Reference · 50 terms

Glossary

Short, precise definitions of the words and symbols used in this module, each linked to the lesson where it is taught. Filter the list as you type, jump to a letter, or look a symbol up in Symbols at a glance.

Showing all 50 terms

A

Applied force \(P\): A force that someone or something controls, such as a push, a rope's pull or a motor's thrust. For a constant force \(P\) at the angle \(\alpha\) to a straight path, its work is \({U_P = P\cos\alpha\,d}\). Applied forces are nonconservative: their work depends on how and where they act, not only on the end points.; See: Lesson 2 Lesson 4Related: Work \(U\), Nonconservative force, Power \(P\)
Area under the force–displacement graph: The work of a force that varies along the path is the signed area between its \({F\cos\theta}\)–\(s\) graph and the \(s\)-axis: \({U = \int F\cos\theta\,ds}\). Areas below the axis count as negative work. For straight-line graphs the area splits into rectangles and triangles.; See: Lesson 2Related: Work \(U\), Variable force, Spring work

C

Closed path: A path that ends where it started. The work of a conservative force around any closed path is zero; the work of friction around a closed path is never zero, because friction opposes the motion on every leg.; See: Lesson 6Related: Conservative force, Path independence
Conservation of energy (mechanical): When only conservative forces (weights and springs) do work, the sum \({T + V_g + V_e}\) stays the same: \({T_1 + V_1 = T_2 + V_2}\). With other forces, \({T_1 + V_1 + \sum U_{nc} = T_2 + V_2}\), where \({\sum U_{nc}}\) is their work. It is the principle of work and energy with the conservative works written as potential energies.; See: Lesson 7Related: Mechanical energy, Principle of work and energy, Nonconservative force
Conservative force: A force whose work between two positions depends only on those positions, not on the path between them; equivalently, its work around any closed path is zero. The weight, the force of a linear spring and gravity at any distance are conservative, and each has a potential energy with \({U_{1\to2} = V_1 - V_2}\).; See: Lesson 6Related: Potential energy, Closed path, Nonconservative force
Connected bodies: Particles joined by inextensible cords, links or contact. Applying work and energy to the whole system is often simplest: the internal forces (the tension at both ends of a light cord over a smooth pulley) do equal and opposite work and cancel, and every particle's kinetic energy is added.; See: Lesson 4 Lesson 7Related: System of particles, Inextensible cord, Tension \(T_s\)
Curvature \(\kappa\): How sharply a path bends at a point: \({\kappa = 1/\rho}\), where \(\rho\) is the radius of curvature, in 1/m. A straight line has \({\kappa = 0}\). The normal force on a particle following a curved track must supply part of \({m v^2 \kappa}\) toward the center of curvature.; See: Lesson 8Related: Radius of curvature \(\rho\), Normal force \(N\)

D

Datum: The level, chosen freely, from which heights \(y\) are measured upward for the gravitational potential energy \({V_g = Wy}\). Below the datum \(V_g\) is negative. Moving the datum adds the same constant to every \(V_g\), so differences, and every answer, are unchanged.; See: Lesson 6Related: Gravitational potential energy \(V_g\), Potential energy
Deformation \(s\) (of a spring): How far a spring is stretched or compressed from its unstretched length, \({s = L - L_0}\) in absolute value. Both the spring force \({ks}\) and the elastic energy \({\tfrac12 ks^2}\) use it, so it must be measured from the unstretched length, not from where the motion starts.; See: Lesson 3Related: Unstretched length, Linear spring, Elastic potential energy \(V_e\)
Drag (air resistance): The resisting force of the air on a moving body. Like friction it always opposes the motion, so its work is negative and it is nonconservative. In car problems it is often combined with rolling resistance into one resisting force.; See: Lesson 5Related: Rolling resistance, Nonconservative force

E

Efficiency \(\varepsilon\): The fraction of the power put into a machine that comes out as useful power: \({\varepsilon = P_\text{out}/P_\text{in}}\), always less than 1 because of friction and other losses, which end as heat. For machines in series the efficiencies multiply.; See: Lesson 5Related: Power \(P\), Heat
Elastic potential energy \(V_e\): The energy stored in a deformed linear spring: \({V_e = \tfrac12 ks^2}\), with \(s\) the deformation from the unstretched length. It is never negative, and is the same for a stretch or a compression of the same size.; See: Lesson 6Related: Deformation \(s\), Linear spring, Potential energy
Energy methods: Ways of solving dynamics problems that work with scalar energies instead of vector accelerations: the principle of work and energy, conservation of energy and power. They come from integrating Newton's second law along the path, so they give speeds at positions directly, without finding the acceleration first.; See: Lesson 1Related: Principle of work and energy, Conservation of energy

