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&amp;gt;, o&amp;gt; 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 



icos-l{-f(a)-o) ) 2 /(co 2 + co 2)}. 



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 15V&amp;gt;l6J(ag), 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 2&amp;lt;x is a side of either cube, and X Q 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 + v/3) a I V after turning through an angle 



VI + tan- V*}. 



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 D 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 D will come to relative rest after 

 moving over distances (3-^/5) a on B @ and BC, and that the sides CD 

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



18 

 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 



