80 APPLIED MECHANICS 
then T= P(a+b)=PR. If P is in pounds and R is in inches, then T will 
be in inch-pounds. The inch-pound is generally the most convenient unit 
for the torque on a shaft, but the foot-pound, foot-ton, and inch-ton 
are also used. 
If T is the torque on a shaft in inch-pounds, N the number of revolu- 
tions per minute, and H the horse-power transmitted, then it follows that 
we _27RPN «TN 
~ 12x 33000 6 x 33000 ° 
95. Angle of Twist of a Shaft.—Let a shaft of length J, radius 7, 
or diameter d be subjected to pure torsion by torques each equal to T 
applied at its ends, as shown in Fig. 100. A straight line AM drawn on 
the surface of the shaft and parallel to the axis when the shaft is 
unstrained, will become a helix when the shaft is twisted. This follows 
from the following consideration. If the shaft be divided into a number 
of parts each of unit length by planes perpendicular to the axis each part 
will be subjected to the same torque, and the angular movement of one 
end of each part of unit length relative to the other end will be the same, 
and therefore the angular movement of one end of the shaft relative to 
the other end will be the sum of the angular movements due to each 
part, and therefore the movement of A relative to M, namely, the are 
AB, will be proportional to 7. 
Tf a small square MN be drawn on the surface of the shaft when the 
latter is unstrained and having a side on AM, this square, shown 
enlarged to the left of Fig. 100, will become a parallelogram. If this 
p 
pe p< Tt 
i fe gat ate rene AIS STi LS: eet 
=o w= U A 
as eee ie Get ieee Pk 3 
‘a . Ce ery 
meat Sie ‘ 
ee 
P 
Fia. 100. 
square be the outer face of a thin layer of material on the shaft, then the 
edges or narrow faces of this layer are subjected to shear stress of, say, an 
intensity f, and the shear strain is «’//’=a/l where « is the length of the 
. shear stress 
are AB. But shear strain = ——, therefore x/l=7/C or 
modulus of rigidity 
x=fl/C. If @ is the angle of twist in circular measure, then 0 = x/r = 22/d, 
but «= f7/C, therefore 0 = 4 di 
If » is the angle RY Bcict in degrees, dies since @/r=n/180, 
_ 36071 
~ ad * 
