248 MOTION OF A KIGID BODY IN TWO DIMENSIONS. [CHAP. XI. 



6. Two circular rings, each of radius a, are firmly joined together so 

 that their planes contain an angle 2a and are placed on a rough horizontal 

 plane. Prove that the length of the simple equivalent pendulum is 



\a cos a cosec 2 a (1+3 cos 2 a). 



7. A thin uniform rod one end of which can turn about a smooth hinge 

 is allowed to fall from a horizontal position. Prove that, when the horizontal 

 component of the pressure on the hinge is a maximum, the vertical com- 

 ponent is ^ of the weight of the rod. 



8. A uniform sphere of mass M and radius a oscillates under gravity 

 about a fixed horizontal tangent as axis. Given the angular velocity CD of the 

 sphere in the lowest position, find the pressure on the axis in any position, 

 and prove that in a position of rest the resultant pressure will be perpen- 

 dicular to the radius drawn to the axis if co 2 = ^g/a. 



9. A uniform rectangular block of mass M stands on a railway truck 

 with two faces perpendicular to the direction of motion, the lower edge of 

 the front face being hinged to the floor of the truck. If the truck is 

 suddenly stopped, find its previous velocity so that the block may just turn 

 over. Prove that in this case the horizontal and vertical pressures on the 

 hinge vanish when the angle the plane through the hinge and the centre of 

 inertia of the block makes with the horizontal has the values sin -1 and 

 sin -1 ^ respectively, and that the total pressure is a minimum, and equal 

 to ^MgJ^i, when the angle is sin" 1 ^. 



10. The door of a railway carriage which has its hinges (supposed 

 smooth) towards the engine stands open at right angles to the length of the 

 train when the train starts with an acceleration f. Prove that the door closes 



in time ^/(^g^ / Jfinffj' with an an S ular velocitv v/{2a//( 2 + 2 )}, 

 where 2a is the breadth of the door, and k the radius of gyration about a 

 vertical axis through the centre of inertia. 



11. Two bodies of masses P and Q are attached by ropes to a wheel and 

 axle of mass M and radius of gyration k about its axis, the axle being 

 supported in two rough sockets symmetrically placed which it just fits and 

 touches along a horizontal line. Prove that when Q has ascended a height 

 x its velocity V is given by the equation 



where P' is the mass of a body which substituted for P would keep equilibrium. 



12. A particle is placed on a rough plane lamina which is initially hori- 

 zontal, and which is free to turn about a horizontal axis through its centre of 

 inertia. Show that the particle will begin to slip when the plane has turned 

 through an angle 



tan ~ i {jjLMa*/(Ma* + 9mc 2 )}, 



p, being the coefficient of friction, 2a the length of the plane perpendicular to 

 the axis, c the distance of the particle from that axis, and M, m the masses 

 of the lamina and the particle. 



