Reference · 51 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 51 terms

A

Angle of repose: The steepest slope on which a body can rest without sliding: \({\tan\theta = \mu_s}\). Below it, static friction holds the body; above it, the body slides. Tilting a board until a block just slips is a simple way to measure \(\mu_s\).; See: Lesson 3Related: Coefficient of static friction \(\mu_s\), Incline, Slip check
Applied force \(P\): A push or pull exerted on a body by a person or a machine. Drawn in amber. A push ends on the face it acts on; a pull, like a cord, starts there. A force at an angle has a component across the surface that changes the normal force.; See: Lesson 1 Lesson 3Related: Normal force \(N\), Tension \(T\), Free-body diagram
Atwood machine: Two bodies hanging from the ends of one cord over a fixed pulley. The heavier one descends with \({a = \frac{m_B - m_A}{m_A + m_B}g}\) and the cord carries \({T = \frac{2m_Am_B}{m_A + m_B}g}\), a value between the two weights.; See: Lesson 4Related: Connected particles, Tension \(T\), Smooth pulley

B

Banked curve: A curve whose road surface is tilted toward the center, so that the normal force has a component toward the center and less friction is needed to turn. At the design speed no friction is needed at all.; See: Lesson 7Related: Design speed \(v_0\), Normal force \(N\), Normal acceleration \(a_n\)

C

Center of curvature: The center of the circle that best fits the path at a point, at distance \(\rho\) from it on the inside of the curve. The normal axis \(n\) points toward it, and so does the normal acceleration.; See: Lesson 6Related: Radius of curvature \(\rho\), Normal acceleration \(a_n\), Normal and tangential axes \(n\), \(t\)
“Centripetal force” (a direction, not a force): A name for the net force toward the center of curvature, \(mv^2/\rho\). It is not an extra force: on a free-body diagram some real force, such as friction, a normal force or a tension, must point that way. There is no outward “centrifugal” force on a particle seen from an inertial frame.; See: Lesson 6Related: Normal acceleration \(a_n\), Kinetic diagram, Inertial frame
Choosing a method: \(\sum\Fvec = m\avec\) relates forces to accelerations at an instant. Integrated over distance it becomes the principle of work and energy, best for speeds at positions; integrated over time, the principle of impulse and momentum, best for speeds after a time and for impacts. Many problems use two.; See: Lesson 8Related: Equation of motion, Separation of variables
Coefficient of kinetic friction \(\mu_k\): The ratio of the friction force to the normal force while two surfaces slide: \({F_f = \mu_k N}\), opposing the relative motion. It is usually a little smaller than \(\mu_s\) and nearly independent of the speed.; See: Lesson 3Related: Coefficient of static friction \(\mu_s\), Friction force \(F_f\), Slip check
Coefficient of static friction \(\mu_s\): Sets the largest friction two surfaces can exert without slipping: \({F_f \le \mu_s N}\). Below that limit, static friction is whatever the equations of motion need, in whichever direction they need it.; See: Lesson 3Related: Coefficient of kinetic friction \(\mu_k\), Slip check, Angle of repose
Conical pendulum: A ball on a cord swinging in a horizontal circle, the cord at a steady angle \(\theta\) to the vertical. \({T\cos\theta = mg}\) and \({T\sin\theta = mv^2/\rho}\) with \(\rho = L\sin\theta\), so \(T = mg/\cos\theta\) and \(v = \sqrt{gL\sin\theta\tan\theta}\).; See: Lesson 6Related: Normal acceleration \(a_n\), Tension \(T\), Radius of curvature \(\rho\)
Connected particles: Bodies joined by cords, links or contact, whose motions are tied together. Draw one free-body diagram per body; the forces between them act equally and oppositely on each, and the cord or the contact gives an equation between their accelerations.; See: Lesson 4Related: Tension \(T\), Dependent motion, System of particles
Crest (of a road): The top of a hill on a road. The center of curvature is below, so \({N = m(g - v^2/\rho)}\): the road pushes up with less than the weight, and at \(v = \sqrt{g\rho}\) not at all, so a faster car leaves the road.; See: Lesson 7Related: Dip, Normal acceleration \(a_n\), Normal force \(N\)
Cylindrical components: Equations of motion along \(r\), \(\theta\) and \(z\), for a particle guided by a rotating arm or rod: \({\sum F_r = m(\ddot r - r\dot\theta^2)}\), \({\sum F_\theta = m(r\ddot\theta + 2\dot r\dot\theta)}\), \({\sum F_z = m\ddot z}\). The accelerations come from the Cylindrical Coordinates module.; See: Lesson 8Related: Equation of motion, Rectangular components, Normal and tangential axes \(n\), \(t\)

