272 MOTION UNDER A VARIABLE FORCE 



drawn from its position of equilibrium and projected in any way 

 will always describe an ellipse in the horizontal plane in which it 

 is free to move, having the point immediately below its point of 

 suspension as center. 



An arrangement may sometimes be seen at village fairs in England, in 

 which the showman ingeniously takes advantage of this result. A weight 

 is suspended by a string, and a skittle is placed on the floor exactly under 

 the point of suspension of the weight. Passers-by are invited to pay an 

 entrance fee and compete for a prize which is awarded to any one who can 

 project the weight so that on its return it knocks the skittle over. The 

 problem is, of course, as impossible as that of describing an ellipse which 

 shall pass through its own center. 



217. Another way in which motion under a force proportional 

 to the direct distance may be realized, is as follows: An elastic 

 string of natural length I has one end fastened to a small particle 

 which is free to move on a smooth horizontal table ; the other end, 

 after passing through a small hole in the table, is fastened to a 

 fixed point at a distance I from the hole. If the particle is pulled 

 away from the hole to a point distant r from it, the total length 



T 



of the string is I + r, so that its tension is - X, where X is its 



i 



modulus of elasticity. The force acting on the particle, namely 

 the tension of the string, is therefore proportional to the distance 

 of the particle from a fixed point namely the hole in the table 

 and its direction is toward the hole. Thus the particle will 

 move in elliptic motion on the table. 



EXAMPLES 



1. The point P is describing an ellipse under an attractive force to the center, 

 and p is the corresponding point on the auxiliary circle. Show that p moves 

 round the auxiliary circle with uniform velocity. 



2. A particle describes an ellipse about a center of force, the attraction 

 being that of the direct distance. Show that the radius vector from the center 

 of the ellipse to the particle sweeps out equal areas in equal times. 



3. A particle is describing an ellipse under a force proportional to the dis- 

 tance, when it receives a blow in a direction parallel to the major axis of the 

 ellipse. Show that the minor axis of the new orbit is the same as that of the 

 old, and show how to find the change produced in the major axis. 