F

Free-body diagram: A sketch of the particle alone with every force acting on it. In energy problems it shows which forces do work (and with what sign) and which do none, and it gives the normal force needed for friction.; See: Lesson 4Related: Principle of work and energy, Normal force \(N\)
Friction (kinetic): The force of a surface opposing sliding: \({F_f = \mu_k N}\), against the motion. Its work over a sliding distance \(d\) with constant \(N\) is \({U_f = -\mu_k N\,d}\), always negative. The coefficient \({\mu_k}\) depends on the two surfaces; \(N\) comes from the free-body diagram.; See: Lesson 3Related: Normal force \(N\), Static friction, Heat

G

Gravitational potential energy \(V_g\): The potential energy of the weight: \({V_g = Wy = mgy}\), with \(y\) measured upward from a datum. Far from the Earth's surface, where \(g\) is not constant, \({V_g = -GM_em/r}\) instead.; See: Lesson 6Related: Datum, Weight \(W\), Potential energy

H

Heat (energy lost to friction): Where the energy taken by friction and drag goes. Mechanical energy decreases by the size of the nonconservative work, but total energy is not destroyed: it warms the surfaces and the air.; See: Lesson 7Related: Friction, Mechanical energy, Efficiency \(\varepsilon\)
Horsepower (hp): A unit of power still used for engines: \({1\ \text{hp} = 550\ \text{ft}\cdot\text{lb/s} = 745.7\ \text{W}}\).; See: Lesson 5Related: Power \(P\), Watt

I

Inextensible cord: A cord that does not stretch. Two bodies joined by a taut inextensible cord move the same distance along it, so their speeds are equal, and the tension does equal and opposite work on them: no net work on the system.; See: Lesson 4Related: Connected bodies, Tension \(T_s\)

J

Joule (J): The SI unit of work and energy: \({1\ \text{J} = 1\ \text{N}\cdot\text{m} = 1\ \text{kg}\cdot\text{m}^2/\text{s}^2}\), the work of a \(1\ \text{N}\) force acting through \(1\ \text{m}\) along its direction.; See: Lesson 2Related: Work \(U\), Watt

K

Kilowatt-hour (\(\text{kW}\cdot\text{h}\)): A unit of energy, not power: the energy delivered in one hour at \(1\ \text{kW}\), \({1\ \text{kW}\cdot\text{h} = 3.6\ \text{MJ}}\).; See: Lesson 5Related: Joule, Power \(P\)
Kinetic energy \(T\): The energy a particle has because it moves: \({T = \tfrac12 mv^2}\). It is a scalar that depends only on the speed, never on the direction of motion, and it is never negative. The total work done on a particle equals the change in its kinetic energy.; See: Lesson 1 Lesson 4Related: Principle of work and energy, Speed \(v\)

L

Linear spring: A spring whose force is proportional to its deformation: \({F_s = ks}\), always toward the unstretched length. Its work is \({U_s = -\left(\tfrac12 ks_2^2 - \tfrac12 ks_1^2\right)}\): negative while the deformation grows, positive while it shrinks.; See: Lesson 3Related: Spring stiffness \(k\), Deformation \(s\), Spring work
Loop-the-loop: A vertical circular loop in a track. A particle starting from rest on a smooth track stays on at the top only if \({N \ge 0}\) there, which needs \({v^2 \ge gR}\) at the top and a start at least \({h = \tfrac52 R}\) above the bottom.; See: Lesson 8Related: Normal force \(N\), Radius of curvature \(\rho\)

M

Mechanical energy: The sum of a particle's kinetic and potential energies, \({E = T + V_g + V_e}\). It stays constant when only conservative forces do work, and falls by the size of the nonconservative work when friction or drag acts.; See: Lesson 7Related: Conservation of energy, Kinetic energy \(T\), Potential energy

N

Nonconservative force: A force whose work depends on the path, such as friction, drag or an applied push. It has no potential energy; its work \({U_{nc}}\) is added on the left of \({T_1 + V_1 + \sum U_{nc} = T_2 + V_2}\).; See: Lesson 6 Lesson 7Related: Conservative force, Friction, Applied force \(P\)
Normal force \(N\): The push of a surface on a particle, perpendicular to the surface. On a fixed surface it does no work, but it sets the friction (\({\mu_k N}\)). It is \(mg\) only on a level surface with no other vertical forces; on a curved track it also supplies part of \({mv^2/\rho}\). A surface can only push, so \(N\) cannot be negative.; See: Lesson 3 Lesson 8Related: Friction, Curvature \(\kappa\), Loop-the-loop