D

Dependent motion: Motions of bodies tied together by a cord of fixed length. Write the length in position coordinates and differentiate: through one movable pulley, \({s_A + 2s_B = \text{const}}\) gives \(v_A = -2v_B\) and \(a_A = -2a_B\).; See: Lesson 4Related: Position coordinate, Movable pulley, Connected particles
Design speed \(v_0\) (of a banked curve): The speed at which a car can take a banked curve with no friction at all: \({v_0 = \sqrt{g\rho\tan\theta}}\). Faster, friction acts down the bank; slower, up the bank.; See: Lesson 7Related: Banked curve, Friction force \(F_f\)
Dip (in a road): The bottom of a sag in a road, or of a vertical circle. The center of curvature is above, so \({N = m(g + v^2/\rho)}\): you are pressed into the seat. Pilots pulling out of a dive feel several times their weight.; See: Lesson 7Related: Crest, Normal acceleration \(a_n\), Loop
Drag \(F_D\): The force of a fluid on a body moving through it, opposing the velocity. Nearly proportional to the speed, \(F_D = cv\), for small slow bodies; closer to its square, \(F_D = kv^2\), for cars, balls and people. It makes the force depend on velocity.; See: Lesson 5Related: Terminal velocity \(v_t\), Separation of variables, Force that varies

E

Equation of motion: Newton\'s second law for a particle: \({\sum\Fvec = m\avec}\). The resultant of every force on the particle equals its mass times its acceleration, measured in an inertial frame. It holds component by component.; See: Lesson 1Related: Newton's laws of motion, Resultant force, Kinetic diagram
External force: A force on a system exerted by something outside it, such as gravity, a support, the road or a hand. Only external forces appear on the free-body diagram of a system, and they alone determine the acceleration of its mass center.; See: Lesson 8Related: Internal force, System of particles, Mass center \(G\)

F

Force that varies: A force given as a function of time, position or velocity, such as a growing push, a spring or drag. The acceleration then varies too, so the constant-acceleration formulas do not apply: separate variables and integrate.; See: Lesson 5Related: Separation of variables, Drag \(F_D\), Spring force \(F_s\)
Free-body diagram: A sketch of a body isolated from its surroundings, showing every force they exert on it: the weight, and one force at each point of contact. In this module the weight ends at the center of gravity, pushes end on the face they push, and friction and tensions start where they act.; See: Lesson 2Related: Kinetic diagram, Procedure for analysis, Weight \(W = mg\)
Friction force \(F_f\): The component of a contact force along the surfaces, opposing slipping. Without slip it can be anywhere up to \(\mu_s N\); while slipping it is \(\mu_k N\), against the relative motion. Drawn thick, in orange.; See: Lesson 3Related: Coefficient of static friction \(\mu_s\), Coefficient of kinetic friction \(\mu_k\), Slip check

I

Incline: A sloping surface. Axes along and normal to it make the acceleration one-dimensional: \(N = mg\cos\theta\) (with no other force across the slope) and, sliding down, \({a = g(\sin\theta - \mu_k\cos\theta)}\).; See: Lesson 3Related: Angle of repose, Rectangular components, Normal force \(N\)
Inertial frame: A frame of reference that is not accelerating or rotating, in which Newton\'s laws hold as written. A frame fixed to the ground is close enough for engineering; accelerations measured relative to an accelerating vehicle are not.; See: Lesson 1Related: Equation of motion, Newton's laws of motion
Internal force: A force that one part of a system exerts on another, such as a coupling or a contact between crates. By the third law they come in equal and opposite pairs and cancel in \(\sum\Fvec\) for the system. Cut the system to see one.; See: Lesson 8Related: External force, System of particles, Third law

K

Kinetic diagram: A sketch of the body showing the vector \(m\avec\), or its components, beside the free-body diagram. Newton\'s second law says the two diagrams are equivalent. \(m\avec\) is not a force, and never belongs on the free-body diagram.; See: Lesson 2Related: Free-body diagram, Equation of motion, “Centripetal force”

L

Loop (vertical): A vertical circle traveled on the inside of a track, or on a cord. At the top, \({N + mg = mv^2/r}\), and the track (or cord) can only push (or pull), so \(v_\text{top} \ge \sqrt{gr}\); from the bottom that needs \(v_b \ge \sqrt{5gr}\).; See: Lesson 7Related: Dip, Normal acceleration \(a_n\), Tension \(T\)

