Reference · 63 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 63 terms

A

Acceleration (in cylindrical components): The rate of change of the velocity, written along the unit vectors at the particle: \(\avec = {(\ddot r - r\dot\theta^2)\,\er} + {(r\ddot\theta + 2\dot r\dot\theta)\,\et} + {\ddot z\,\ez}\). The centripetal term \({-r\dot\theta^2}\) and the Coriolis term \({2\dot r\dot\theta}\) appear because \(\er\) and \(\et\) turn, so a particle can accelerate even when \(r\), \(\theta\) and \(z\) all change at constant rates.; See: Lesson 5Related: Centripetal acceleration, Coriolis acceleration, Velocity
Angular acceleration \(\ddot\theta\): The rate of change of the angular velocity, \({\ddot\theta = d\dot\theta/dt}\), in \(\text{rad/s}^2\). It gives the \({r\ddot\theta}\) part of the transverse acceleration, which is nonzero only while the turning rate changes, as when an arm speeds up.; See: Lesson 5Related: Angular velocity \(\dot\theta\), Transverse component
Angular coordinate \(\theta\): The angle from the fixed reference line (usually the \({+x}\) axis) to the line \(OP\), positive counter-clockwise as seen from \({+z}\), in radians. For a moving particle \(\theta\) keeps counting past \({2\pi}\): an arm that has made one and a half turns has \({\theta = 3\pi}\).; See: Lesson 2Related: Radial coordinate \(r\), Radian, Quadrant
Angular momentum (about \(O\)): For a particle moving in a plane, the moment of its linear momentum about \(O\): \({H_O = m r v_\theta = m r^2\dot\theta}\). Only the transverse component of the force can change it, so it stays constant under a central force. The impulse and momentum methods later in the course build on this.; See: Lesson 8Related: Central force, Kepler's second law
Angular velocity \(\dot\theta\): The rate at which the line \(OP\) turns, \({\dot\theta = d\theta/dt}\), in rad/s. It gives the transverse velocity \({v_\theta = r\dot\theta}\) (the familiar \({v = \omega r}\)), so at the same \(\dot\theta\) a point farther out moves faster. Convert motor speeds first: \(N\) rpm is \({2\pi N/60}\) rad/s.; See: Lesson 2 Lesson 4Related: rpm, Velocity, Angular acceleration \(\ddot\theta\)
Arc length: The distance along a circular arc of radius \(r\) that spans an angle \(\Delta\theta\) in radians: \({s = r\,\Delta\theta}\). Divided by the time taken, it gives the speed of going round, \({v = r\dot\theta}\) (Example 1.1).; See: Lesson 1Related: Radian, Angular velocity \(\dot\theta\)
Archimedean spiral: The path \({r = b\theta}\): the distance from \(O\) grows by \({2\pi b}\) every turn. Here \({dr/d\theta = b}\), \({d^2r/d\theta^2 = 0}\) and \({\tan\psi = \theta}\), so the spiral crosses the radial lines more and more squarely as it winds out. Spiral feeders and scroll grooves use it.; See: Lesson 6Related: Logarithmic spiral, Tangent angle \(\psi\), Path \(r = f(\theta)\)
atan2 (two-argument arctangent): The function \({\atantwo(y, x)}\) returns the angle of the point \({(x, y)}\) from the signs of both coordinates, so it always lands in the right quadrant. Its result lies between \({-\pi}\) and \(\pi\): add \({2\pi}\) to a negative result for \({0 \le \theta \lt 2\pi}\). Python, NumPy and MATLAB take \({(y, x)}\); Excel's ATAN2 takes \({(x, y)}\).; See: Lesson 2Related: Quadrant, Angular coordinate \(\theta\)
Axial component: The component along the \(z\)-axis, in the direction of \(\ez\): \({v_z = \dot z}\) and \({a_z = \ddot z}\), exactly as in rectangular coordinates, because \({\ez = \khat}\) never turns.; See: Lesson 3 Lesson 5Related: \(z\)-coordinate, Radial component, Transverse component

