293-] BODY WITH FIXED AXIS. 157 



III. Rigid Body ^vith a Fixed Axis. 



291. A rigid body with a fixed axis has but one degree of 

 freedom. Its motion is fully determined by the motion of any 

 one of its points (not situated on the axis), and any such point 

 must move in a circle about the axis. Any particular position 

 of the body is, therefore, determined by a single variable, or 

 co-ordinate, such as the angle of rotation. Just as the equi- 

 librium of such a body depends on a single condition (see Part 

 IL, Art. 227), so its motion is given by a single equation. 



292. The equation of motion can be derived directly from 

 the proposition of angular momentum (Art. 224). Let r be 

 the distance of any particle m of the body from the fixed axis, 

 to the angular velocity at the time t ; then mwr is the momentum 

 of the particle, and mwr 1 its moment, or the angular momentum 

 of the particle, about the axis. At any given instant t, CD has 

 the same value for all particles. Hence, the angular momentum 

 of the body is w^mr* = o>7, where I=^mr^ is the moment of 

 inertia of the body for the fixed axis. 



Now, by Art. 224, the rate at which the angular momentum 

 of the body about the axis changes with the time is equal to 

 the sum of the moments of all the external forces about the 

 .same axis. Denoting this resulting moment by H, and con- 

 sidering that the moment of inertia for the fixed axis is inde- 

 pendent of the time, we have the equation of motion 



dv_H. (} 



dt~ T 



i.e. the angular acceleration about the fixed axis is equal to the 

 moment of all the external forces about this axis, divided by the 

 moment of inertia of the body for the same axis. 



293. The same result can of course be obtained from any 

 one of the equations (6) or (7), Art. 224. Thus, taking the 

 fixed line as axis of 2, the third of the equation (7), viz. 



^ 

 at 



