294 MOTION OF RIGID BODIES 



KOUTH'S RULE 



241. The following convenient rule, given by Dr. Routh, (Rigid 

 Dynamics, 8), provides an easy way of remembering the values 

 of several radii of gyration. The rule applies to linear, plane, and 

 solid bodies which are 



(a) rectangular (rod, lamina, or parallelepiped) ; 



(b) elliptical or circular (disk or lamina) ; 



(c) ellipsoidal, spheroidal, or spherical (solid) ; 



and states that the radius of gyration about an axis of symmetry 

 through the center of gravity is given by 



,2 _ sum of squares of perpendicular semi-axes 



3, 4, or 5 



where the denominator is 3, 4, or 5 according as the body comes 

 under headings (a), (6), or (c) of the above classification. 



ILLUSTRATIVE EXAMPLE 



A coin rolls down an inclined plane. Find its velocity after any distance and 

 also its acceleration. 



Let the coin be treated as a uniform circular disk, and let a be its radius. 

 When its velocity down the plane is F, its angular velocity will be V/a. The 



axis of rotation is perpendicular to the plane of 

 the coin. Its semi-axes of symmetry, regarding it 

 as a lamina, will be a, a. The radius of gyration 

 about the axis of rotation through its center is, 

 by Routh's rule, 



so that the kinetic energy is 



F,o.U5 



After rolling a distance s down the plane, the center of gravity of the coin 

 has fallen a distance s sin a, so that from the conservation of energy 



and therefore the velocity is given by 



F 2 = | sg sin a. 



Comparing with the formula (48), V 2 - 2/s, for motion under uniform accel- 

 eration, we see that the disk rolls down the plane with a uniform acceleration 

 fflr sin a. 



