a 
3 
WORK AND ENERGY 31 
etc., whose distances from the axis of rotation are 7,, 7, r;, etc., respec. 
tively, then the kinetic energy of the whole body is 
Ww? _ Tw* 
29 29 
where & is the i 2 of gyration, and I the moment of inertia of the body 
about the axis of rotation. If these expressions give the kinetic energy 
in ft.-Ibs., then W must be in lbs., & in feet, and I in Ib. and foot units, 
49. Total Kinetic Energy of a Body. —If a body of weight W 
rotates about an axis through its centre of gravity with an angular 
velocity , and if the radius of gyration of the body about that axis 
is &, then its kinetic energy due to its rotary motion is — . If the 
centre of gravity of this body has a linear velocity v, then its kinetic 
2 
energy due to its motion of translation is S . If the body has both 
kinds of motion simultaneously, then its total kinetic energy is 
Work? | Wr? 
po, 
2g og 
50. Mechanical Equivalent of Heat.—Heat and work are mutually 
convertible the one into the other. In a heat engine the heat produced 
by the combustion of the fuel used is converted into the work done by the 
engine. When the brakes are applied to the wheels of a moving train, in 
5 g (iri + WN" + war's + etc.) = , 
order to bring it to rest, the kinetic energy of the train is converted into 
heat at the rubbing surfaces of the brake blocks and wheels, or if the wheels 
skid the heat is produced at the rubbing surfaces of the wheels and rails. 
Careful experiments have shown that 778 ft.-lbs. of work are equiva- 
lent to one British thermal unit (B.Th.U.) of heat, or the heat required to 
raise the temperature of 1 Ib. of water 1° Fahrenheit. The number 778 
is called the mechanical equivalent of heat. In terms of the lb.-degree 
centigrade unit of heat the mechanical equivalent of heat is 1400 ft.-lbs. 
51. Analogies of Linear and Angular Motions.—The student 
would do well to study the analogies of linear and angular motions 
exhibited in the following table :— 
“ 7 
QUANTITY. LINEAR. ANGULAR. 
Time . : t t 
Distance or Aisplacement : 8 0 
Velocity . v w 
Acceleration DB a 
Inertia Mass, M =1/g Moment of inertia, I 
Effort . ' Force, P= Mf Torque, T=Ia 
Momentum. Mv Iw 
Impulse , . mb Pt= Mv Tt=Iw 
Work done=U . as Ps ; Té 
Space average of effort. 5 U+s U+0 
Time average of effort ; 4 Mv+-t Iw+t 
Kinetic energy . : 4Mv? }Iw* 
Kinematical equations for uni- v=ft w=at 
form acceleration from rest s=4 ft? 0=}at* 
in time ¢ ws 2fs w* = 2a0 
x} 
in thine is = 
