243, 244] RIGID BODIES UNDER IMPULSES. 277 



We thus obtain 



P, } 



') = P(x + a cos a). J 



Again, we can form three equations containing R and R by resolving for 

 CD at right angles to BC, and taking moments for EC and AD about B and 

 A respectively. We thus obtain 



m(vcosa a,Q>') = 

 m [(v - aa>) a cos a 

 m [(v + aa>) a cos a 



from which, on elimination of R and R, we get 



a a,Q>') = R + R, \ 

 a 2 a>'] = - 2aR, > 

 - a 2 a>'] = - 2a/',) 



Hence v = %P/m, <a = P^/ma 2 , a>'=f Pcosa/wa. 



*244. Examples. 



1. Two equal rods AB, A C freely jointed at A are at rest with the angle 

 BAG a right angle, and AC is struck at C by an impulse in a direction 

 parallel to AB. Prove that the velocities of the centres of inertia of AB 

 and AC in the direction of AB are in the ratio 2 : 7. 



2. Two equal uniform rods freely hinged at a common end are laid out 

 straight, and one end of one of them is struck by an impulse at right angles 

 to their length. Prove that the kinetic energy generated is greater than it 

 would be if the rods were firmly fastened together so as to form a single rigid 

 body in the ratio 7 : 4. 



3. Four equal uniform rods are freely hinged together so as to form a 

 rhombus of side 2a with one diagonal vertical, and the system falling in a 

 vertical plane with velocity V strikes against a fixed horizontal plane. 

 Taking a to be the angle which each rod makes with the vertical and assum- 

 ing no restitution, prove that (i) the impulsive action between the two upper 

 rods is directed horizontally, (ii) the angular velocity of each rod after the 

 impulse is f ( V/a) sin a/(l +3 sin 2 a), (iii) the impulsive action between the two 

 upper rods is to the momentum of the system before impact in the ratio 



sin a (3 cos 2 a ~ 1) : 8 cos a (1 + 3 sin 2 a), 



(iv) the impulsive action at either of the hinges in the horizontal diagonal 

 makes with the horizontal an angle tan" 1 {(3 cos 2 a ~ 1) cot a}. 



4. In Example 3, prove that, if the coefficient of restitution between the 

 rhombus and the ground is e, the angular velocity of each rod after the 

 impulse is f (1 +e) ( V/a) sin a/(l +3 sin 2 a). 



5. A square framework ABCD is formed of uniform rods freely jointed 

 at B, C, and D, the ends at A being in contact but free. Prove that, if AB 

 is struck by a blow at A in direction DA, the initial velocity of A is 79 times 

 that of Z). 



