Unit 3 Beta

Work, Energy, and Power: the one-page sheet

3.1 Translational Kinetic Energy

Kinetic energy is K = ½mv², in joules. It is a scalar, never negative, independent of direction and dependent on the reference frame. Because the speed is squared, doubling the speed quadruples K; and a change in K is ½mv_f² − ½mv_i², not ½m(Δv)².

Energy is a scalar quantity a system has; it can change form and move between objects. Its SI unit is the joule: 1 J = 1 kg·m²/s².
Kinetic energy, K = ½mv², is the energy of motion. It depends on mass and on the speed squared.
K is never negative and has no direction. A cart moving left at 3 m/s has the same K as one moving right at 3 m/s.
ΔK = ½mv_f² − ½mv_i², which is not ½m(Δv)². Square each speed first, then subtract.
Kinetic energy depends on the reference frame, because speed does.

it has kinetic energy K = ½mv², measured in joules (kg·m²/s²) doubling the speed makes K four times as large, while doubling the mass only doubles it K is a scalar that is never negative and does not depend on direction the same object has different kinetic energies in different frames its kinetic energy is the sum of the parts' kinetic energies, which never cancel

energy: A scalar quantity a system has that can change form and move between objects; measured in joules.
joule: The SI unit of energy: 1 J = 1 kg·m²/s² = 1 N·m.
kinetic energy: The energy an object has because it moves: K = ½mv². A scalar, never negative, that depends on the reference frame.

3.2 Work

Work is W = F d cos θ: energy carried into an object (positive) or out of it (negative) by a force along a displacement. Perpendicular forces do none. The net work equals ΔK, and a varying force's work is the area under its force-position graph.

Work is energy moved into or out of a system by a force acting along a displacement: W = F d cos θ, with θ between the force and the displacement. It is a scalar, in joules.
Positive work adds energy, negative work removes it, and a force perpendicular to the motion does none.
The work-energy theorem: the net work done on an object equals its change in kinetic energy, W_net = ΔK.
Work by a varying force is the area under its force-position graph.
Kinetic friction does negative work that grows with the path length, even on a round trip.

the force does work W = F d cos θ, using only the component along the displacement the work is positive and energy enters the object the work is negative and energy leaves the object it does no work and does not change the speed the net work equals the change in kinetic energy: W_net = ΔK

work (physics): Energy moved into or out of a system by a force acting along a displacement: W = F d cos θ, a scalar in joules that can be positive, negative or zero.
work-energy theorem: The net work done on an object equals its change in kinetic energy: W_net = ΔK.
force-position graph: A graph of the force along the motion against position; the area under it is the work done by the force.
energy transfer: Energy moving into or out of a system across its boundary, for example by work done by an external force.

3.3 Potential Energy

Potential energy is stored in the configuration of interacting objects and belongs to the system. ΔU_g = mgΔy near Earth, U_s = ½k(Δx)² for a spring, U_G = −Gm₁m₂/r far away. Only conservative forces have one; the zero is a free choice because only changes matter.

Potential energy is energy stored in the configuration of a system of interacting objects. One object alone has none.
Near Earth: ΔU_g = mgΔy. For an ideal spring: U_s = ½k(Δx)², with Δx measured from the relaxed length.
A conservative force does work that depends only on the start and end positions; for it, W = −ΔU. Kinetic friction is nonconservative.
The zero of potential energy is a choice. Values change with the choice; changes do not.
Far from a planet: U_G = −Gm₁m₂/r, zero at infinite separation and negative everywhere else.

the work the force does depends only on where the objects start and end the system can store energy in its configuration: potential energy, with ΔU = −W_conservative the system's potential energy increases: ΔU_g = mgΔy, U_s = ½k(Δx)² the zero of potential energy can be placed wherever is convenient it is nonconservative and has no potential energy

potential energy: Energy stored in the configuration of a system of objects that interact through conservative forces; it belongs to the system, not to one object.
conservative force: A force whose work depends only on the start and end positions (gravity, an ideal spring). A force whose work depends on the path, like kinetic friction, is nonconservative.
gravitational potential energy: The potential energy of an object-Earth system near Earth's surface: ΔU_g = mgΔy.
elastic potential energy: The potential energy stored in a stretched or compressed ideal spring: U_s = ½k(Δx)², with Δx measured from the relaxed length.
zero of potential energy: The configuration chosen to have zero potential energy; any choice works because only changes in potential energy matter.
gravitational potential energy far from a planet: The gravitational potential energy of two masses at any separation: U_G = −Gm₁m₂/r, zero at infinite separation.

3.4 Conservation of Energy

Pick the system first. Inside it energy changes form; across its boundary it moves as work by external forces. With no external work and no friction, K + U is constant. Friction turns mechanical energy into thermal energy, and energy bar charts must add up to match the equation.

A one-object system has only kinetic energy. Potential energy appears only when the interacting objects (Earth, a spring) are inside the system.
Mechanical energy is K + U. It is constant in a closed system with no external work and no internal nonconservative forces.
Conservation of energy: total energy never changes; it moves across the boundary as work or changes form inside.
Kinetic friction turns mechanical energy into thermal energy; only mechanical energy is calculated, but the loss equals f_k d.
An energy bar chart shows each energy at two moments; the bars must add up (with any work) and match the situation.

every energy is either inside it (K, U, thermal) or crossing the boundary as work done by external forces the system's energy changes by exactly that work: ΔE_system = W_external mechanical energy K + U is constant: K_i + U_i = K_f + U_f mechanical energy decreases and thermal energy increases by the same amount the bars before plus any work in equal the bars after: the bar chart is the equation

mechanical energy: The sum of a system's kinetic and potential energies, K + U.
closed system: A system that no energy enters or leaves: no external force does net work on it. An open system exchanges energy with its surroundings.
conservation of energy: Energy is never created or destroyed: a system's energy changes only by the energy transferred across its boundary.
energy bar chart: Bars showing each form of a system's energy at two moments, plus any work across the boundary; the bars must balance.
thermal energy: Mechanical energy turned into thermal energy (and sound) by nonconservative forces such as kinetic friction; it is still energy, just no longer mechanical.

3.5 Power

Power is how fast energy is transferred: P = W/Δt in watts, or P = F_∥v at an instant. The same work in less time means more power; at constant power a faster object gets a smaller force. A kilowatt-hour is a unit of energy.

Power is the rate of energy transfer: P_avg = W/Δt = ΔE/Δt. Its unit is the watt, 1 W = 1 J/s.
Same work in half the time is twice the power. Work tells how much; power tells how fast.
Instantaneous power of a force is P = F_∥v, the force component along the velocity times the speed.
A kilowatt-hour is energy, not power: 3.6 × 10⁶ J.

its average power is P = W/Δt, in watts (J/s) the power is larger, in inverse proportion to the time its instantaneous power is P = Fv (F_∥v in general) the force it can supply falls as the speed rises: F = P/v

power (physics): The rate at which energy is transferred or work is done: P = W/Δt, or P = F_∥v at an instant; measured in watts (J/s).