EXAMPLES. 333 



equilibrium of the balls by a small distance x. Prove that the density of 



the spheres is 



where y is the constant of gravitation. 



140. A number of uniformly distributed particles move with the same 

 velocity v in the same direction ; in this medium is placed a body of any form 

 and such that all the particles impinging on it adhere. Show that, if M is the 

 mass of the body at any time, and u its velocity, then M(v-u] will remain 

 constant. 



141. An umbrella whose surface is smooth and spherical is held in rain 

 which falls vertically with velocity v and the umbrella itself is drawn 

 vertically downwards with velocity V(&amp;lt;v}. Prove that the average pressure 

 per unit area of the rain falling on the umbrella at a point whose distance 

 from the highest point is 6 is p cos 2 6 (v V) 2 /v 2 , where p is the average 

 pressure per unit area of the rain falling on a fixed horizontal plane. 



142. Three equal particles are attached at equal intervals to a thread, 

 and, when the thread is straight, the two extreme ones are projected in 

 parallel directions with the same velocity v at right angles to the thread. 

 Prove that, if there are no external forces, the angular velocity of the 

 portions of the thread when they have turned through an angle 6 is 

 l/V(l+2sin 2 0) of its initial value. 



143. Two particles on a smooth table are connected by an elastic thread 

 of natural length a and are initially at rest at a distance a apart. One 

 particle is projected at right angles to the thread. Prove that, if the 

 greatest length of the thread during the subsequent motion is 2a, the velocity 

 of projection is ^/(SaX/Sm), where X is the modulus of elasticity of the thread, 

 and m is the harmonic mean between the masses of the particles. 



144. An equilateral wedge of mass M is placed on a smooth table with 

 one of its lower edges in contact with a smooth vertical wall, and a smooth 

 ball of mass M is placed in contact with the wall and with one face of the 

 wedge so that motion ensues without rotation of the wedge. Prove that the 

 ball will descend with acceleration 



145. Two particles A, B of masses 2m and m are attached at equal 

 intervals to an inextensible thread OAB, and lie on a smooth table with 

 the thread straight and the end fixed. The particle B is projected on the 

 table at right angles to AB. Prove that in the subsequent motion the angle 

 OAB is never less than a right angle, and that when OAB is again a straight 

 line the velocity of B is half that of A. 



146. Two particles of masses m, m are placed close together on a smooth 

 horizontal plane and are connected by an elastic thread of modulus X which 

 passes round a smooth peg in the plane, and is of its natural length a. The 

 two particles are projected away from the peg with equal momenta. Prove 



