ATWOOD'S MACHINE 195 



EXAMPLES 



1. A body is projected with a velocity of 20 feet per second up an inclined 

 plane of angle 45. Find how high up the plane it will go, and how long it will 

 take in going up. 



2. Two particles slide down the two faces of a double inclined plane, the 

 angles being a and /3. Compare the times they take to reach the bottom, and 

 the velocities they acquire. 



3. A body is projected down an inclined plane of length I and height &, from 

 the summit, at the same instant as another is let fall vertically from the same 

 point. Prove that if they strike the base at the same time, the velocity of 

 projection of the first must be 



[7 

 \2h' 



4. Give a construction for finding the line of quickest descent from a fixed 

 point to a circle in the same vertical plane. 



5. Particles are sliding down a number of wires which meet in a point, all 

 having started from rest simultaneously at this point. Prove that at any instant 

 their velocities are in the same ratio as the distances they have described. 



^/6. A railway carriage is observed to run with a uniform velocity of 10 miles 

 an hour down an incline of 1 in 250, and on reaching the foot of the incline 

 runs on the level. Find how many yards it will run before coming to rest, assum- 

 ing the resistance to be constant and the same in each stage of the motion. 



7. Prove that if a motor car going at 100 kilometers an hour can be stopped 

 in 200 meters, the brakes can hold the car on an incline of about 1 in 6 ; and 

 determine the time required to stop the car. 



8. A carriage weighing 12 tons becomes uncoupled from a train which is 

 running down an incline of 1 in 250 at a rate of 40 miles per hour. The f ric- 

 tional resistance is 14 pounds weight per ton. Find how far the carriage will 

 go before coming to rest. 



9. The pull of the locomotive exceeds the ordinary resistances to the motion 

 of a train by -fa of its whole weight ; and when the brakes are full on there is a 

 total resistance of T ^ of its whole weight. Find the least time in which the 

 train could travel between two stopping stations on the level 3 miles apart. 



10. In the last question, find the time if the track is down a gradient of 

 1 in 100. 



ATWOOD'S MACHINE 



161. It is difficult to measure the acceleration produced by 

 gravity from direct observations on a body falling freely, because 

 either the distance fallen must be very great or else the time 

 of falling very small. These difficulties are to some extent obviated 

 in a machine designed by Atwood. 



