aa MOTION AND FORCE 21 
the force in pounds, the torque is measured in inch-pounds. Other units 
of torque are—the inch-ton, the foot-pound, and the foot-ton. 
36. Rotational Inertia—Moment of Inertia.—Consider a small 
body A (Fig. 18), whose mass is m, revolving about an axis O under 
the action of a force P, whose line of action is tan- 
gential to the path of A. If / is the linear accele- 
ration produced in A by P, then (Art. 28) P=m/. 
If a is the angular acceleration, then f= 7a ; therefore 
_P=mra, and Pr=mr*a, or T=Ia, where T is the 
torque causing rotation, and I is called the moment 
of inertia of the body A about the axis O. Fia. 18. 
If a large body, revolving about an axis, be ; 
divided into small parts, whose masses are 1, m,, mg, etc., and whose 
distances from the axis are 1,, 7, 7, ete., respectively, then T=(m,r} 
+ Mg + Mgr; +etc.)a=Ia, where T is the torque causing the rotation of 
the body, and I=m,r} +m,r; + mgr; +etc. is the moment of inertia of 
the body. 
If the whole mass M of the body be placed at a distance k from the 
axis without altering its moment of inertia, then I = MA*, and k is called 
the radius of gyration of the body. 
37. Moment of Momentum—Angular Momentum.—Referring to 
the small body A of the preceding Article and Fig. 18, if v is its linear 
velocity and w its angular velocity, then its linear momentum is 
mv=mro. The moment of this momentum about the axis O is 
mro=Iw. This moment of momentum of the body about the axis O 
is also called the angular momentum of the body about that axis. 
For a large body made up of small parts, whose masses are m), Mo, 
Ms, etc., and whose distances from the axis about which the body 1s 
rotating are 7, 7 713, etc., respectively, the total linear momentum is 
evidently (m,r, + mr. + mg’, + etc.)o, and the sum of the moments of 
momenta, or the total angular momentum, is 
(myr} + mary + mgr; + ete.)w =e, 
where I is the moment of inertia of the whole body. 
Since T=Ia, it follows that if the torque T acts on the body for 
_ t seconds, Tt = Iat =I, where w is the increase in the angular velocity in 
the time #, and Iw is the increase in the angular momentum in that time. 
Hence, equal torques acting during equal times will produce equal 
amounts of angular momentum. 
Exercises II. 
Take 1 metre=3°281 feet. 
1. Express the following velocities in feet per second: 45 miles per hour, 
225 feet per minute, 11} knots, and 150 metres per minute. 
2. Express the following velocities in feet per minute: 3°5 feet per second, 
18 miles per hour, 18 knots, and 15 metres per second. 
3. Express the following velocities in miles per hour: 33 feet per second, 
3080 feet per minute, 16} knots, and 48 kilometres per hour. 
4. Express the following velocities in radians per second: 5 revolutions per 
second, 270 revolutions per minute. 
5. Convert a velocity of 63 radians per second into revolutions per minute. 
6. What is the angular velocity, in radians per second, of a train when 
running round a curve of 18 chains radius at the rate of 36 miles per hour? 
1 chain =22 yards. 
= 
