EXAMPLES. 343 



200. A square formed of four similar uniform rods freely jointed at their 

 extremities, is laid on a smooth horizontal table, one of its corners being 

 fixed; show that, if angular velocities o>, o>' in the plane of the table are 

 communicated to the rods that meet at this corner, the greatest value of the 

 angle between them is 



201. A homogeneous hemisphere of radius a and mass M falls from rest 

 with its base vertical on to a smooth horizontal plane (no restitution). Prove 

 that its pressure on the plane when its base is horizontal is equal to 



where V is the velocity with which it strikes the plane. 



Prove that the hemisphere will leave the plane immediately upon its base 

 becoming vertical if 157>16V(o00, and " that, if 675F 2 /(10247ra#) is an 

 integer, the hemisphere will again strike the plane with its base vertical. 



202. Two equal homogeneous cubes are moving on a smooth table with 

 equal and opposite velocities F in parallel lines, and impinge so that finite 

 portions of opposing faces come into contact ; show that so long as they 

 remain in contact the line joining their centres meets the opposing faces at 

 a distance x from the centres of the faces where 



where 2a is a side of either cube, and # is the initial value of x. 



Prove further that, if the line joining the centres at the instant of impact 

 cuts the opposing faces at an angle \TT, then while the faces are in contact 

 they slip with uniform relative velocity, and separate after an interval 

 (1 +\/3) a/ V after turning through an angle 



203. Two equal rigid inelastic uniform hooks A BCD, A'B'C'D' each in 

 the form of three sides of a square of side 2a moving with equal velocities V 

 in opposite directions parallel to AB or A'B' impinge, so that the points D 

 and jy strike the middle points of B'C' and BC. Show that they separate 

 immediately after impact with the velocities of their centres of inertia reduced 

 in the ratio 9 : 53. 



Prove that if the ends D and D' are provided with apparatus for clipping 

 the sides B'C' and BC so that they can slide on these sides without friction, 

 then in the subsequent motion D and Lf will come to relative rest after 

 moving over distances (3 ^5) a on .B'C'' and BC, and that the sides CD 

 and C'D' will impinge upon A'B' and AB after an interval 



from the instant when D and D' were at rest relative to the hooks. 



204. Two smooth rigid uniform hemispheres, each of radius a and of 

 equal masses, moving at right angles to their bases with the same velocity F, 

 impinge so that lengths fa of the diameters of their bases in the plane 



