208 HANDBOOK OF MECHANICAL DESIGN 



SIGNIFICANCE OF WR^ 



In Drives for Machinery 



Any moving body has stored in it kinetic energy, the magnitude of which is pro- 

 portional to the mass of the body and to the square of its velocity. Whenever the 

 speed of a body is changed, the amount of kinetic energy is increased, and the increase 

 in energy must be supplied from a source within the system. If the speed is decreased, 

 the kinetic energy of the body is decreased, and the energy lost must be absorbed by 

 some other part of the system. 



In a body of mass M moving with a hnear velocity V ft. per sec, the kinetic 

 energy E in foot-pounds is 



E = ImV' = 1(^j)v' (35) 



where W is the weight of the body, in lb., and g is the acceleration of gravity, in ft. 

 per sec. per sec. 



In a body rotating at N r.p.m., the kinetic energy of the mass as actually dis- 

 tributed is the same as an equivalent mass concentrated at a point distant from the 

 axis of rotation equal to the radius of gyration R of the body, the equivalent mass hav- 

 ing the same speed of rotation N. The kinetic energy E in foot-pounds then becomes 



Note that the term WR^ is a physical term applying to a specific body; the term 

 involves the weight W of the body and a radius of gyration R which is determined by 

 the shape and dimensions of the body. The kinetic energy stored in a rotating body, 

 therefore, is proportional to its WR- and to the square of A'', its rotational speed. 



Since Eq. (36) represents the kinetic energy stored in the body after speed A'' is 

 attained, this equation also represents the energy that must be suppUed from some 

 source, to accelerate the body from rest to A^ r.p.m. In mechanical-drive problems, 

 however, energy as such is of little interest; the major concern deals with the torque 

 required to produce the acceleration. It can be easily demonstrated that the torque 

 T in pound-feet required to accelerate a body from rest to a speed of N r.p.m. in / 

 sec. is 



- - W <-) 



From Eq. (37), it is obvious that the term WR^ is also an important factor in 

 determining the torque required to produce a given acceleration. 

 By making use of the familiar equation 



„ torque X N /„cs 



^P = 5,250 ^^^^ 



and Eq. (37), it is simple to determine the horsepower H required to accelerate uni- 