C

Cardioid: The heart-shaped path \({r = a(1 + \cos\theta)}\), with \({dr/d\theta = -a\sin\theta}\) and \({d^2r/d\theta^2 = -a\cos\theta}\). It passes through \(O\) at \({\theta = \pi}\) and is a common shape for cam grooves (Example 6.1).; See: Lesson 6Related: Path \(r = f(\theta)\), Chain rule
Central force: A force that always points along the line through a fixed point \(O\), such as the pull of a cord through a hole at \(O\) or a planet's gravity. It has no transverse component, so \({\sum F_\theta = 0}\) gives \({r^2\dot\theta = h}\), a constant: the particle turns faster as it comes closer, with \({\dot\theta = h/r^2}\) and \({v_\theta = h/r}\).; See: Lesson 8Related: Angular momentum, Kepler's second law, Orbit
Centripetal acceleration: The term \({-r\dot\theta^2\,\er}\) of the acceleration, pointing toward the axis. It is present whenever the particle turns (\({\dot\theta \ne 0}\)) away from the axis, even at a constant \(\dot\theta\). On a circle about \(O\) it has the magnitude \({v^2/r}\), the normal acceleration of path coordinates.; See: Lesson 5Related: Acceleration, Circular motion, Coriolis acceleration
Chain rule (on a path): On a path \({r = f(\theta)}\), \(r\) depends on time through \(\theta\), so \({\dot r = f'\,\dot\theta}\) and \({\ddot r = f''\,\dot\theta^2 + f'\,\ddot\theta}\), with \({f' = dr/d\theta}\) and \({f'' = d^2r/d\theta^2}\) evaluated at the current \(\theta\). The \({f''\dot\theta^2}\) term remains even when \(\dot\theta\) is constant.; See: Lesson 6Related: Path \(r = f(\theta)\), Product rule
Circle through \(O\): The path \({r = d\cos\theta}\): a circle of diameter \(d\) along the \(x\)-axis that passes through \(O\). It describes a pin in a fixed circular slot that runs through the pivot of a turning arm (Example 6.3).; See: Lesson 6Related: Circular motion, Path \(r = f(\theta)\)
Circular motion (about \(O\)): Motion on a circle centered on \(O\): \({\dot r = \ddot r = 0}\), so \({a_r = -r\dot\theta^2 = -v^2/r}\) and \({a_\theta = r\ddot\theta = \dot v}\). For this path only, the polar and path components are the same (\({a_n = -a_r}\) and \({a_t = a_\theta}\)).; See: Lesson 5Related: Centripetal acceleration, Path coordinates
Collar: A ring or sleeve that slides along a rod or shaft. In problems it is a particle that can move only along the rod: the rod sets \(\theta\), and the collar's place along it sets \(r\). A smooth rod pushes the collar only across itself.; See: Lesson 1 Lesson 7Related: Smooth, Slotted arm
Coriolis acceleration: The term \({2\dot r\dot\theta\,\et}\) of the acceleration, nonzero when the particle moves in or out (\({\dot r \ne 0}\)) while it turns (\({\dot\theta \ne 0}\)). Half of it comes from \(\er\) turning, the other half from \({v_\theta = r\dot\theta}\) growing as \(r\) grows. A spinning rod must push a sliding collar sideways to provide it.; See: Lesson 5 Lesson 7Related: Acceleration, Centripetal acceleration, Transverse component
Curvilinear motion: Motion along a curved path, in a plane or in space. It can be described with rectangular components (\(x\), \(y\), \(z\)), path components (\(t\), \(n\)) or cylindrical components (\(r\), \(\theta\), \(z\)); all three give the same velocity and acceleration vectors.; See: Lesson 1Related: Path coordinates, Polar coordinates, Cylindrical coordinates
Cylindrical coordinates: The three numbers \({(\colR{r}, \colT{\theta}, \colZ{z})}\) that locate a point in space: \(r\) is the distance from the \(z\)-axis, \(\theta\) the angle around it from the \({+x}\) axis, and \(z\) the height along it. In a plane (\(z\) constant) they reduce to polar coordinates.; See: Lesson 2Related: Polar coordinates, Position vector, \(z\)-coordinate
Cylindrical robot: A robot whose three joints each set one cylindrical coordinate of the gripper: a turning base sets \(\theta\), a lifting carriage sets \(z\) and a telescoping arm sets \(r\). The joint rates are directly \(\dot\theta\), \(\dot z\) and \(\dot r\) (Examples 4.1 and 5.1).; See: Lesson 1 Lesson 4Related: Cylindrical coordinates, Slewing

