Work, , and efficiency
is a quantity that can be transferred between systems and stored in different ways. Work is an transfer associated with a acting through a . For a constant in the direction of motion. If the is perpendicular to the , it does no work on the object in that direction, despite possibly changing the direction of motion.
Kinetic is the associated with motion and is given by Eₖ = ½mv². The square on is important: doubling quadruples kinetic if mass stays constant. Near Earth’s surface, a change in gravitational potential can be calculated using ΔEₚ = mgΔh. Choose a reference height and use consistent units of kilograms, metres and seconds.
is conserved: it cannot be created or destroyed, although it can move between stores and spread into the surroundings. When a ball falls, gravitational potential can become kinetic . Air transfers some to the surroundings as internal . A statement that is ‘lost’ usually means it is less available for the intended useful task, rather than that it has vanished.
is the rate of transfer, and is measured in watts. One watt is one joule per second. Two machines can perform the same amount of work, but the one that finishes in a shorter time has the greater average . is not the same as or total ; identify which quantity the question actually requests.
Efficiency compares useful output or useful output with the corresponding total input. Efficiency = useful output/input, and multiplying by 100 expresses it as a percentage. Keep both quantities as or both as . A motor receiving 500 J and delivering 350 J of useful mechanical has efficiency 70%, while the remaining 150 J is transferred elsewhere.
In calculations, first name the stores at the start and end, then identify transfers and any stated assumptions. For ideal lifting at constant , the minimum useful work is mgh. For real equipment, the electrical input must be larger if efficiency is below 100%. A result exceeding 100% efficiency under ordinary definitions indicates a calculation, unit or interpretation error that should be checked.
Defining the system and the transfer
Begin by deciding what belongs to the system. Lifting a bag increases the gravitational potential of the bag–Earth system. between a sliding box and a floor transfers into internal stores in the box and floor. A useful account names initial stores, final stores and any crossing the system boundary. diagrams can track these transfers without suggesting is a substance that disappears. In a closed account total input equals total output plus the change in stored . Mechanical alone can decrease even though total is conserved.
at an angle and variable forces
For a constant F at angle θ to a straight s, work by that is cosθ. Only the component parallel to contributes. A person holding a stationary heavy bag does no mechanical work on the bag during that interval because is zero, even though muscles use internally. A resistive can do negative work when it opposes . For a varying with position, the signed area under a – graph gives work. A triangular graph can be evaluated geometrically without pretending its maximum acted over the whole distance.
The work–kinetic- connection
Net work by all forces on a body equals its change in kinetic . Starting from v² −u² =2as and using F =ma gives Fs =½mv² −½mu² for constant net along motion. This avoids finding time when a braking or distance is required. Apply the resultant , or account for each 's work separately. If a car's doubles at the same mass, kinetic becomes four times as large. With approximately constant braking , braking distance also becomes four times as large under that ideal model.
Elastic and Hooke's law
Within its linear elastic range a spring obeys, where x is extension from natural length and k is spring constant in N/m. Extension is not total length. Work to stretch it slowly from zero to x is the triangular area under its –extension graph,½kx². This assumes the spring stays in its elastic linear range. Beyond the limit of proportionality the graph is no longer straight, and beyond the elastic limit permanent deformation may remain. A suspended mass at rest satisfies balance is different from the spring- formula.
at steady
is work per time. For steady parallel to motion, substituting into P =W/t gives P =Fv. A vehicle climbing a slope at constant supplies against gravity and . Constant means zero resultant , not zero transfer. Its driving balances opposing components while doing positive work. Distinguish average over a changing journey from instantaneous at one . A machine rated in kilowatts states how quickly it transfers under specified conditions, not how much it can transfer over unlimited time.
Efficiency in combined processes
Use the same interval and matching categories in efficiency calculations. A pump raising water delivers useful mgh while input may be VIt. If a motor is80% efficient and its transmission75% efficient, combined efficiency is0.80 ×0.75 =0.60, or60%. Adding the percentages gives an impossible interpretation. Input required for a specified useful output is useful output divided by decimal efficiency. Efficiency can improve by reducing and unwanted heating, but devices still obey conservation and practical limits. A percentage above100 under ordinary definitions means the account or calculation needs checking.
resources and society
Renewable sources are replenished on relevant human timescales, while fossil fuels and nuclear fuel resources are finite. Solar photovoltaic cells convert incoming radiation into electrical output, whereas a solar thermal collector transfers into a heated fluid. Hydroelectric systems use the gravitational of water passing through turbines; practical dam siting considers reliable water , height difference, geology and environmental effects. Wind generation depends on moving air and suitable equipment. Compare resources using reliability, storage, transmission, cost boundaries and environmental effects, not the word renewable alone. Diversification can reduce dependence on one source without removing every constraint.
Units, bills and self-check
One kilowatt-hour is1,000. A60 W lamp for five hours uses0.30 kWh; multiply by the tariff provided rather than inventing a price. A4 kg object falling5 m from rest with negligible and kinetic , giving speed10 m/s. If transfers40 J elsewhere, only160 J becomes kinetic and is√80 ≈8.94 m/s. For a motor receiving1,200 J and lifting6 kg through10 m, useful output is600 J and efficiency50%. State assumptions and define useful output before substituting.