M

Mass \(m\): A property of a body, its resistance to acceleration, the same everywhere. Measured in kilograms. A scale measures the force it pushes with, not the mass.; See: Lesson 1Related: Weight \(W = mg\), Newton (N), Equation of motion
Mass center \(G\): The point of a system whose position is the mass-weighted average of its particles\' positions, \(m\rvec_G = \sum m_i\rvec_i\). It moves as a single particle with the whole mass, acted on by all the external forces: \({\sum\Fvec_\text{ext} = m\avec_G}\).; See: Lesson 8Related: System of particles, External force, Internal force
Movable pulley: A pulley that moves with the load it carries. Two lengths of cord hold it, so it feels \(2T\), and the cord\'s free end moves twice as far as the pulley.; See: Lesson 4Related: Dependent motion, Smooth pulley, Tension \(T\)

N

Newton (N) (unit): The SI unit of force, defined by the second law: \(1\ \text{N}\) gives \(1\ \text{kg}\) an acceleration of \(1\ \text{m/s}^2\). A \(1\ \text{kg}\) mass weighs \(9.81\ \text{N}\) on Earth.; See: Lesson 1Related: Mass \(m\), Weight \(W = mg\)
Newton's laws of motion: First: a particle stays at rest or moves at constant velocity unless an unbalanced force acts. Second: \(\sum\Fvec = m\avec\). Third: the forces two bodies exert on each other are equal, opposite and collinear.; See: Lesson 1Related: Equation of motion, Third law, Inertial frame
Normal acceleration \(a_n\): The component of acceleration toward the center of curvature, \({a_n = v^2/\rho}\), present whenever the path curves, even at constant speed. It changes the direction of the velocity; \(\sum F_n = ma_n\).; See: Lesson 6Related: Tangential acceleration \(a_t\), Radius of curvature \(\rho\), “Centripetal force”
Normal and tangential axes \(n\), \(t\): Axes attached to the particle\'s path: \(t\) along the velocity, \(n\) toward the center of curvature, and \(b\) perpendicular to both. The equations of motion become \(\sum F_t = m\,dv/dt\), \(\sum F_n = mv^2/\rho\), \(\sum F_b = 0\).; See: Lesson 6Related: Normal acceleration \(a_n\), Tangential acceleration \(a_t\), Center of curvature
Normal force \(N\): The component of a contact force perpendicular to the surfaces: a push. It is whatever the equation across the motion makes it, which is \(mg\) only on a level surface with nothing else pushing or pulling across it. Find it from its own equation before using \(\mu N\).; See: Lesson 2 Lesson 3Related: Friction force \(F_f\), Applied force \(P\), Incline

P

Particle: A body whose size and rotation do not matter to the question asked, so that its motion is that of a single point with all its mass. Cars on curves, crates on ramps and satellites in orbit are often treated as particles.; See: Lesson 1Related: Equation of motion, Mass center \(G\)
Position coordinate: A coordinate that locates a body along its line of motion, measured from a fixed datum. Writing a cord\'s length in position coordinates gives the constraint between the bodies\' velocities and accelerations.; See: Lesson 4Related: Dependent motion, Movable pulley
Procedure for analysis: (1) Free-body diagram; (2) kinetic diagram, assuming directions where they are unknown; (3) axes and one equation of motion per direction, plus friction and constraint equations; (4) kinematics for the motion. A negative result means the assumed direction was wrong.; See: Lesson 2Related: Free-body diagram, Kinetic diagram, Equation of motion

R

Radius of curvature \(\rho\): The radius of the circle that best fits the path at a point. Large \(\rho\) means a gentle curve; a straight line has \(\rho = \infty\). It sets the normal acceleration, \(a_n = v^2/\rho\).; See: Lesson 6Related: Center of curvature, Normal acceleration \(a_n\)
Rectangular components: The equation of motion resolved along fixed axes: \(\sum F_x = ma_x\), \(\sum F_y = ma_y\), \(\sum F_z = ma_z\). Put one axis along the motion, so the equation across it, with zero acceleration, gives the normal force.; See: Lesson 3Related: Equation of motion, Incline, Normal and tangential axes \(n\), \(t\)
Resultant force: The vector sum of all the forces on a body, \(\sum\Fvec\). It, and not any single force, sets the acceleration. When it is zero the body does not accelerate: the forces balance.; See: Lesson 1Related: Equation of motion, Free-body diagram

