Abstract
Here we extend our analysis of rigid body motion to include dynamics—i.e., the forces and torques that produce the complicated translational and rotational motion of a rigid body as it moves through space. In particular, we derive Euler’s equations for rigid body motion, which extend the familiar first-year physics equations to general rotations in three dimensions around an axis that can change its orientation in response to external forces. We then apply Euler’s equations to several examples.
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Notes
- 1.
Although we will often take \(O'\) to be at the center of mass of the body, it doesn’t have to be there, so we will keep things general at this stage of the calculation.
- 2.
- 3.
There is also the support force that balances the weight of the top, but since it acts at O it does not exert a torque on the system about this point.
- 4.
Alternatively, we can replace the discrete mass points \(m_I\) at locations \(\mathbf {r}'_I\) by infinitesimal masses \(\mathrm {d}m = \rho \mathrm {d}V\) at locations \(\mathbf {r}'\), where \(\rho \) is the mass density of the Earth (assumed to be constant), and \(\mathrm {d}V\) is an infinitesimal volume element centered at \(\mathbf {r}'\).
- 5.
Recall that the first three Legendre polynomials are given by
$$\begin{aligned} P_0(x)=1\,, \qquad P_1(x)=x\,, \qquad P_2(x)=\frac{1}{2}(3x^2-1)\,, \end{aligned}$$which are normalized so that \(P_n(1)=1\).
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Benacquista, M.J., Romano, J.D. (2018). Rigid Body Dynamics. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68780-3_7
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DOI: https://doi.org/10.1007/978-3-319-68780-3_7
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Publisher Name: Springer, Cham
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Online ISBN: 978-3-319-68780-3
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