D

Dot notation: A dot over a quantity means its time derivative, \({\dot r = dr/dt}\) and \({\dot\theta = d\theta/dt}\); two dots mean the second derivative, \({\ddot r = d^2r/dt^2}\). Do not confuse \(\ddot r\), the second derivative of the coordinate \(r\), with the acceleration component \({a_r = \ddot r - r\dot\theta^2}\).; See: Lesson 4Related: Angular velocity \(\dot\theta\), Radial component

E

\(\mathbf{e}_r, \mathbf{e}_\theta\) notation: Beer & Johnston and Meriam & Kraige write the polar unit vectors as \(\mathbf{e}_r\) and \(\mathbf{e}_\theta\). They are the same vectors as Hibbeler's \(\er\) and \(\et\), used in this module, and those books call the \(\theta\) components transverse components.; See: Lesson 1Related: Unit vectors \(\er, \et, \ez\), Transverse component
Equations of motion (cylindrical components): Newton's second law written along \(\er\), \(\et\) and \(\ez\): \({\sum F_r = m(\ddot r - r\dot\theta^2)}\), \({\sum F_\theta = m(r\ddot\theta + 2\dot r\dot\theta)}\) and \({\sum F_z = m\ddot z}\). Three scalar equations, so a problem can have at most three unknowns, such as forces or acceleration terms.; See: Lesson 7Related: Newton's second law, Free-body diagram, Kinetic diagram

F

Free-body diagram: A sketch of the particle on its own with every force that acts on it, drawn with \(\er\) and \(\et\) at the particle's current position. With the kinetic diagram it sets up the equations of motion: along each unit vector, the force components equal the \(m\avec\) components.; See: Lesson 7Related: Kinetic diagram, Equations of motion
Friction (kinetic): The part of a contact force that acts along the surfaces when they slide: magnitude \({\mu_k N}\), opposite to the velocity of sliding. A smooth surface has no friction. Example 7.3 finds \({\mu_k \approx 0.31}\) for a spiral slide.; See: Lesson 7Related: Normal force, Smooth

G

Guide (smooth, fixed): A fixed groove, slot or track that holds a particle on a path \({r = f(\theta)}\). A smooth guide pushes perpendicular to the path, at the angle \(\psi\) from \(\et\): \({\mathbf{N} = N(-\sin\psi\,\er + \cos\psi\,\et)}\). When an arm pushes the particle along it, \({N = -m a_r/\sin\psi}\).; See: Lesson 6 Lesson 8Related: Tangent angle \(\psi\), Slotted arm, Normal force

H

Helix: A path that winds around an axis at a constant radius while climbing or falling steadily: \(r\), \(\dot\theta\) and \(\dot z\) all constant. Spiral slides, ramps and chutes are helices. The acceleration is purely centripetal, \({-r\dot\theta^2\,\er}\), although the particle moves in space (Example 5.2).; See: Lesson 1 Lesson 5 Lesson 7Related: Cylindrical coordinates, Centripetal acceleration

