INITIAL MOTION OF FREE BODY. 



205 



F 



(3) In Ex. (2), if the lamina be an ellipse of semi-axes a, Ja, at 

 what point of the major axis must it be struck at right angles to this 

 axis in order to rotate initially about a focus ? 



(4) A rigid body of mass M receives an impulse F parallel to the 

 principal axis Gy, and meeting the principal axis Gx at the distance 



GO' = Xx from the centroid. Determine the initial motion (Fig. 52). 



The impulse F at O' is equivalent to an equal and parallel impulse 

 F through the centroid G, in combination with the couple H '=. Fx^ 

 whose vector is parallel to the principal 

 axis Gz. This couple produces, there- 

 fore, rotation about the centroidal prin- 

 cipal axis Gz, or 7, of angular velocity 

 <o = Fxi/Mq%, where q z is the radius of 

 inertia for /. The instantaneous axis / 

 is parallel to 7, and meets the axis Gx at 

 a point O such that GO- GO 1 = ? 3 2 , or 

 putting GO = x, GO' = x 1} such that 

 jcxi = q j it can be constructed as in 

 .Ex. (2). v Fig. 52. 



(5) Determine the impulse "F imparted to a body of mass M when 

 struck at the point O' (Fig. 53) of a principal axis Gx by a particle of 

 mass m moving with a velocity u parallel to another principal axis Gy. 

 The impact is assumed to be inelastic (compare Art. 305). 



It has been shown in Art. 19 that if a particle of mass m moving with 

 a velocity u impinges upon a particle of mass M at rest, the two parti- 

 cles will, after inelastic impact, move 

 on together with the common veloc- 

 ity v =mu/(m -\- M). Similarly in 

 our case, as soon as the impact 

 has taken place, the two masses m 

 and M may be regarded as forming 

 a whole, and as moving together. 

 The impulse mu acting on this mass 

 M+ m at O' imparts to this point 

 a certain velocity v which can be 

 determined. As the particle m as- 



Fig. 53. 



sumes this velocity v after impact, it loses the momentum mu mv, 

 owing to the impact ; and this is the impulse 



= m(u v) 

 of the blow transmitted to the body. 



