302 Mr. A. S. Davis on the Vibrations which Heated 



rocker by means of a long thin rod, and so arranged, that every 

 vertical slice of the body of the rocker has as nearly as possible 

 the same motion. Let M be the mass of the rocker, M& 2 the 

 moment of inertia about an axis through the centre of gravity 

 parallel to the length of the rocker, 2a the distance between the 

 ridges, 2b the depth, and 2c the breadth of the rocker. Let 

 o), on' be the angular velocities just before and just after the im- 

 pact of the rocker upon the point 0. Let R, Q be the hori« 

 zontal and vertical components of the impulse, u, u 1 the hori- 

 zontal, ?;, v' the vertical component velocities of the centre G, 

 co, co' the angular velocities of the rocker before and after impact, 

 the inclination of the rocker. The equations of impulsive 

 motion are 



M(w'— «) = R, 



M(v ! -v) =Q, 



MF( ffl '-fi))=R.y-Q.^ 



Eliminating R and Q, we have 



& 2 (o/ — &}=y(u' — u)—x{ri—v). .... (5) 



If x, y be the coordinates of G, and H the height of 0' above 

 X, the ..geometrical relations are, before impact, 

 2a—x~acos -\-b sin 6?, 



i/-~H = 5cos 0— asin0 ; 



whence, by differentiation, 



— w= ( — a sin + b cos 0)co, 



v— -jr = (—5 sin — acos 0)co. 



After impact the geometrical relations are 



% = a cos — b sin 0, 



y = asin + bcos0; 

 whence 



u'=. (— a sin 0— b cos 0) o>', 



v t =( a cos 0—b sin 0)co f . 



Substituting these values in (5) and neglecting very small quan- 

 tities, we obtain 



We have now to determine the motion between two impacts. 

 If the vibrations are continuous, the angular velocity acquired 

 just before the next impact upon the other point of support 0' 

 must be equal to — co. Immediately after the impact upon O, 

 the point begins to rise. Let h be the height to which it has 



