Reference · 36 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 36 terms

A

Angular impulse: The time integral of a moment, \(\int_{t_1}^{t_2}\Mvec_O\,dt\), in N·m·s. It changes the angular momentum about the same point: \(\Hvec_{O1} + \sum\int\Mvec_O\,dt = \Hvec_{O2}\).; See: Lesson 2, Lesson 5Related: Angular momentum \(\Hvec_O\), Impulse–momentum principle, Conservation of angular momentum
Angular momentum \(\Hvec_O\) (of a particle): The moment of a particle's linear momentum about a point \(O\): \(\Hvec_O = \rvec \times m\vvec\), in kg·m²/s. It is perpendicular to \(\rvec\) and \(\vvec\); in the plane its magnitude is \(m v d\), and in polar coordinates \(m r^2\dot\theta\).; See: Lesson 1, Lesson 1Related: Linear momentum \(\Lvec\), Right-hand rule, Angular momentum of a rigid body
Angular momentum of a rigid body: In plane motion, \(H_G = I_G\omega\) and, about any other point, \(H_P = I_G\omega \pm m v_G d\). In 3D, \(\Hvec_G = \Imat_G\wvec\), which is parallel to \(\wvec\) only for rotation about a principal axis.; See: Lesson 4, Lesson 6Related: Inertia tensor \(\Imat\), Angular momentum \(\Hvec_O\), Principal axes
Angular velocity \(\wvec\): The rate and axis of rotation of a rigid body, in rad/s, directed along the axis by the right-hand rule. In plane motion \(\wvec = \omega\khat\); for a spinning top it combines precession, nutation and spin.; See: Lesson 6, Lesson 8Related: Angular momentum of a rigid body, Euler angles
Areal velocity: The rate at which the line from \(O\) to a particle sweeps out area: \(\dot A = \tfrac12 r^2\dot\theta = H_O/2m\). Under a central force it is constant, which is Kepler's second law.; See: Lesson 2Related: Central force, Conservation of angular momentum

C

Center of mass \(G\): The mass-weighted average position of a system, \(\rvec_G = \sum m_i\rvec_i / m\). Moments about \(G\) obey \(\sum\Mvec_G = \dot{\Hvec}_G\) even when \(G\) accelerates, and \(\Hvec_O = \rvec_G \times m\vvec_G + \Hvec_G\).; See: Lesson 3Related: System of particles, Moment equation
Center of percussion: The point on a pinned body where a blow causes no impulsive reaction at the pin: a distance \(h_P = k_O^2/r_G\) from the pin. For a slender rod pinned at one end it is \(\tfrac23\) of the way along.; See: Lesson 5Related: Impulsive force, Angular impulse
Central force: A force whose line of action always passes through one fixed point \(O\), such as a cord pulled through a hole or a planet's gravity. Its moment about \(O\) is zero, so the particle's \(H_O\) is conserved.; See: Lesson 2Related: Conservation of angular momentum, Areal velocity
Conservation of angular momentum: If the external forces have no net angular impulse about a point, the angular momentum about that point is constant: \(\Hvec_{O1} = \Hvec_{O2}\). Internal forces and the impulse of a pin at the chosen point do not spoil it; the kinetic energy may still change.; See: Lesson 2, Lesson 3, Lesson 5Related: Angular impulse, Central force, Impulsive force

D

Direct precession: Torque-free precession of a long axisymmetric body (\(I > I_z\)) in which the spin \(\dot\psi\) and the precession \(\dot\phi\) have the same sense, as for a thrown football or a rocket.; See: Lesson 8Related: Retrograde precession, Torque-free motion, Precession

E

Euler angles: Three angles that fix the orientation of an axisymmetric body: precession \(\phi\) about the fixed vertical, nutation (tilt) \(\theta\), and spin \(\psi\) about the body's own axis. Then \(\wvec = \dot\theta\,\ihat + \dot\phi\sin\theta\,\jhat + (\dot\phi\cos\theta + \dot\psi)\,\khat\).; See: Lesson 8Related: Precession, Nutation, Spin
Euler's equations: The rotational equations of a rigid body in principal axes fixed in it: \(\sum M_x = I_x\dot\omega_x - (I_y - I_z)\omega_y\omega_z\) and cyclically. Moments are about \(G\) or a fixed point \(O\).; See: Lesson 7Related: Modified Euler equations, Rotating axes, Principal axes

F

Fixed-axis rotation: Rotation of a body about a fixed pin \(O\): \(H_O = I_O\omega\), \(\sum M_O = I_O\alpha\), with \(I_O = I_G + m r_G^2\). The same form holds about the contact point of a wheel rolling without slipping.; See: Lesson 4Related: Angular momentum of a rigid body, Mass moment of inertia \(I\)

