Formula Sheet · Angular Momentum

A particle

Angular momentum about \(O\)

\[ \Hvec_O = \rvec \times m\vvec \]

Units: \(\text{kg·m}^2/\text{s}\) (= N·m·s).

In the plane; polar coordinates

\[ H_O = \pm m v d \] \[ H_O = m r v_\theta = m r^2\dot\theta \]

\(d\) = perpendicular distance from \(O\) to the line of \(\vvec\). Only \(v_\theta\) contributes; the radial velocity passes through \(O\).
Components: \(\Hvec_O = m\,[\,(y v_z - z v_y),\ (z v_x - x v_z),\ (x v_y - y v_x)\,]\), perpendicular to \(\rvec\) and \(\vvec\).

Angular impulse and momentum

Moment equation and impulse–momentum (\(O\) fixed)

\[ \sum\Mvec_O = \dot{\Hvec}_O \] \[ \Hvec_{O1} + \sum\int_{t_1}^{t_2}\Mvec_O\,dt = \Hvec_{O2} \]

Conservation: \(\Hvec_{O1} = \Hvec_{O2}\) when the angular impulse about \(O\) is zero.
Central force (cord through a hole, gravity of a planet): \(r_1 (v_\theta)_1 = r_2 (v_\theta)_2\); areal rate \(\dot A = \tfrac12 r^2\dot\theta = H_O/2m\) is constant (Kepler's second law).
Energy is not conserved just because \(H_O\) is: a pulled cord does work.

Systems of particles

About a fixed point, and about \(G\)

\[ \sum\Mvec_O = \dot{\Hvec}_O, \qquad \sum\Mvec_G = \dot{\Hvec}_G \] \[ \Hvec_O = \rvec_G \times m\vvec_G + \Hvec_G \]

\(\Hvec_O = \sum \rvec_i \times m_i\vvec_i\); only external forces contribute moments (internal ones cancel in pairs).
\(\sum\Mvec_G = \dot{\Hvec}_G\) holds even when \(G\) accelerates; about an arbitrary moving point it does not.
Internal forces can change the kinetic energy (skaters pulling on a rope) but not the total angular momentum.

Rigid bodies in plane motion

About \(G\), and about any point \(P\)

\[ H_G = I_G\,\omega \] \[ H_P = I_G\,\omega \pm m v_G d \]

Fixed axis, or the rolling contact \(C\)

\[ H_O = I_O\,\omega \] \[ H_C = I_C\,\omega \]

\(d\) = perpendicular distance from \(P\) to the line of \(m\vvec_G\) through \(G\); take each sign from its sense about \(P\).
Kinetics: \(\sum M_G = I_G\alpha\); \(\ \sum M_O = I_O\alpha\) (fixed axis); \(\ \sum M_P = I_G\alpha \pm m a_G d\) (the kinetic diagram).
Standard \(I_G\): rod \(\tfrac1{12}ml^2\), disk \(\tfrac12 mr^2\), ring \(mr^2\), sphere \(\tfrac25 mr^2\); parallel axes \(I_O = I_G + m d^2\).

Impulse, momentum and impact of rigid bodies

Plane motion

\[ m\vvec_{G1} + \sum\int\Fvec\,dt = m\vvec_{G2} \] \[ I_G\,\omega_1 + \sum\int M_G\,dt = I_G\,\omega_2 \]

Fixed axis: \(I_O\omega_1 + \sum\int M_O\,dt = I_O\omega_2\). About any point \(P\): use \(H_P\), including \(m v_G d\).
Clutch (shafts on one axis): \(I_A\omega_A + I_B\omega_B = (I_A + I_B)\,\omega\); fraction of \(T\) lost \(= I_B/(I_A + I_B)\) if \(B\) starts at rest.
Impact on a pinned body: \(H\) about the pin is conserved (the pin impulse has no moment); linear momentum is not.
Center of percussion: \(h_P = I_O/(m r_G) = k_O^2/r_G\); rod pinned at an end: \(\tfrac23 L\). Struck there, the pin feels no impulse.

