100] GENERAL MOTION OF ROLLING HOOP. 311 



The equations of the motion of the center of mass thus become 

 M ( c TJT ~l~ aq 2 ~\- PQ (Q>T CJP)) = RX ~}~ Mg sin -9 1 , 



OAON n/r I dp dr \ -n 



zOZ) Mi c~ a~j cr o<l a P < l) == ^yj 



a -^ + P ( C P ar ) + cq 2 ] = E s Mg cos &. 



We may verify that these equations are satisfied by the steady motion 

 in' which q = 0, p and r are constants. For the case of the hoop, 

 in which c = 0, they thus become 



Mar Q r = R x + Mg sin #, 



203) = R tJ , 



- Mapr = R z Mg cos #, 

 while the second of 200) becomes 



204) (Cr-Ar )p = aR z . 

 Eliminating R z we have the equation for the steady motion 



205) (Cr Ar Q )p + aM(apr gcos&) = 



which , on inserting the values of p, r, and r from 197) is the same 

 as equation 196). From the first equation 203) we may calculate R x 

 from which the tendency to slip, ^cos^ is found. If this is greater 

 than Mgii the hoop will slip. The slipping of the bicycle may be 

 similarly dealt with. 



In order to treat the general motion, let us eliminate R y from 

 the first and third of equations 200) and then from the third of 200) 

 and the second of 202), obtaining 



n/r ( dp dr , 



Ma \ c Tt - a ~di - ( cr 

 If in these equations we substitute from 197), 



we find that each term contains dt in the denominator, so that we 

 may change the variable from t to # by multiplying by ^> giving 



Aajl + Cc~ + Car + Aapcin& = 0, 

 207) 



- Ma*p + Macpctnfr = 0, 