G

Gyroscope: A fast-spinning rotor whose axis precesses instead of falling under a moment. With the axis horizontal, \(\sum M_x = I_z\Omega\,\omega_s\), so a weight \(mg\) at arm \(r\) gives \(\Omega = m g r/(I_z\omega_s)\).; See: Lesson 8Related: Gyroscopic moment, Steady precession
Gyroscopic moment: The moment needed to turn the axis of a spinning rotor: for steady motion \(\sum\Mvec = \Wvec \times \Hvec\), of size about \(I_s\omega_s\Omega\). It acts perpendicular to both the spin and the turning axis and loads the bearings.; See: Lesson 7Related: Gyroscope, Modified Euler equations

I

Impulse–momentum principle (angular): Initial angular momentum plus angular impulse equals final angular momentum, about a fixed point or about \(G\). For a rigid body in plane motion it is used together with \(m\vvec_{G1} + \sum\int\Fvec\,dt = m\vvec_{G2}\).; See: Lesson 2, Lesson 5, Lesson 6Related: Angular impulse, Conservation of angular momentum
Impulsive force: A large force that acts for a very short time, as in an impact. Its impulse is finite while that of ordinary forces such as weight is negligible, so only impulsive forces enter the momentum balance.; See: Lesson 5Related: Conservation of angular momentum, Center of percussion
Inertia tensor \(\Imat\): The symmetric \(3\times3\) matrix of moments of inertia (diagonal) and negative products of inertia (off-diagonal). It gives \(\Hvec = \Imat\wvec\) and \(T = \tfrac12\wvec^\mathsf{T}\Imat\wvec\).; See: Lesson 6Related: Product of inertia, Principal axes, Angular momentum of a rigid body
Intermediate-axis theorem: Free rotation about the axis of maximum or minimum moment of inertia is stable; about the intermediate axis a small wobble grows and the body flips over and over, while \(\Hvec_G\) stays fixed.; See: Lesson 7Related: Euler's equations, Principal axes, Torque-free motion

K

Kinetic diagram: A sketch showing \(m\mathbf{a}_G\) at \(G\) and \(I_G\alpha\) as a couple, set equal to the free-body diagram. Moments about any point \(P\) then give \(\sum M_P = I_G\alpha \pm m a_G d\).; See: Lesson 4Related: Moment equation, Angular momentum of a rigid body
Kinetic energy \(T\): For a rigid body \(T = \tfrac12 m v_G^2 + \tfrac12\wvec\cdot\Hvec_G\); about a fixed point, \(T = \tfrac12\wvec\cdot\Hvec_O\). It is often not conserved when angular momentum is, as in clutches and impacts.; See: Lesson 6Related: Inertia tensor \(\Imat\), Conservation of angular momentum

L

Linear momentum \(\Lvec\): \(\Lvec = m\vvec\), in kg·m/s; \(\sum\Fvec = \dot{\Lvec}\). Angular momentum is its moment about a point, just as a moment is the moment of a force.; See: Lesson 1Related: Angular momentum \(\Hvec_O\)

M

Mass moment of inertia \(I\): \(I = \int r^2\,dm\) about an axis, in kg·m². About a parallel axis a distance \(d\) from \(G\), \(I = I_G + m d^2\). Standard values: rod \(\tfrac1{12}ml^2\), disk \(\tfrac12 mr^2\), sphere \(\tfrac25 mr^2\).; See: Lesson 4Related: Fixed-axis rotation, Inertia tensor \(\Imat\)
Modified Euler equations: For an axisymmetric body, axes that follow the symmetry axis but not the spin: \(\sum\Mvec = (\dot{\Hvec})_{xyz} + \Wvec \times \Hvec\), with \(\wvec = \Wvec + \omega_s\khat\). Used for gyroscopes and tops.; See: Lesson 7Related: Euler's equations, Rotating axes, Gyroscopic moment
Moment equation: \(\sum\Mvec_O = \dot{\Hvec}_O\): the resultant external moment about a fixed point equals the rate of change of angular momentum about it. It also holds about \(G\), but not about an arbitrary moving point.; See: Lesson 2, Lesson 3Related: Angular momentum \(\Hvec_O\), Center of mass \(G\)

N

Nutation: Change in the tilt \(\theta\) of a spinning body's axis. A top released without the right precession rate nods up and down while it precesses; its axis traces loops or cusps.; See: Lesson 8, Lesson 8Related: Euler angles, Precession, Steady precession

