282 MOTION UNDER A VARIABLE FORCE 



5. Taking the mass of Jupiter to be T J ?7 of the mass of the sun, and its 

 greatest distance from the sun to be 498| million miles, show that, on account of 

 Jupiter's attraction, the sun will describe an ellipse of semi major-axis equal to 

 about 461,000 miles, and find the length of Jupiter's year. 



6. The maximum velocity attained by the earth in its orbit is 3,000,000 

 centimeters per second, and the minimum velocity is 2,920,000 centimeters per 

 second. Find the eccentricity of the earth's orbit. 



GENERAL EXAMPLES 



1. A particle, attached by a string to a point, has just sufficient energy 

 to make complete revolutions in a vertical circle. Show that the tension 

 of the string is zero and six times the weight of the particle respectively, 

 when the particle is at the highest and lowest points of its path. 



2. A particle moves under gravity in a vertical circle, sliding down the 

 convex side of a smooth circular arc. If its velocity is that due to a height 

 h above the center, show that it will fly off the circle when at a height f h 

 above the center. 



3. If the angle a through which a simple pendulum swings on each 

 side of the vertical is small, but not infinitesimal, show that to a first 

 approximation the time of oscillation is 



Deduce that a pendulum, which beats seconds accurately when performing 

 infinitesimal oscillations, would lose about 40 seconds a day when attached 

 to a clock which caused it to oscillate to 5 degrees on each side of the 

 vertical. 



4. A train is moving uniformly round a curve at 60 miles an hour, and 

 in one of the carriages a seconds pendulum is found to beat 121 times 

 in two minutes. Show that the radius of the curve is about a quarter 

 of a mile. 



5. One end of an elastic string, natural length a, modulus X, is tied to 

 a fixed point on a smooth horizontal table, and the other end is tied to a 

 particle of mass m which rests on the table. If the mass is pulled to a 

 distance 2 a from the other end of the string, and is then let go, show that 



it will return to its original position at regular intervals 2(?r + 2)-i/ 



\ A 



6. Two balls weighing W 1 and W 2 pounds are connected by a thread a 

 feet long ; and W l is held in the hand while W 2 is whirled round. Deter- 

 mine the motion which ensues if W l is released from rest when W 2 