Angular momentum in three dimensions

Inertia tensor, angular momentum and kinetic energy

\[ \Hvec_G = \Imat_G\,\wvec = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{xy} & I_{yy} & -I_{yz} \\ -I_{xz} & -I_{yz} & I_{zz} \end{bmatrix}\!\wvec \] \[ T = \tfrac12 m v_G^2 + \tfrac12\wvec\cdot\Hvec_G \]

About a fixed point \(O\) of the body: \(\Hvec_O = \Imat_O\,\wvec\), \(\ T = \tfrac12\wvec\cdot\Hvec_O\). In general \(\Hvec_P = \rvec_{G/P} \times m\vvec_G + \Hvec_G\).
\(\Hvec \parallel \wvec\) only for spin about a principal axis; in principal axes \(\Hvec = (I_x\omega_x,\ I_y\omega_y,\ I_z\omega_z)\).
Impulse–momentum in 3D: \(\Hvec_{G1} + \sum\int\Mvec_G\,dt = \Hvec_{G2}\), vector by vector.

Euler's equations

Rotating axes; principal axes fixed in the body (\(\Wvec = \wvec\))

\[ \sum\Mvec = (\dot{\Hvec})_{xyz} + \Wvec \times \Hvec \] \[ \begin{aligned} \sum M_x &= I_x\dot\omega_x - (I_y - I_z)\,\omega_y\omega_z \\ \sum M_y &= I_y\dot\omega_y - (I_z - I_x)\,\omega_z\omega_x \\ \sum M_z &= I_z\dot\omega_z - (I_x - I_y)\,\omega_x\omega_y \end{aligned} \]

Axisymmetric body with axes following the axle but not the spin: \(\wvec = \Wvec + \omega_s\khat\), and \(\sum\Mvec = (\dot{\Hvec})_{xyz} + \Wvec \times \Hvec\). Steady: \(\sum\Mvec = \Wvec \times \Hvec\).
Moments about \(G\), or about a fixed point \(O\) with the moments of inertia about \(O\).
Free spin is stable about the max and min principal axes, unstable about the intermediate one; \(\Hvec_G\) and \(T\) stay constant.

Gyroscopic motion

Euler angles and steady precession (\(\theta\), \(\dot\phi\), \(\dot\psi\) constant)

\[ \wvec = \dot\theta\,\ihat + \dot\phi\sin\theta\,\jhat + (\dot\phi\cos\theta + \dot\psi)\,\khat \] \[ \begin{aligned} \sum M_x = &-I\dot\phi^2\sin\theta\cos\theta \\ &+ I_z\,\dot\phi\sin\theta\,(\dot\phi\cos\theta + \dot\psi) \end{aligned} \]

Gyroscope (\(\theta = 90^\circ\))

\[ \sum M_x = I_z\,\Omega\,\omega_s \] \[ \Omega = \frac{m g r}{I_z\,\omega_s} \]

Torque-free (about \(G\))

\[ \dot\phi = \frac{H_G}{I} \] \[ \dot\psi = \frac{I - I_z}{I\,I_z}\,H_G\cos\theta \]

Top pivoted at \(O\), \(G\) a distance \(r\) up the axis: \(\sum M_x = m g r \sin\theta\); \(I\) and \(I_z\) about \(O\).
The spin axis swings toward the moment vector. Torque-free: direct precession if \(I > I_z\) (long), retrograde if \(I \lt I_z\) (flat).

Common mistakes

Mixing reference points. Take \(\Hvec\) and \(\sum\Mvec\) about the same point, fixed or \(G\).
Forgetting \(m v_G d\). About a point other than \(G\), \(H_P = I_G\omega \pm m v_G d\), not \(I_G\omega\).
Assuming energy is conserved. Clutches, impacts and pulled cords conserve \(H\) but not \(T\).
Conserving \(H\) about a point with an impulsive reaction. Choose the pin or contact point so its impulse has no moment.
\(\Hvec\) parallel to \(\wvec\). Only about a principal axis; otherwise use \(\Imat\wvec\).
Differentiating components in rotating axes. Add \(\Wvec \times \Hvec\).
Moments of inertia about the wrong point. For a top, \(I\) is about the pivot: \(I_G + m r^2\).
The gyroscope “falls”. It precesses about the vertical instead: the axis moves toward \(\Mvec\).