Work, Energy and Power

7.1 Scientific Definition of Work

Work, in a scientific context, is defined as the product of force and displacement in the direction of that force. It is mathematically expressed as W = F \times d \times \cos(\theta), where F is the force, d is the displacement, and \theta is the angle between the force and displacement vectors. For work to be done, both a force must be exerted and there must be displacement in the direction of the force.

7.2 Net Work and the Work-Energy Theorem

Work done on a system results in energy transfer. Net work (W{\text{net}}) is the sum of work done by all forces acting on an object. The Work-Energy Theorem states that the net work done on a system equals the change in its kinetic energy (W{\text{net}} = \Delta KE). Kinetic energy (KE) is the energy of motion, given by KE = \frac{1}{2}mv^2.

7.3 Gravitational Potential Energy and Conservative Forces

Work done against gravity, such as lifting an object, results in gravitational potential energy (PEg), defined as PEg = mgh. This energy is stored in the object-Earth system due to its position. A conservative force is one for which the work done depends only on the initial and final positions, not the path taken (e.g., gravitational force, spring force). For any conservative force, a potential energy (PE) can be defined. The potential energy stored in a spring is defined as PEs = \frac{1}{2}kx^2. When only conservative forces act, the total mechanical energy (KE + PE) of a system remains constant, expressed as the conservation of mechanical energy principle: KEi + PEi = KEf + PE_f.

7.6 Law of Conservation of Energy

The Law of Conservation of Energy states that total energy is constant in any process, undergoing transformations or transfers but never being created or destroyed. This can be broadly expressed as KEi + PEi + W{\text{nc}} + OEi = KEf + PEf + OEf, where OE represents other forms of energy (electrical, chemical, radiant, nuclear, thermal) and W{\text{nc}} is work done by nonconservative forces. Efficiency is a measure of useful energy output compared to total energy input: Eff = \frac{\text{useful energy or work output}}{\text{total energy input}}.

7.7 Power

Power (P) is the rate at which work is done or energy is expended, given by the formula P = \frac{W}{t}. The SI unit for power is the watt (W), where 1 \text{ W} = 1 \text{ J/s}. Energy consumed can be calculated as E=Pt, often measured in kilowatt-hours (kW\cdot{}h) for electrical consumption.

10.4 Rotational Work

Work done in rotational motion is analogous to translational work. For a net torque (\text{net } \tau) causing an angular displacement (\theta), the rotational work done is given by W_{\text{net}} = (\text{net } \tau)\theta.