| Fig. 95. |
§ 35. Application to a Pair of Shifting Pieces.—In fig. 95, let P1P2 be the line of connexion of a pair of pieces, each of which has a motion of translation or shifting. Through any point T in that line draw TV1, TV2, respectively parallel to the simultaneous direction of motion of the pieces; through any other point A in the line of connexion draw a plane perpendicular to that line, cutting TV1, TV2 in V1, V2; then, velocity of piece 1 : velocity of piece 2 :: TV1 : TV2. Also TA represents the equal components of the velocities of the pieces parallel to their line of connexion, and the line V1V2 represents their velocity relatively to each other.
§ 36. Application to a Pair of Turning Pieces.—Let α1, α2 be the angular velocities of a pair of turning pieces; θ1, θ2 the angles which their line of connexion makes with their respective planes of rotation; r1, r2 the common perpendiculars let fall from the line of connexion upon the respective axes of rotation of the pieces. Then the equal components, along the line of connexion, of the velocities of the points where those perpendiculars meet that line are—
α1r1 cos θ1 = α2r2 cos θ2;
consequently, the comparative motion of the pieces is given by the equation
| α2 | = | r1 cos θ1 | . |
| α1 | r2 cos θ2 |
(15)
§ 37. Application to a Shifting Piece and a Turning Piece.—Let a shifting piece be connected with a turning piece, and at a given instant let α1 be the angular velocity of the turning piece, r1 the common perpendicular of its axis of rotation and the line of connexion, θ1 the angle made by the line of connexion with the plane of rotation, θ2 the angle made by the line of connexion with the direction of motion of the shifting piece, v2 the linear velocity of that piece. Then
α1r1 cos θ1 = v2 cos θ2;
(16)