P

Precession: Rotation of a spinning body's axis about another axis, such as the vertical for a top (rate \(\dot\phi\)) or \(\Hvec_G\) for a body in free flight.; See: Lesson 8Related: Steady precession, Nutation, Spin
Principal axes: Axes for which all products of inertia vanish, so \(\Imat\) is diagonal and \(\Hvec = (I_x\omega_x, I_y\omega_y, I_z\omega_z)\). Rotation about a principal axis is the only case with \(\Hvec \parallel \wvec\).; See: Lesson 6, Lesson 7Related: Inertia tensor \(\Imat\), Euler's equations, Intermediate-axis theorem
Product of inertia: \(I_{xy} = \int xy\,dm\) (and similarly \(I_{yz}\), \(I_{zx}\)). Products appear with a minus sign in the inertia tensor and make \(\Hvec\) tilt away from \(\wvec\).; See: Lesson 6Related: Inertia tensor \(\Imat\), Principal axes

R

Retrograde precession: Torque-free precession of a flat axisymmetric body (\(I \lt I_z\)) in which the spin \(\dot\psi\) and the precession \(\dot\phi\) have opposite senses, as for a wobbling coin or the Earth.; See: Lesson 8Related: Direct precession, Torque-free motion
Right-hand rule: Curl the fingers of the right hand in the sense of rotation; the thumb gives the direction of the vector. It fixes the direction of \(\rvec \times m\vvec\), of \(\wvec\) and of moments.; See: Lesson 1Related: Angular momentum \(\Hvec_O\)
Rotating axes (derivative in): For a vector written in axes turning at \(\Wvec\), \(\dot{\mathbf{A}} = (\dot{\mathbf{A}})_{xyz} + \Wvec \times \mathbf{A}\). Applied to \(\Hvec\) it gives Euler's equations.; See: Lesson 7Related: Euler's equations, Modified Euler equations

S

Spin: Rotation of a body about its own symmetry axis, at rate \(\dot\psi\) relative to the precessing axes. The total spin component is \(\omega_z = \dot\phi\cos\theta + \dot\psi\).; See: Lesson 8Related: Euler angles, Precession
Steady precession: Motion with \(\theta\), \(\dot\phi\) and \(\dot\psi\) all constant. It needs \(\sum M_x = -I\dot\phi^2\sin\theta\cos\theta + I_z\dot\phi\sin\theta\,(\dot\phi\cos\theta + \dot\psi)\); for a top, \(\sum M_x = m g r\sin\theta\), which gives a slow and a fast rate.; See: Lesson 8Related: Gyroscope, Precession, Nutation
System of particles: Any collection of particles treated together. Internal forces cancel in pairs in \(\sum\Mvec_O\), so only external moments change the total \(\Hvec_O = \sum\rvec_i \times m_i\vvec_i\).; See: Lesson 3Related: Moment equation, Center of mass \(G\), Conservation of angular momentum

T

Torque-free motion: Motion of a body with no moment about \(G\): \(\Hvec_G\) and \(T\) are constant. An axisymmetric body precesses about \(\Hvec_G\) at \(\dot\phi = H_G/I\) with constant tilt.; See: Lesson 8Related: Direct precession, Retrograde precession, Intermediate-axis theorem

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
Symbol	Meaning	Entry
\(\Hvec_O,\ \Hvec_G\)	Angular momentum about a fixed point, about \(G\)	Angular momentum \(\Hvec_O\)
\(\Lvec = m\vvec\)	Linear momentum	Linear momentum \(\Lvec\)
\(\Mvec_O,\ \Mvec_G\)	Moment about \(O\), about \(G\)	Moment equation
\(\int\Mvec\,dt\)	Angular impulse	Angular impulse
\(\wvec,\ \omega\)	Angular velocity and its magnitude	Angular velocity \(\wvec\)
\(\Wvec\)	Angular velocity of the reference axes	Rotating axes
\(I_G,\ I_O\)	Moment of inertia about \(G\), about \(O\)	Mass moment of inertia \(I\)
\(\Imat\)	Inertia tensor	Inertia tensor \(\Imat\)
\(I_x,\ I_y,\ I_z\)	Principal moments of inertia	Principal axes
\(I,\ I_z\)	Transverse and axial moments of an axisymmetric body	Steady precession
\(d\)	Perpendicular distance to the line of \(m\vvec\) or \(m\vvec_G\)	Angular momentum of a rigid body
\(h_P\)	Distance from the pin to the center of percussion	Center of percussion
\(\phi,\ \theta,\ \psi\)	Euler angles: precession, nutation, spin	Euler angles
\(\omega_s,\ \Omega\)	Spin rate and precession rate of a gyroscope	Gyroscope
\(T\)	Kinetic energy	Kinetic energy \(T\)

Notation follows Hibbeler: \(\Hvec\) for angular momentum (some books use \(\mathbf{L}\) or \(\mathbf{h}\)), and \(\phi\), \(\theta\), \(\psi\) for the Euler angles of precession, nutation and spin.