P

Particle: A body modeled as a point with mass: its size and rotation are ignored. This module treats every body as a particle; a rolling ball's spin energy, for example, is left out.; See: Lesson 1Related: Kinetic energy \(T\)
Path independence: The property of a conservative force that its work depends only on the starting and ending positions. The weight does the same work on a skier who slides straight down as on one who zigzags, if both drop the same height.; See: Lesson 3 Lesson 6Related: Conservative force, Closed path
Pendulum: A bob on a string or rod swinging about a fixed pivot. The string is always perpendicular to the motion and does no work, so \({v^2 = 2gL(\cos\theta - \cos\theta_0)}\). The tension follows from \({\sum F_n = mv^2/L}\): \({T_s = mg(3\cos\theta - 2\cos\theta_0)}\).; See: Lesson 7 Lesson 8Related: Tension \(T_s\), Conservation of energy
Potential energy: Energy that depends only on position, defined for a conservative force so that its work is the drop in potential energy: \({U_{1\to2} = V_1 - V_2}\). The two used here are \({V_g = Wy}\) and \({V_e = \tfrac12 ks^2}\). The force is \({F = -dV/ds}\).; See: Lesson 6Related: Gravitational potential energy \(V_g\), Elastic potential energy \(V_e\), Potential-energy diagram
Potential-energy diagram: A graph of \(V\) against position. A particle with total energy \(E\) can only be where \({V \le E}\); it turns back where \({V = E}\), and the force at any point is minus the slope, \({F = -dV/ds}\). Minima are stable equilibrium positions.; See: Lesson 6Related: Potential energy, Mechanical energy
Power \(P\): The rate at which a force does work: \({P = dU/dt = \Fvec\cdot\vvec = Fv\cos\theta}\), in watts. A motor's power output at an instant uses the force at that instant, so an accelerating load needs more power than a steady one.; See: Lesson 5Related: Efficiency \(\varepsilon\), Watt, Horsepower
Principle of work and energy: For a particle moving from position 1 to position 2, \({T_1 + \sum U_{1\to2} = T_2}\): the kinetic energy at the start plus the work of every force equals the kinetic energy at the end. It comes from integrating \({\sum F_t = m\,dv/dt}\) along the path.; See: Lesson 4Related: Kinetic energy \(T\), Work \(U\), Conservation of energy

R

Radius of curvature \(\rho\): The radius of the circle that best fits the path at a point; for a circular arc, its radius. Moving at speed \(v\), a particle needs a net force \({mv^2/\rho}\) toward the center of curvature.; See: Lesson 8Related: Curvature \(\kappa\), Normal force \(N\)
Rolling resistance: The resisting force on a rolling wheel from the deformation of the tire and the road. In vehicle problems it is usually given as a force, or modeled like friction as a coefficient times the normal force, and combined with drag.; See: Lesson 5Related: Drag, Power \(P\)

S

Scalar: A quantity with size but no direction. Work, energy and power are scalars, which is why energy methods add numbers rather than vectors, and why one energy equation can find only one unknown.; See: Lesson 2Related: Work \(U\), Energy methods
Speed \(v\): The magnitude of the velocity, in m/s. Energy methods give \(v^2\), and so the speed, but not the direction of motion. Convert km/h by dividing by 3.6 before squaring.; See: Lesson 4Related: Kinetic energy \(T\)
Spring stiffness \(k\) (spring constant): The force per unit deformation of a linear spring, in N/m (or kN/m). A stiffer spring stores more energy for the same deformation, \({\tfrac12 ks^2}\), and stops a moving body in a shorter distance.; See: Lesson 3Related: Linear spring, Elastic potential energy \(V_e\)
Spring work: The work of a linear spring on the body attached to it: \({U_s = -\left(\tfrac12 ks_2^2 - \tfrac12 ks_1^2\right)}\). Square each deformation, then subtract; \({\tfrac12 k(s_2 - s_1)^2}\) is right only when \({s_1 = 0}\).; See: Lesson 3Related: Linear spring, Deformation \(s\), Area under the force–displacement graph
Static friction: Friction between surfaces that do not slide, up to \({\mu_s N}\). On a wheel rolling without slipping, or a fixed body, it does no work because its point of application does not move; on a box riding on an accelerating truck, it does work on the box.; See: Lesson 2 Lesson 3Related: Friction, Work \(U\)
System of particles: Several particles treated together. The principle becomes \({\sum T_1 + \sum U_{1\to2} = \sum T_2}\), with the works of all forces, internal and external; internal forces that do equal and opposite work, like a cord's tension, cancel.; See: Lesson 4 Lesson 7Related: Connected bodies, Inextensible cord

