Kinetics of rigid bodies

Euler's laws of motion

The laws of motion for rigid bodies were formulated by Leonhard Euler, which extended Newton's equations for point-masses to rigid bodies.

Euler's first law relates the mass $m$ and acceleration of the center of mass $\vec{a}_C$ of the body to the total force $\vec{F}$ on the body (sum of all external forces). It describes the translational motion of the body.

Euler's first law (acceleration form).

\[ \vec{F} = m\vec{a}_C \]

Euler's second law relates the moment of inertia about the center of mass $I_C$ and the angular acceleration of the body to the total moment $\vec{M}_C$ (sum of all external moments) about the center of mass. It describes the rotational motion of the body.

Euler's second law (angular acceleration form).

\[ \vec{M}_C = I_C\vec\alpha \]

Recall that the total force on a particle is related to the linear momentum of the particle. Thus, if we extend that to rigid bodies, we have an alternative way to express Euler's first law:

Euler's first law (momentum form).

\[ \vec{F} = \frac{d\vec{p}_C}{dt} = \frac{d}{dt}(m\vec{v}_C) \]

Similarly, the total moment on a particle about a certain point is related to the angular momentum of the particle about the same base point. We can do the same thing as above, and extend it to rigid bodies, and obtain an alternative form to Euler's second law:

Euler's second law (momentum form).

\[ \vec{M}_C = \frac{d\vec{L}_C}{dt} = \frac{d}{dt}(I_C \, \vec\omega) \]

Rotation about arbitrary reference points

In some cases, using the center of mass as a reference point is not ideal, and we might encounter situations in which we would need to consider the dynamics about another point. We will begin by finding the angular momentum of a rigid body about any arbitrary point, and extend that from there.

Angular momentum of a rigid body about an arbitrary point $P$.

\[ \vec{L}_P = \vec{r}_{PC} \times m\vec{v}_C + I_C \, \vec\omega \]

We can differentiate the above expression with respect to time and obtain the time derivative of the angular momentum of the rigid body about point $P$. This will yield two important special cases of rotations.

Rate of change of angular momentum about an arbitrary point $P$.

\[ \frac{d\vec{L}_P}{dt} = \vec{M}_P - \vec{r}_{PC} \times m\vec{a}_P \]

The important special cases are outlined below.

Special cases of the angular momentum rate of change.

case result consequence
$P = C$ $\vec{M}_P = \vec{M}_C = \frac{d\vec{L}_C}{dt}$ This is Euler's 2nd law.
$\vec{a}_P = \vec{0}$ $\vec{M}_P = \frac{d\vec{L}_P}{dt}$ The rigid body is rotating about a fixed point $P$. This is another form of Euler's 2nd law.