84 APPLIED MECHANICS 
semicircle and perpendicular to its plane, the wire will be under a torque 
equal to PR. The length of the wire is wR. Hence the total angular 
4 2 ‘ 
movement of the two arms is 0= fo: , and the relative movement of 
. 32PR3 
A and C is 8=R0= Gat 
The half ring shown in Fig. 103 corresponds very approximately with 
a half coil of a helical spring (Fig. 104), in which the slope of the coils 
is small. 
Extending the above formule for the half ring (Fig. 103) to the 
helical spring (Fig. 104), if m is the number of complete coils, then since 
; 3 : aie 64PR en 
for one half coil 8=— the extension for » coils is b= ae f 
ars _4eRYn ng pitas _ Cas 
Also, PR= ie? >f, Therefore also, 5 Ga” and P 16k 64h" 
The length of wire, /= 27Rn, nearly. 
The above formule also apply when the load is reversed, so that the 
spring as a whole is in compression. 
Helical springs are also called cylindrical spiral springs 
Exercises VIc. 
1. Find the twisting moment in inch-pounds on a shaft which transmits 
50 horse-power at a speed of 120 revolutions per minute, and calculate its © 
diameter, taking the maximum stress at 9000 lbs. per square inch. 
2. The turbine shaft of a 5 horse-power De Laval steam turbine is 0:2(3 
inch in diameter, and its speed is 30,000 revolutions per minute. What is the 
maximum stress on this shaft when transmitting 5 horse-power ? 
3. A shaft transmits 100 horse-power at 120 revolutions per minute. The 
maximum torque is 1°4 times the mean torque, and the maximum stress is 
9500 lbs. per square inch. Find the diameter of the shaft. 
4. The diameter of ashaft is 94 inches. Determine the horse-power trans- 
mitted when the maximum stress is 9000 lbs, per square inch, and the speed is 
130 revolutions per minute, The couplings of this shaft have each six bolts 
2? inches diameter, whose centres lie on a circle 144 inches diameter. Find the 
average shear stress on the bolts. 
5. A cast-iron flanged shaft coupling has six bolts 1} inches diameter. The 
axis of each bolt is 73 inches from the axis of the shaft. Diameter of shaft, 
53 inches, When the maximum shearing stress on the shaft, due to the twisting 
moment, is 10,000 lbs, per square inch, what is the average shearing stress on 
the bolts ? 
6. If a shaft 2 inches diameter safely supports a torque of 15,000 inch- 
pounds, what torque would a shaft of the same material 54 inches diameter 
support with the same factor of safety? What horse-power would the former 
shaft transmit at 150 revolutions per minute, and what should be the speed 
of the latter to transmit 600 horse-power. 
7. Determine the horse-power transmitted by a hollow steel shaft whose 
external diameter is 18 inches, and internal diameter 12 inches. The speed 
is ao revolutions per minute, and the maximum stress 10,000 lbs. per square 
inch. 
8. A hollow steel shaft, external diameter d, internal diameter 3d, is subjected 
to pure twisting, and transmits 8000 horse-power at a speed of 110 revolutions 
Lad ~% Taking the maximum shear stress at 9000 lbs. per square inch, 
nd d. 
9. A hollow steel shaft is made to replace a solid wrought-iron one of the 
same diameter, the material being 35 per cent. stronger than the iron; find 
what fraction of the outside diameter the internal diameter may be, and, 
