## Rigid body motion (notes from an audited phy354 classical mechanics course (Taught by Prof. Erich Poppitz))

Posted by peeterjoot on March 8, 2012

# Rigid body motion.

## Setup

We’ll consider either rigid bodies as in the connected by sticks figure (\ref{fig:rigidBodyMotion:rigidBodyMotionFig1}) or a body consisting of a continuous mass as in figure (\ref{fig:rigidBodyMotion:rigidBodyMotionFig2})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{rigidBodyMotionFig1}

\caption{Rigid body of point masses.}

\end{figure}

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{rigidBodyMotionFig2}

\caption{Rigid solid body of continuous mass.}

\end{figure}

In the first figure our mass is made of discrete particles

whereas in the second figure with mass density and a volume element , our total mass is

## Degrees of freedom

How many numbers do we need to describe fixed body motion. Consider figure (\ref{fig:rigidBodyMotion:rigidBodyMotionFig3})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{rigidBodyMotionFig3}

\caption{Body local coordinate system with vector to a fixed point in the body}

\end{figure}

We will need to use six different numbers to describe the motion of a rigid body. We need three for the position of the body as a whole. We also need three degrees of freedom (in general) for the motion of the body at that point in space (how our local coordinate system at the body move at that point), describing the change of the orientation of the body as a function of time.

Note that the angle hasn’t been included in any of the pictures because it is too messy with all the rest. Picture something like figure (\ref{fig:rigidBodyMotion:rigidBodyMotionFig5})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{rigidBodyMotionFig5}

\caption{Rotation angle and normal in the body}

\end{figure}

Let’s express the position of the body in terms of that body’s center of mass

or for continuous masses

We consider the motion of point , an arbitrary point in the body as in figure (\ref{fig:rigidBodyMotion:rigidBodyMotionFig4}), whos motion consists of

\begin{enumerate}

\item displacement of the CM

\item rotation of around some axis going through CM on some angle . (here is a unit vector).

\end{enumerate}

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{rigidBodyMotionFig4}

\caption{A point in the body relative to the center of mass.}

\end{figure}

From the picture we have

where

Dividing by we have

The total velocity of this point in the body is then

where

This circular motion is illustrated in figure (\ref{fig:rigidBodyMotion:rigidBodyMotionFig6})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{rigidBodyMotionFig6}

\caption{circular motion.}

\end{figure}

Note that is the velocity of the particle with respect to the unprimed system.

We’ll spend a lot of time figuring out how to express .

Now let’s consider a second point as in figure (\ref{fig:rigidBodyMotion:rigidBodyMotionFig7})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{rigidBodyMotionFig7}

\caption{Two points in a rigid body.}

\end{figure}

we have

Have another way that we can use to express the position of the point

or

Equating with above, and noting that this holds for all , and noting that if

hence

or

The moral of the story is that the angular velocity is a characteristic of the system. It does not matter if it is calculated with respect to the center of mass or not.

See some examples in the notes.

# Kinetic energy

For all in the body we have

here is an arbitrary fixed point in the body as in figure (\ref{fig:rigidBodyMotion:rigidBodyMotionFig8})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{rigidBodyMotionFig8}

\caption{Kinetic energy setup relative to point in the body.}

\end{figure}

The kinetic energy is

We see that if we take to be the center of mass then our cross term

which vanishes. With

we have

with

or

Forgetting about the dependent term for now we have

Expanding this out with

and

we have

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