K

Kepler's second law: The line from the Sun to a planet sweeps out equal areas in equal times. The area rate is \({\tfrac12 r^2\dot\theta = h/2}\), constant because gravity is a central force.; See: Lesson 8Related: Central force, Orbit
Kinematics: The study of the geometry of motion, position, velocity and acceleration, without regard to the forces that cause it. Lessons 2 to 6 cover the kinematics of a particle in cylindrical coordinates.; See: Lesson 1Related: Kinetics, Velocity, Acceleration
Kinetic diagram: A sketch of the particle showing \({m a_r}\) along \(\er\), \({m a_\theta}\) along \(\et\) and \({m a_z}\) along \(\ez\), drawn in the positive directions. It is the \({m\avec}\) side of Newton's second law, placed next to the free-body diagram.; See: Lesson 7Related: Free-body diagram, Equations of motion
Kinetics: The study of how forces change the motion of a body, through Newton's second law. Lessons 7 and 8 cover the kinetics of a particle in cylindrical coordinates.; See: Lesson 1 Lesson 7Related: Kinematics, Equations of motion

L

Logarithmic spiral (equiangular spiral): The path \({r = a\,e^{k\theta}}\), with \({dr/d\theta = kr}\) and \({d^2r/d\theta^2 = k^2 r}\). Its tangent angle is the same everywhere, \({\tan\psi = 1/k}\), which is why cutters and cams that must meet their work at a constant angle use it.; See: Lesson 6Related: Archimedean spiral, Tangent angle \(\psi\)

N

Newton's second law: The resultant force on a particle equals its mass times its acceleration, \({\sum\Fvec = m\avec}\). It holds in any set of unit vectors; written along \(\er\), \(\et\) and \(\ez\), it gives the equations of motion in cylindrical components.; See: Lesson 7Related: Equations of motion, Kinetics
Normal force: The part of a contact force perpendicular to the surfaces in contact. A smooth rod that turns with the arm pushes only across itself, \({N\,\et}\) (plus a vertical part); a smooth fixed guide pushes perpendicular to the path. A negative answer means the other wall pushes.; See: Lesson 7 Lesson 8Related: Smooth, Guide, Friction

O

Orbit: The path of a satellite or planet under gravity alone. Gravity is a central force, so \({r\,v_\theta}\) is constant. At perigee and apogee the velocity is perpendicular to the radial line, so \({r_p v_p = r_a v_a}\) (Example 8.3).; See: Lesson 8Related: Central force, Perigee and apogee, Kepler's second law
Origin \(O\) (pole): The fixed point from which \(r\) is measured and about which \(\theta\) turns, also called the pole. Put it at the pivot of the arm, the foot of the crane, the radar or the center of attraction: then the motion and the forces look simplest.; See: Lesson 2Related: Polar coordinates, Radial coordinate \(r\)

