Work, Energy, and Power

Work, Energy, and Power

Work and energy are central concepts in physics that describe the transfer of energy between systems.

Work Done by Constant and Variable Forces

Work ($W$): The product of force and displacement.

• Constant Force: $W = \mathbf{F} \cdot \mathbf{d} = F d \cos \theta$
• Variable Force: $W = \int_{x_i}^{x_f} F(x) \, dx$

Work is a scalar quantity. The unit is the Joule (J).

Kinetic and Potential Energy

Kinetic Energy (KE)

The energy of motion. $KE = \frac{1}{2} m v^2$

Potential Energy (PE)

The energy stored in a system due to its configuration.

• Gravitational PE: $PE_g = mgh$ (near Earth's surface).
• Elastic PE: $PE_s = \frac{1}{2} k x^2$.

Conservation of Mechanical Energy

The total mechanical energy ($E$) of a system is the sum of its kinetic and potential energies: $E = KE + PE$

Conservation Principle: If only conservative forces (like gravity and spring force) do work, the total mechanical energy is conserved. $E_i = E_f$ $KE_i + PE_i = KE_f + PE_f$

Non-conservative forces (like friction) dissipate energy as heat or other forms. In this case: $\Delta E = W_{nc}$ where $W_{nc}$ is the work done by non-conservative forces.

Power

Power ($P$) is the rate at which work is done or energy is transferred.

• Average Power: $\bar{P} = \frac{W}{\Delta t}$
• Instantaneous Power: $P = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v}$

The unit of power is the Watt (W). $1 \text{ W} = 1 \text{ J/s}$ Common conversions: $1 \text{ hp} = 746 \text{ W}$

In civil engineering, power calculations are essential for sizing motors for pumps, cranes, and other construction equipment.