}$. For an object moving at constant velocity under a force, = Fv$. Power is measured in watts (W), where 1 W = 1 J/s.

Energy stored in a compressed or stretched spring, given by = frac{1}{2}kx^2$, where $ is the spring constant and $ is the displacement from equilibrium. This energy can be fully recovered as kinetic energy when the spring returns to its natural length.

Forces like friction and air resistance that convert mechanical energy into thermal energy. When non-conservative forces act, total mechanical energy decreases: $Delta E_{mech} = W_{nc}$, where {nc}$ is negative work done by dissipative forces.