P

Particle: A body whose size and rotation do not matter for the question asked, so it is treated as a point with mass \(m\). The collars, pins, grippers, cars and satellites in this module are all particles.; See: Lesson 1Related: Kinematics, Kinetics
Path coordinates (\(t\), \(n\)): Components along the path (\(\mathbf{u}_t\)) and toward its center of curvature (\(\mathbf{u}_n\)), with \({a_t = \dot v}\) and \({a_n = v^2/\rho}\). Best when the path and the speed along it are known. On a circle centered on \(O\), \({a_n = -a_r}\) and \({a_t = a_\theta}\).; See: Lesson 1 Lesson 5 Lesson 8Related: Radius of curvature \(\rho\), Circular motion, Curvilinear motion
Path \(r = f(\theta)\): A path described by how the distance from \(O\) depends on the angle, set by a fixed groove, slot or cam. Once \(\theta(t)\) is known, usually from a turning arm, the motion along the path is fixed: find \(\dot r\) and \(\ddot r\) with the chain rule.; See: Lesson 6Related: Chain rule, Guide, Tangent angle \(\psi\)
Perigee and apogee: The closest (perigee) and farthest (apogee) points of an orbit around the Earth. There the velocity is perpendicular to the radial line (\({v_r = 0}\)), so \({r_p v_p = r_a v_a}\): the satellite is fastest where it is closest.; See: Lesson 8Related: Orbit, Central force
Polar coordinates: The pair \({(\colR{r}, \colT{\theta})}\) that locates a point in a plane: the distance \(r\) from the origin \(O\) and the angle \(\theta\) from the \({+x}\) axis. Convert with \({x = r\cos\theta}\), \({y = r\sin\theta}\), \({r = \sqrt{x^2 + y^2}}\) and \({\theta = \atantwo(y, x)}\).; See: Lesson 2Related: Cylindrical coordinates, atan2, Quadrant
Position vector: The vector from \(O\) to the particle, \({\rvec = r\,\er + z\,\ez}\) (in a plane, \({\rvec = r\,\er}\)), with magnitude \({\sqrt{r^2 + z^2}}\). It has no \(\et\) term: the angle is hidden in the direction of \(\er\).; See: Lesson 2Related: Unit vectors \(\er, \et, \ez\), Velocity
Product rule: \({\tfrac{d}{dt}(fg) = \dot f g + f\dot g}\). Applied to \({r\,\er}\), whose length and direction both change, it gives \({\dot r\,\er + r\dot\theta\,\et}\); applied to \({r\dot\theta\,\et}\), a product of three changing factors, it gives three terms.; See: Lesson 4 Lesson 5Related: Time derivatives of the unit vectors, Velocity, Chain rule

Q

Quadrant: One of the four regions into which the \(x\)- and \(y\)-axes divide the plane. A calculator's \({\tan^{-1}(y/x)}\) returns only \({-90^\circ}\) to \({90^\circ}\), so it is off by \({180^\circ}\) whenever \({x \lt 0}\): sketch the point, or use \(\atantwo\).; See: Lesson 2Related: atan2, Angular coordinate \(\theta\)

R

Radar tracking: Following a target by measuring its range \(r\), the angle \(\theta\) of the line of sight and their rates. The velocity follows from \({v_r = \dot r}\) and \({v_\theta = r\dot\theta}\); to follow a known velocity, a tracker needs \({\dot r = v_r}\) and \({\dot\theta = v_\theta/r}\).; See: Lesson 4Related: Velocity, Radial component
Radial component: The component of a vector along \(\er\), away from the axis: \({F_r = \Fvec\cdot\er = F_x\cos\theta + F_y\sin\theta}\). For the motion, \({v_r = \dot r}\) and \({a_r = \ddot r - r\dot\theta^2}\).; See: Lesson 3 Lesson 4Related: Transverse component, Axial component
Radial coordinate \(r\): The distance of the particle from \(O\) (in a plane) or from the \(z\)-axis (in space); never negative. In space it is not the distance from \(O\), which is \({\sqrt{r^2 + z^2}}\).; See: Lesson 2Related: Angular coordinate \(\theta\), \(z\)-coordinate
Radian: The angle unit in which an arc of radius \(r\) has length \({s = r\,\Delta\theta}\); \({\pi\ \text{rad} = 180^\circ}\). Every formula with \(\theta\), \(\dot\theta\) or \(\ddot\theta\) in this module, such as \({r\dot\theta}\), \({r\dot\theta^2}\) or \({f(\theta)}\), needs radians and rad/s. The radian is dimensionless, so m × rad/s is m/s.; See: Lesson 1 Lesson 2Related: Arc length, Angular velocity \(\dot\theta\), rpm
Radius of curvature \(\rho\): The radius of the circle that best fits the path at a point, used in path coordinates: \({a_n = v^2/\rho}\). For a circle about \(O\), \({\rho = r}\). On any other path \(\rho\) and \(r\) are different lengths and \(\mathbf{u}_n\) is not along \(-\er\), so the path and polar components differ.; See: Lesson 5Related: Path coordinates, Centripetal acceleration
rpm (revolutions per minute): A common unit of turning speed. Convert it before using it in a formula: \(N\) rpm is \({\dot\theta = 2\pi N/60}\) rad/s, so \(60\) rpm is \({2\pi \approx 6.283}\) rad/s.; See: Lesson 2Related: Angular velocity \(\dot\theta\), Radian

