Work, Energy & Power

Work, Energy, and Power connect forces to motion through scalar quantities, offering an alternative to Newton's Laws that is often more elegant and efficient for solving problems, especially when forces are variable or when only the initial and final states matter.

Work (W): The scalar product of force and displacement: W = F .
s = Fscos($\theta$), where $\theta$ is the angle between F and s.
SI unit: Joule (J) = Nm = kg*$\frac{m^{2}}{s^{2}}$. Dimension: [$ML^{2}$T^{-2}].

Work can be positive (force along displacement, e.g., pushing a car), negative (force opposite to displacement, e.g., friction), or zero (force perpendicular to displacement, e.g., normal force on horizontal surface, centripetal force in circular motion).

For variable force: W = integral of F.ds (area under F-x graph).

Work-Energy Theorem: The net work done on a body equals its change in kinetic energy: $W_{net}$ = $\delta_{KE}$ = ($\frac{1}{2}$)$mv^{2}$ - ($\frac{1}{2}$)$\mu$². This is a scalar equation — no direction headaches.

Kinetic Energy (KE): KE = ($\frac{1}{2}$)$mv^{2}$ = $p^{2}$/(2m). Dimension: [$ML^{2}$T^{-2}]. Always non-negative.

Potential Energy (PE): Energy due to position/configuration.

• Gravitational: U = mgh (taking reference level where U = 0)
• Spring/Elastic: U = ($\frac{1}{2}$)$kx^{2}$
• PE is defined only for conservative forces.

Conservation of Mechanical Energy: If only conservative forces do work: KE + PE = constant.
Or: ($\frac{1}{2}$)$mv_{1}^{2}$ + $U_{1}$ = ($\frac{1}{2}$)$mv_{2}^{2}$ + $U_{2}$.

Conservative vs Non-conservative Forces:

• Conservative: work done is path-independent (gravity, spring force, electrostatic). W in closed loop = 0. F = -dU/dx.
• Non-conservative: work is path-dependent (friction, air resistance, viscous force). Energy is dissipated.

Power: Rate of doing work: P = dW/dt = F.v (instantaneous). SI unit: Watt (W) = J/s = kg*$\frac{m^{2}}{s^{3}}$. Dimension: [$ML^{2}$T^{-3}]. Average power: $P_{avg}$ = W/t.

The key problem-solving concept is: use energy methods when you need to relate velocities at two points without caring about the path; use Newton's Laws when you need forces and accelerations explicitly.

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