T

Tension \(T_s\) (in a string or cord): The pull of a string or cord, along it. A string can only pull, so a computed negative tension means the string has gone slack. The pendulum string's tension does no work, because it is perpendicular to the motion.; See: Lesson 4 Lesson 8Related: Pendulum, Inextensible cord
Total work: The sum of the works of all the forces on a particle, \({\sum U_{1\to2}}\), which equals the change in its kinetic energy. Find each force's work separately, with its sign, and add.; See: Lesson 4Related: Work \(U\), Principle of work and energy

U

Unstretched length (free length): The length of a spring when it carries no force. Deformations, spring forces and elastic energies are all measured from it.; See: Lesson 3Related: Deformation \(s\), Linear spring

V

Variable force: A force whose size or direction changes along the path. Its work must be found by integration, \({U = \int F\cos\theta\,ds}\), or as the area under its force–displacement graph.; See: Lesson 2Related: Area under the force–displacement graph, Work \(U\)

W

Watt (W): The SI unit of power: \({1\ \text{W} = 1\ \text{J/s} = 1\ \text{N}\cdot\text{m/s}}\).; See: Lesson 5Related: Power \(P\), Joule
Weight \(W\): The force of gravity on a particle, \({W = mg}\), straight down. Its work depends only on the change in height: \({U_W = -W\,\Delta y}\), positive when the particle goes down.; See: Lesson 3Related: Gravitational potential energy \(V_g\), Path independence
Work \(U\) (of a force): The scalar \({U_{1\to2} = \int \Fvec\cdot d\rvec = \int F\cos\theta\,ds}\): the part of a force along the motion, times the distance moved. It is positive when the force helps the motion, negative when it opposes it, and zero when the force is perpendicular to the motion or its point of application does not move.; See: Lesson 2Related: Total work, Joule, Scalar
Work ledger: The module's picture of the principle of work and energy: start from \(T_1\), add the work of each force as a bar that floats from the running total, and land on \(T_2\).; See: Lesson 4Related: Principle of work and energy, Total work

Symbols at a glance

Met a symbol in a lesson or in another textbook and not sure what it stands for? Find it here, then follow the link to its entry.

Symbols used in this module and in other books
Symbol	Meaning	Entry
\(\colKE{T}\)	Kinetic energy \(\tfrac12 mv^2\)	Kinetic energy \(T\)
\(U_{1\to2}\), \(U_{1-2}\)	Work done from position 1 to position 2	Work \(U\), Total work
\(\colVg{U_W},\ \colVe{U_s},\ \colF{U_f},\ \colP{U_P}\)	Work of the weight, a spring, friction, an applied force	Weight \(W\), Spring work, Friction, Applied force \(P\)
\(U_{nc}\)	Work of the nonconservative forces	Nonconservative force
\(\colVg{V_g}\)	Gravitational potential energy \(Wy\)	Gravitational potential energy \(V_g\)
\(\colVe{V_e}\)	Elastic potential energy \(\tfrac12 ks^2\)	Elastic potential energy \(V_e\)
\(E\)	Mechanical energy \(T + V\)	Mechanical energy
\(\colP{P}\)	Power (or, in a figure, an applied force)	Power \(P\), Applied force \(P\)
\(\varepsilon\), \(\eta\)	Efficiency (\(\eta\) in some books)	Efficiency \(\varepsilon\)
\(k\)	Spring stiffness	Spring stiffness \(k\)
\(s\)	Spring deformation; also distance along a path	Deformation \(s\)
\(\mu_k,\ \mu_s\)	Coefficients of kinetic and static friction	Friction, Static friction
\(N\)	Normal force	Normal force \(N\)
\(T_s\)	Tension in a string (to keep it apart from \(T\))	Tension \(T_s\)
\(\rho,\ \kappa\)	Radius of curvature and curvature \(1/\rho\)	Radius of curvature \(\rho\), Curvature \(\kappa\)
\(y\)	Height above the datum, measured upward	Datum