S

Slewing: Turning a crane's jib, or a robot's base, about its vertical axis, which changes only \(\theta\). The slew rate is \(\dot\theta\); a load at radius \(r\) moves at \({r\dot\theta}\) (Example 1.1).; See: Lesson 1Related: Angular velocity \(\dot\theta\), Cylindrical robot
Slotted arm: An arm with a slot along its length, turning about \(O\). A pin in the slot can slide along the arm, so a smooth slotted arm pushes it only across the slot, with a force \({F\,\et}\). Together with a fixed guide it drives the pin along a path \({r = f(\theta)}\).; See: Lesson 6 Lesson 8Related: Guide, Collar, Smooth
Smooth (surface, rod or guide): Frictionless: a smooth surface can push only perpendicular to itself. So a smooth rod that turns with the arm exerts no force along the rod (no \(\er\) component), and a smooth guide pushes perpendicular to the path.; See: Lesson 7Related: Normal force, Friction
Speed: The magnitude of the velocity, \({v = |\vvec| = \sqrt{\dot r^2 + (r\dot\theta)^2 + \dot z^2}}\). A constant speed does not mean zero acceleration: on a circle or a helix the direction of \(\vvec\) changes.; See: Lesson 4Related: Velocity, Tangent angle \(\psi\)
Spring force: A linear spring of stiffness \(k\) along the arm, attached at \(O\), exerts \({-k(r - r_0)\,\er}\), where \({r_0}\) is its unstretched length: a stretched spring pulls toward \(O\). Figure 7.2 balances it against \({m r\dot\theta^2}\), as in a centrifugal clutch.; See: Lesson 7Related: Equations of motion, Tension \(T\)

T

Tangent angle \(\psi\): The angle from the extended radial line (\(\er\)) to the tangent of the path, positive toward \(\et\): \({\tan\psi = v_\theta/v_r = r/(dr/d\theta)}\). It depends only on the shape of the path, and it sets the direction of a smooth guide's push (Lesson 8).; See: Lesson 4 Lesson 6Related: Path \(r = f(\theta)\), Guide, Logarithmic spiral
Tension \(T\): The pull of a cord, rope or cable. A cord pulled through a hole at \(O\) exerts \({-T\,\er}\) on the particle. A cord can only pull, so \({T \ge 0}\): if the equations give \({T \lt 0}\), the cord goes slack and the assumed motion cannot happen.; See: Lesson 7 Lesson 8Related: Central force, Equations of motion
Time derivatives of the unit vectors: \({\dot{\mathbf{u}}_r = \dot\theta\,\et}\), \({\dot{\mathbf{u}}_\theta = -\dot\theta\,\er}\) and \({\dot{\mathbf{u}}_z = \mathbf{0}}\). The unit vectors turn only when \(\theta\) changes, and these two results produce every new term in the velocity and the acceleration.; See: Lesson 3Related: Unit vectors \(\er, \et, \ez\), Product rule
Transverse component: The component of a vector along \(\et\), around the axis in the direction of increasing \(\theta\): \({F_\theta = -F_x\sin\theta + F_y\cos\theta}\). For the motion, \({v_\theta = r\dot\theta}\) and \({a_\theta = r\ddot\theta + 2\dot r\dot\theta}\). It is not the tangential component of path coordinates, except on a circle about \(O\).; See: Lesson 3 Lesson 4Related: Radial component, \(\mathbf{e}_r, \mathbf{e}_\theta\) notation

