Work, Energy, and Power
Students apply conservation of energy to explain transfers among kinetic, gravitational, elastic, and thermal energy in physical systems.
Illustrations are auto-generated and may be placeholders. They can be refreshed to match the narration.
Work and Energy Transfer
Work is a transfer of energy that occurs when a force acts through a displacement. For a constant force, work is calculated with W = Fd cos θ, where F is force, d is displacement, and θ is the angle between them. Work and energy are measured in joules. Positive work adds energy to a system, negative work removes energy, and a perpendicular force does no work. For example, a student pulls a crate 5.0 meters with a 40-newton horizontal force. The work done on the crate is W = (40 N)(5.0 m) = 200 J. If friction does −75 J of work, the net work is 125 J. Defining the system boundary is important because energy crossing that boundary can be tracked as work, heating, or another transfer process.
Kinetic Energy
Kinetic energy is the energy an object has because of its motion. For an object moving much slower than the speed of light, K = 1/2 mv², where m is mass and v is speed. The work-energy theorem states that the net work on an object equals its change in kinetic energy: Wnet = ΔK. Because speed is squared, doubling speed produces four times as much kinetic energy. Consider a 1,000-kilogram car traveling at 20 meters per second. Its kinetic energy is 1/2(1,000 kg)(20 m/s)² = 200,000 J. Increasing its speed to 40 meters per second raises its kinetic energy to 800,000 J. This relationship helps explain why higher-speed vehicles require much greater stopping distances when braking forces are similar.
Potential Energy
Potential energy is stored energy associated with the arrangement of a system. Near Earth's surface, gravitational potential energy is Ug = mgh, where h is measured from a chosen zero level. Raising a 2.0-kilogram backpack 3.0 meters increases its gravitational potential energy by (2.0 kg)(9.8 m/s²)(3.0 m) = 58.8 J. Elastic potential energy is stored when an ideal spring is stretched or compressed: Us = 1/2 kx², where k is the spring constant and x is the change from equilibrium length. Stretching a 200-newton-per-meter spring by 0.10 meter stores 1.0 J. Potential energy belongs to the interacting system, such as the backpack-Earth system or spring-object system. Only differences in potential energy affect motion, so the zero level may be selected for convenience.
Conservation of Mechanical Energy
In an isolated system, total energy is conserved: energy changes form but is neither created nor destroyed. Mechanical energy is the sum of kinetic and potential energies. If only conservative forces act, Ki + Ui = Kf + Uf. For a 2.0-kilogram cart released from rest 5.0 meters above the bottom of a frictionless track, its initial gravitational potential energy is 98 J. At the bottom, that energy has become 98 J of kinetic energy, giving a speed of about 9.9 meters per second. On a real track, friction transforms some mechanical energy into thermal energy. A more complete model is Ki + Ui = Kf + Uf + ΔEth. If 18 J becomes thermal energy, only 80 J remains as kinetic energy at the bottom. Total energy is still conserved even though mechanical energy decreases.
Power and Efficiency
Power describes how quickly energy is transferred or work is done. Average power is P = W/Δt = ΔE/Δt and is measured in watts, where 1 W = 1 J/s. A motor that lifts a 600-newton load 4.0 meters in 8.0 seconds does 2,400 J of work and delivers 300 W of useful output power. Efficiency compares useful output energy or power with total input: efficiency = useful output/input × 100%. If the motor draws 500 W, its efficiency is 300/500 × 100% = 60%; the remaining power is transferred mainly as thermal energy and sound. Efficiency influences economic choices. Lower operating costs, rebates, fuel prices, and emissions rules can encourage consumers and companies to choose efficient motors or renewable technologies, even when their purchase prices are higher. Incentives therefore affect which energy resources and technologies are adopted.