EXAMPLES. 255 



46. A uniform rod has its lower end on a smooth table and is released 

 from rest in any position. Show that the velocity of its centre on arriving 

 at the table is &amp;gt;J(gh\ where h is the height through which the centre has 

 fallen, and that at the instant when the centre reaches the table the pressure 

 on the table is one quarter of the weight of the rod. 



47. If a particle is moving in a circular tube held at rest on a smooth 

 horizontal plane, and the tube is let go, the centre of the tube will describe a 

 trochoid. 



48. A uniform sphere of mass m is rolling on the horizontal upper 

 surface of a wedge of mass M whose under surface slides without friction on 

 a fixed plane inclined at an angle a to the horizontal. Assuming the system 

 to move from rest, and the whole motion to be in a vertical plane, prove that, 

 if at time t the wedge has slipped a length x along the plane, and the sphere 

 has rolled a length s along the surface of the wedge, then 



_ 7 



- 



49. A wheel can turn freely about a horizontal axis, and a fly of mass 

 m is at rest at the lowest point ; if the fly suddenly starts off to walk along 

 the rim of the wheel with constant velocity F relative to the rim, show that 

 he cannot ever get to the highest point of the rim unless F is at least as 

 great as 



2 \/{ga (maP/MK 2 ) (1 -\-ma 2 IMK 2 )}, 



where a is the radius of the wheel, and NK 2 its moment of inertia about its 

 axis. 



50. A hollow thin cylinder of radius a and mass M is maintained at rest 

 in a horizontal position on a rough plane of inclination a, and an insect of 

 mass m is at rest in the cylinder on the line of contact with the plane. The 

 insect starts to crawl up the cylinder with velocity F, and the cylinder is 

 released at the same instant. Prove that if the relative velocity is maintained 

 and the cylinder rolls uphill, then it will come to instantaneous rest when 

 the angle the radius through the insect makes with the vertical is given by 

 the equation 



F 2 (1 - cos (6 - a)} + ag (cos a cos 6) = (I + M\m) ag (6 - a) sin a. 



51. A rigid square A BCD of four uniform rods each of length 2a lies on 

 a smooth horizontal table and can turn freely about one angular point A 

 which is fixed. An insect whose mass is equal to that of either rod starts 

 from the corner B to crawl along the rod BC with uniform velocity F relative 

 to the rod. Prove that in any time t before the insect reaches C the angle 

 turned through by the square is 



V? 