U

Unit vectors \(\er, \et, \ez\): Vectors of length 1 at the particle, each pointing where one coordinate increases: \({\er = \cos\theta\,\ihat + \sin\theta\,\jhat}\) (away from the axis), \({\et = -\sin\theta\,\ihat + \cos\theta\,\jhat}\) (around it) and \({\ez = \khat}\) (along it). They form a right-handed set, \({\er \times \et = \ez}\), and the first two turn with the particle.; See: Lesson 3Related: Time derivatives of the unit vectors, \(\mathbf{e}_r, \mathbf{e}_\theta\) notation

V

Velocity (in cylindrical components): The rate of change of the position, always tangent to the path: \({\vvec = \dot r\,\er + r\dot\theta\,\et + \dot z\,\ez}\). The \({r\dot\theta}\) term comes from differentiating the turning unit vector \(\er\).; See: Lesson 4Related: Speed, Acceleration, Tangent angle \(\psi\)

W

Weight (in polar components): The weight \(mg\) acts straight down. For motion in a horizontal plane it is \({-mg\,\ez}\), balanced by a vertical support. In a vertical plane, with \(\theta\) measured from the horizontal \(x\)-axis and \(y\) up, \({W_r = -mg\sin\theta}\) and \({W_\theta = -mg\cos\theta}\).; See: Lesson 3 Lesson 7Related: Radial component, Transverse component

Z

\(z\)-coordinate (height): The height of the particle along the \(z\)-axis, the same \(z\) as in rectangular coordinates. The \(z\)-axis is the axis that \(r\) is measured from and \(\theta\) turns about, usually vertical: a crane's mast, a robot's column, the pole of a spiral slide.; See: Lesson 2Related: Cylindrical coordinates, Axial component

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
\(\colR{r}\)	Distance from \(O\), or from the \(z\)-axis in space	Radial coordinate \(r\)
\(\colT{\theta}\)	Angle from the \(+x\) axis, counter-clockwise	Angular coordinate \(\theta\)
\(\colZ{z}\)	Height along the \(z\)-axis	\(z\)-coordinate
\(\er,\ \et,\ \ez\)	Cylindrical unit vectors (Hibbeler)	Unit vectors \(\er, \et, \ez\)
\(\mathbf{e}_r,\ \mathbf{e}_\theta\)	The same, in Beer & Johnston and Meriam & Kraige	\(\mathbf{e}_r, \mathbf{e}_\theta\) notation
\(\ihat,\ \jhat,\ \khat\)	Rectangular unit vectors	Unit vectors \(\er, \et, \ez\)
\(\mathbf{u}_t,\ \mathbf{u}_n\)	Path unit vectors: along the path, toward its center of curvature	Path coordinates
\(\rvec\)	Position vector	Position vector
\(\dot r,\ \dot\theta,\ \dot z\)	Rates of change with time	Dot notation, Angular velocity \(\dot\theta\)
\(\ddot r,\ \ddot\theta,\ \ddot z\)	Second time derivatives	Dot notation, Angular acceleration \(\ddot\theta\)
\(\vvec,\ v\)	Velocity and speed	Velocity, Speed
\(v_r,\ v_\theta,\ v_z\)	Velocity components	Radial component, Transverse component, Axial component
\(\avec;\ a_r,\ a_\theta,\ a_z\)	Acceleration and its components	Acceleration
\(\psi\)	Angle from the radial line to the tangent	Tangent angle \(\psi\)
\(f',\ f''\)	\(dr/d\theta\) and \(d^2r/d\theta^2\) on a path \(r = f(\theta)\)	Chain rule
\(h\)	\(r^2\dot\theta\), constant under a central force	Central force
\(\rho\)	Radius of curvature of the path (not the distance from \(O\))	Radius of curvature \(\rho\)
\(N,\ F,\ T\)	Normal force, force of an arm, cord tension	Normal force, Slotted arm, Tension \(T\)
\(\mu_k\)	Coefficient of kinetic friction	Friction
\(\atantwo(y, x)\)	Quadrant-correct angle of \((x, y)\)	atan2