Rigid Body

7.1 Rigid-Body Kinematics

A rigid body is an idealized system whose interparticle distances remain constant. Its motion combines translation of the center of mass with rotation about the center of mass.

For two points \(A\) and \(B\) fixed in the body,

\[ \mathbf v_B=\mathbf v_A+\boldsymbol\omega\times\mathbf r_{B/A}, \]

where \(\mathbf r_{B/A}=\mathbf r_B-\mathbf r_A\). Differentiating in an inertial frame gives

\[ \mathbf a_B=\mathbf a_A+\boldsymbol\alpha\times\mathbf r_{B/A} +\boldsymbol\omega\times(\boldsymbol\omega\times\mathbf r_{B/A}). \]

For rotation about a fixed axis, the angular variables are

\[ \omega=\frac{d\theta}{dt},\qquad \alpha=\frac{d\omega}{dt},\qquad v_t=\omega r_\perp,\qquad a_t=\alpha r_\perp,\qquad a_n=\omega^2r_\perp. \]

Here \(r_\perp\) is the perpendicular distance to the axis, \(v_t\) and \(a_t\) are signed tangential components, and \(a_n\) is the magnitude of the inward normal acceleration.

7.2 Moment of Inertia

The moment of inertia measures the distribution of mass relative to a specified axis:

\[ I=\sum_i m_i r_{\perp,i}^{\,2} \qquad\text{or}\qquad I=\int r_\perp^2\,dm. \]

Its SI unit is \(\mathrm{kg\,m^2}\). The same body generally has different moments of inertia about different axes.

Theorem 7.1 (Parallel-Axis Theorem) For a body of total mass \(M\), let an axis be parallel to an axis through its center of mass and at perpendicular distance \(d\) from it. Then

\[ I=I_{\mathrm{CM}}+Md^2. \]

For uniform bodies:

Body Axis Moment of inertia
Thin hoop, radius \(R\) Symmetry axis perpendicular to its plane \(MR^2\)
Solid disk or cylinder, radius \(R\) Symmetry axis \(\frac12MR^2\)
Solid sphere, radius \(R\) Any diameter \(\frac25MR^2\)
Thin rod, length \(\ell\) Through its center, perpendicular to the rod \(\frac1{12}M\ell^2\)
Thin rod, length \(\ell\) Through one end, perpendicular to the rod \(\frac13M\ell^2\)

7.3 Rotational Dynamics

The torque of a force about a point \(O\) is

\[ \boldsymbol\tau_O=\mathbf r\times\mathbf F, \qquad |\boldsymbol\tau_O|=rF\sin\phi, \]

where \(\mathbf r\) points from \(O\) to the force’s point of application and \(\phi\) is the angle between \(\mathbf r\) and \(\mathbf F\).

Theorem 7.2 (Newton’s Second Law for Fixed-Axis Rotation) For a rigid body rotating about a fixed axis in an inertial frame,

\[ \sum\tau_z^{\mathrm{ext}}=\frac{dL_z}{dt}=I\alpha, \qquad L_z=I\omega. \]

The subscript \(z\) denotes the component along the rotation axis. The scalar relation \(L_z=I\omega\) does not assert that the full vectors \(\mathbf L\) and \(\boldsymbol\omega\) are always parallel.

Translation is governed by \(\sum\mathbf F^{\mathrm{ext}}=M\mathbf a_{\mathrm{CM}}\). If the external torque component about a fixed axis vanishes, the corresponding angular momentum component is conserved.

7.4 Rotational Work and Energy

For fixed-axis rotation,

\[ K_{\mathrm{rot}}=\frac12I\omega^2,\qquad W=\int_{\theta_i}^{\theta_f}\tau_z^{\mathrm{ext}}\,d\theta =\Delta K_{\mathrm{rot}},\qquad \mathcal P=\tau_z^{\mathrm{ext}}\omega. \]

For planar motion, separating translation and rotation about the center of mass gives

\[ K=\frac12Mv_{\mathrm{CM}}^2+\frac12I_{\mathrm{CM}}\omega^2. \]

7.5 Rolling Without Slipping

For a circular body rolling without slipping on a stationary surface, its contact point is instantaneously at rest relative to the surface. Along a straight surface, the speed and tangential acceleration satisfy

\[ v_{\mathrm{CM}}=R\omega,\qquad a_{\mathrm{CM}}=R\alpha. \]

Static friction enforces this constraint when needed. Its magnitude must satisfy \(|f_s|\le\mu_sN\); it need not equal \(\mu_sN\).

Example: rolling down an incline. For an axisymmetric body rolling freely down a fixed incline of angle \(\beta\), with sufficient static friction and negligible rolling resistance,

\[ Ma=Mg\sin\beta-f_s,\qquad I_{\mathrm{CM}}\alpha=f_sR,\qquad a=R\alpha. \]

Consequently,

\[ a=\frac{g\sin\beta}{1+I_{\mathrm{CM}}/(MR^2)}. \]

A uniform solid cylinder therefore has \(a=\frac23g\sin\beta\).

7.6 References