100. Rotation in General. -It has been shown in Art. 36 that any system of forces acting upon a rigid body may be reduced to a single force and a single couple whose plane is perpendicular to the line of action of the single force. That is, the most complicated cases of rotation consist of an instantaneous translation combined with an instantaneous rotation at right angles to the translation.

Bodies projected into the air while rotating have been mentioned in Art. 92. The projectile rotating about an axis is projected in the direction of the axis. If no forces acted upon it after leaving the gun, it could move in a straight line. It is, however, acted upon by gravity, which causes it to take a somewhat parabolic path. The resistance of the air causes the projectile to drift.

This action of the projectile will probably be most easily explained by a consideration of the motion of a baseball. The modern pitcher when he throws the ball gives it also a motion of rotation. The force of gravity causes the ball to take a path somewhat parabolic and the resistance of the air, due to the rotation, causes the ball to de

flect from the plane in which it started. The combination of the two deflecting forces makes the path of the ball a twisted curve. Different speeds and directions of rotation and different speeds of translation give great variety to the curves produced. The action of the baseball will be best understood by referring to Fig. 121. Let the baseball have an initial angular velocity and an initial linear velocity v in the directions shown. The rotation of the ball causes the air to be more dense at A than at B, so that the ball is pushed constantly from A to B. This action causes it to deviate from the plane in which it initially moved and to take the path indicated by c. As stated above, this action in the case of a projectile

is known as drift.

CHAPTER XII

DYNAMICS OF MACHINERY

101. Statement of D'Alembert's Principle. A body may be considered as made up of a collection of individual particles held together by forces acting between them. The motion of a body concerns the motion of its individual particles. We have seen that in dealing with such problems as the motion of a pendulum it was necessary to consider the body as concentrated at its center of gravity; that is, to consider it as a material point. The principle due to D'Alembert makes the consideration of the motion of bodies an easy matter. Consider a body in motion due to the application of certain external forces or impressed forces. Instead of thinking of the motion as being produced by such impressed forces, imagine the body divided into its individual particles and imagine each of the particles acted upon by such a force as would give it the same motion it has due to the impressed forces. These forces acting upon the individual particles are called the effective forces. D'Alembert's Principle, then, states that the impressed forces will be in equilibrium with the reversed effective forces.

It must be seen by the student that the principle does not deal with the forces acting between the particles of a body; these are considered as being in equilibrium among themselves. We shall see in what follows that this

principle, by assuming a system of effective forces acting upon the particle, enables us in many cases to apply the principles of equilibrium as developed and used in the subject of statics.

102. Simple Translation of a Rigid Body.—The principle of D'Alembert will be best understood by applying it to

the consideration of the simpler motions of a rigid body. Let

P

with x.

FIG. 121 a

us consider the body, Fig. 121 a, and let us assume that it has simple translation parallel to x due to the action of certain impressed forces Pr

-X P., P2, etc.,

making

angles α1, a2, ag, etc., It is seen at once that only the components of P1, P2, P3, etc., parallel to x have any part in producing motion in that direction. We may say, then, that the impressed forces are P1 cos a, P2 cos α2, P3 cos az, etc., and that these produce an acceleration a in the direction

indicated.

1

Imagine the body now divided into small particles each of mass dM, and assume that the system of forces producing the motion of the body consists of a small force dM.a acting on each particle. D'Alembert's Principle then states that these forces reversed are in equilibrium with We have, then, Σx=0,

the impressed forces.

or Picos α1 + P2cos α + Pscos αg + etc. - EdM. a=0,

or

Picos α1+ Pacos az + Pacos az + etc. = ΣdM. a.

But since motion is parallel to x, it is evident that

P1cos α1+P2cos a2+P3cos az + etc.

Therefore, for continuous bodies,

R = a√аM=aM,

Resultant Force-R.

since a is the same for every particle of the body. Consider each particle at a distance y from x and let d be the distance of R from x; then taking moments with respect to an axis through x and perpendicular to it, we have

Rd = a SydM = aÿM,

where y is the distance of the center of gravity of the body from a (Art. 22). Dividing through by R, we find,

d = y;

that is, the resultant force passes through the center of gravity of the body.

103. Simple Rotation of a Rigid Body. We shall now apply D'Alembert's Principle to the case of a rigid body rotating about a fixed axis. Let B in Fig. 122 be the body, and imagine it rotating in the direction indicated about an axis through O perpendicular to the paper. Suppose the rotation due to the action of forces P1, P2, P3, P etc., making angles a1, B1, 71; a2, B2, 72, etc., with a set of axes x, y, z, with origin at O. It is evident that only the components of the forces P1, P2, P3, P4, etc., parallel to