162 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



EXAMPLES. 



1. A particle is suspended from a point by an elastic thread and oscillates 

 in the vertical line through the point of suspension. Prove that the period 

 is the same as that of a simple pendulum of length equal to the excess 

 of the length of the thread in the position of equilibrium above its natural 

 length. 



2. A particle is attached to one end of an elastic thread of natural 

 length I, the other end of which is fixed to a point on a smooth horizontal 

 table. When the particle is at rest on the table, with the thread straight but 

 unstretched, it receives a blow, which, if directed along the thread would make 

 the particle move to a maximum distance 21 from the fixed end. Prove that, 

 if the direction of the blow makes an angle a with the thread, the maximum 

 length of the thread during the motion is the greatest root of the equation 



3. A particle is attached to a fixed point by means of an elastic thread 

 of natural length 3a, whose coefficient of elasticity is six times the weight 

 of the particle. When the thread is at its natural length, and the particle 

 is vertically above the point of attachment, the particle is projected hori- 

 zontally with a velocity S^/^a^); prove that the angular velocity of the 

 thread will be constant, and that the particle will describe the curve 



r = a (4 cos 6}. 



4. A heavy particle is fastened to the free ends of a number of elastic 

 threads which pass through fixed smooth rings, each ring being at a dis- 

 tance from the fixed end of the thread which passes through it equal to the 

 natural length of the thread. Prove that if the particle is projected in any 

 direction it describes an ellipse about its position of equilibrium as centre. 



5. Prove that a body ejected from the Earth with velocity exceeding 

 seven miles per second will not in general return to the Earth, and may leave 

 the solar system. 



6. Prove that the least velocity with which a body could be projected 

 from the North Pole so as to meet the Earth's surface at the Equator is 

 nearly 4| miles per second, and that the angle of elevation is 22^. 



7. A particle is projected from the Earth's surface so as to describe a 

 portion of an ellipse whose axis major is f of the Earth's radius. Prove that, 

 if the direction of projection makes an angle 30 with the vertical, the time of 

 flight is 



where a is the Earth's radius and g is the value of gravity at its surface. 



