Dynamics of rigid Bodies

Appendices

Summary

• The model of the extended rigid body neglects all internal motions (Deformations and vibrations). The center of mass S has the coordinates

  $$\displaystyle\boldsymbol{r}_{\mathrm{S}}=\frac{1}{M}\int_{V}\boldsymbol{r}\varrho(\boldsymbol{r})\,\mathrm{d}V=\frac{1}{V}\int\boldsymbol{r}\,\mathrm{d}V\quad{\text{for}}\quad\varrho=\mathrm{const}{\;}.$$

• The motion of a free rigid body can be always composed of a translation of the center of mass S with the velocity v _s and a rotation of the body around S with the angular velocity ω. The motion of the extended body has therefore 6 degrees of freedom.

• For the motion of an extended body not only magnitude and direction of the force acting on the body are important but also the point of action on the body.

• An arbitrary force acting on an extended body can always be composed of a force acting on the center of mass S (translational acceleration) and a couple of forces causing an accelerated rotation.

• The moment of inertia (rotational inertia) for a rotation about an axis through the center of mass S is $I_{s}=\int r_{\bot}^{2}\varrho\,\mathrm{d}V$ where $r_{\bot}$ is the distance of the volume element $\,\mathrm{d}V$ from the rotation axis. The moment of inertia for a rotation around an arbitrary axis with a distance a from the parallel axis through S is $I=I_{s}+Ma^{2}$ (parallel axis theorem or Steiner’s theorem).

• The kinetic energy of the rotational motion is $E_{\mathrm{rot}}=\tfrac{1}{2}I\omega^{2}$.

• The equation of motion for a body rotating about a space-fixed axis is $D_{\parallel}=I\cdot\,\mathrm{d}\omega/\,\mathrm{d}t$, where $D_{\parallel}$ is the component of the torque parallel to the rotation axis.

• The moment of inertia I _s depends on the direction of the rotation axis relative to a selected axis of the body. It can be described by a tensor. The directions of the axes with the maximum and the minimum inertial moment determine the principal axes system. In this system the tensor is diagonal. The diagonal elements are the principal moments of inertia.

• If two of the principal moments are equal, the body is a symmetric top. If all three are equal the body is a spherical top.

• Angular momentum L and angular velocity $\boldsymbol{\omega}$ are related by $\boldsymbol{L}=I\cdot\boldsymbol{\omega}$, where I is the inertial tensor, which is diagonal in the principal axes system. In the general case L and $\boldsymbol{\omega}$ are not parallel.

• If the body rotates about a principal axis, L and $\boldsymbol{\omega}$ are parallel and without external torque their directions are space-fixed.

• For an arbitrary direction of ω the momentary rotation axis $\boldsymbol{\omega}$ nutates around the angular momentum axis which is space-fixed without external torque.

• Under the action of an external torque the angular momentum axis L precedes around the external force and in addition the momentary rotation axis nutates around L. The relation between L and D is $\boldsymbol{D}=\,\mathrm{d}\boldsymbol{L}/\,\mathrm{d}t$.

• The general motion of a top is completely described by the Euler-equations.

• The earth can be approximately described by a symmetric top, which rotates about the axis of its maximum moment of inertia. The vector sum of the gravity forces exerted by the sun, the moon and the planets results in a torque which causes a periodic precession of the earth axis with a period of 25 850 years. In addition changes of the mass distribution in the earth cause a small difference between symmetry axis and momentary rotation axis. Therefor the earth axis performs an irregular nutation around the symmetry axis.

Problems

5.1

Determine the center of mass of a homogeneous sector of a sphere with radius R and opening angle α.

5.2

What are moment of inertia, angular momentum and rotational energy of our earth

• a) If the density $\varrho$ is constant for the whole earth

• b) If for $r\leq R/2$ the homogeneous density $\varrho_{1}$ is twice the density $\varrho_{2}$ for $r> R/2$?

• c) By how much would the angular velocity of the earth change, if all people on earth ($n=5\cdot 10^{9}$ with $m=70\,\mathrm{k}\mathrm{g}$ each) would gather at the equator and would start at the same time to run into the east direction with an acceleration $a=2\,\mathrm{m}/\mathrm{s}^{2}$?

5.3

A cylindrical disc with radius R and mass M rotates with $\omega=2\pi\cdot 10\,\mathrm{s}^{-1}$ about the symmetry axis ($R=10\,\mathrm{c}\mathrm{m}$, $M=0.1\,\mathrm{k}\mathrm{g}$).

• a) Calculate the angular momentum L and the rotational energy $E_{\mathrm{rot}}$.

• b) a bug with $m=10\,\mathrm{g}$ falls vertical down onto the edge of the disc and holds itself tight. What is the change of L and $E_{\mathrm{rot}}$?

• c) The bug now creeps slowly in radial direction to the center of the disc. How large are now $\omega(r)$, $I(r)$ and $E_{\mathrm{rot}}(r)$ as a function of the distance r from the center r = 0?

5.4

The mass density $\varrho$ of a circular cylinder (radius R, height H) increases in the radial direction as $\varrho(r)=\varrho_{0}(1+(r/R)^{2})$.

• a) How large is its inertial moment for the rotation about the symmetry axis for $R=10\,\mathrm{c}\mathrm{m}$ and $\varrho_{0}=2\,\mathrm{k}\mathrm{g}/\mathrm{d}\mathrm{m}^{3}$?

• b) How long does it take for the cylinder to roll down an inclined plane with $\alpha=10^{\circ}$ from $h=1\,\mathrm{m}$ to h = 0?

5.5

Calculate the rotational energy of the Na${}_{3}$-molecule composed of 3 Na atoms ($m=23\,\mathrm{AMU}$) which form an isosceles triangle with the apex angle $\alpha=79^{\circ}$ and a side length of $d=0.32\,\mathrm{n}\mathrm{m}$ when it rotates around the three principal axes with the angular momentum $L=\smash{\sqrt{l(l+1)}}\cdot\hbar$. Determine at first the three axis and the center of mass.

5.6

A wooden rod with mass $M=1\,\mathrm{k}\mathrm{g}$ and a length $l=0.4\,\mathrm{m}$, which is initially at rest, can freely rotate about a vertical axis through the center of mass. The end of the rod is hit by a bullet ($m=0.01\,\mathrm{k}\mathrm{g}$) with the velocity $v=200\,\mathrm{m}/\mathrm{s}$, which moves in the horizontal plane perpendicular to the rod and to the rotation axis and which gets stuck in the wood.

What are the angular velocity ω and the rotational energy $E_{\mathrm{rot}}$ of the rod after the collision? Which fraction of the kinetic energy of the bullet has been converted to heat?

5.7

A homogeneous circular disc with mass m and radius R rotates with constant velocity ω around a fixed axis through the center of mass S perpendicular to the disc plane. At the time t = 0 a torque $D=D_{0}\cdot e^{-at}$ starts to act on the disc. What is the time dependence $\omega(t)$ of the angular velocity? Numerical example: $\omega_{0}=10\,\mathrm{s}^{-1}$, $m=2\,\mathrm{k}\mathrm{g}$, $R=10\,\mathrm{c}\mathrm{m}$, $a=0.1\,\mathrm{s}^{-1}$, $D_{0}=0.2\,\mathrm{N}\mathrm{m}$.

5.8

A full cylinder and a hollow cylinder with a thin wall and equal outer diameters roll with equal angular velocity $\omega_{0}$ on a horizontal plane and then role up an inclined plane. At which height h do they return? (Friction should be neglected), numerical example: $R=0.1\,\mathrm{m}$, $\omega_{0}=15\,\mathrm{s}^{-1}$.