S

Separation of variables: Writing \(m\,a = F\) with \(a = dv/dt\) (for a force of time or velocity, to find a time) or \(a = v\,dv/ds\) (for a force of position or velocity, to find a distance), putting each variable on its own side, and integrating.; See: Lesson 5Related: Force that varies, Drag \(F_D\), Choosing a method
Slip check (does it slip?): Assume the surfaces do not slip, find the friction this needs, and compare it with \(\mu_s N\). If it fits, the assumption holds; if not, the body slips, friction is \(\mu_k N\) against the slip, and the acceleration is unknown.; See: Lesson 3Related: Coefficient of static friction \(\mu_s\), Friction force \(F_f\), Angle of repose
Smooth pulley (light, ideal): A pulley of negligible mass turning on a frictionless axle. A light cord passing over it has the same tension on both sides.; See: Lesson 4Related: Tension \(T\), Atwood machine, Movable pulley
Spring force \(F_s\): The force of a linear spring, \(F_s = ks\), where \(s\) is its stretch or compression. It depends on position, so the motion it causes is found with \(v\,dv = a\,ds\). Drawn in violet, ending on the face it pushes.; See: Lesson 5Related: Force that varies, Separation of variables
System of particles: Any group of particles you choose to treat together, such as a train or a stack of crates. The free-body diagram of the system shows only external forces, and gives the acceleration of its mass center.; See: Lesson 8Related: Mass center \(G\), External force, Internal force

T

Tangential acceleration \(a_t\): The component of acceleration along the path, \(a_t = dv/dt\) (or \(v\,dv/ds\)). It changes the speed. On a curve the total acceleration is \(\sqrt{a_t^2 + a_n^2}\), and so is the friction a tire must supply per unit mass.; See: Lesson 6Related: Normal acceleration \(a_n\), Normal and tangential axes \(n\), \(t\)
Tension \(T\): The pull of a cord, rope or cable on whatever it is tied to, along the cord and away from the body. A cord can only pull; a negative tension in a solution means the cord has gone slack. Drawn in gray, starting where the cord is tied.; See: Lesson 4Related: Smooth pulley, Connected particles, Atwood machine
Terminal velocity \(v_t\): The steady speed of a falling body at which drag equals the weight and the acceleration is zero: \({v_t = \sqrt{mg/k}}\) for drag \(kv^2\). From rest the body approaches it, \(v = v_t\tanh(gt/v_t)\), but never quite reaches it.; See: Lesson 5Related: Drag \(F_D\), Separation of variables
Third law: The forces two bodies exert on each other are equal in size, opposite in direction and collinear. It is why internal forces cancel in a system, and why the cord pulls each body it joins with the same tension.; See: Lesson 1 Lesson 8Related: Internal force, Newton's laws of motion

W

Weight \(W = mg\): The force of gravity on a body, \(W = mg\) with \(g = 9.81\ \text{m/s}^2\) near the Earth\'s surface. It depends on where the body is; its mass does not. Drawn in blue, ending at the center of gravity.; See: Lesson 1Related: Mass \(m\), Newton (N), Free-body diagram

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
\(\sum\Fvec = m\avec\)	Newton's second law, the equation of motion	Equation of motion
\(\colW{W = mg}\)	Weight	Weight \(W = mg\)
\(N\)	Normal force	Normal force \(N\)
\(\colF{F_f}\), \(\mu_s,\ \mu_k\)	Friction force; coefficients of static and kinetic friction	Friction force \(F_f\), Slip check
\(T\)	Cord tension	Tension \(T\)
\(\colMA{m\avec}\)	Mass times acceleration, on the kinetic diagram	Kinetic diagram
\(s_A,\ s_B\)	Position coordinates along a cord	Position coordinate
\(F_D\), \(v_t\)	Drag; terminal velocity	Drag \(F_D\), Terminal velocity \(v_t\)
\(\rho\), \(a_n = v^2/\rho\)	Radius of curvature; normal acceleration	Radius of curvature \(\rho\), Normal acceleration \(a_n\)
\(n,\ t,\ b\)	Normal, tangential and binormal axes	Normal and tangential axes \(n\), \(t\)
\(v_0\)	Design speed of a banked curve	Design speed \(v_0\)
\(\avec_G\)	Acceleration of the mass center of a system	Mass center \(G\)