550 
THE ROLLING MOTION OF A CYLINDER. 
gravity G and perpendicular to its axis C ; and let M be its point of contact, at any 
time, with the horizontal plane BD on which it is rolling. Assume 
a =AC, h=CG, 6=ACM. 
W=weight of cylinder. W^^=momentum of inertia of the cylinder about an axis 
passing through G and parallel to the axis of the cylinder. 
iy=given value of the angular velocity when ^ has the given value 
^j=given value of 6 when the angular velocity has the given value u. 
/=given value of GM corresponding to the value of 6. 
Then W(A:^+GM^)=W(A:^+a^+A^— cos <?)=moment of inertia about M. Since 
moreover the cylinder may be considered to be in the act of revolving about the point 
M by which it is in contact with the plane, one-half of its vis viva is represented by 
the formula . 
1 W 
2 cos , 
and one-half of the vis viva acquired by it in rolling through the angle by 
1 wr 1 
But the vertical descent of the centre of gravity while the cylinder is passing from 
the one position into the other, is represented by 
A(cOS cos ^i). 
Therefore, by the principle of vis viva*^, 
^ 2flAcos — (F-f /^)(y^|=WA(cos cos 
whence we obtain 
2gh{co?> fl — cos flj) + + 
\dt) -t- — 2«/} cos fl + A® 
) ( 1 .) 
V®/ X/k'^ a h\ 
1 / a h\ 
Let “ = 2(^ + A + «) (2-) 
,/3=cos^,-(-^j^^ (3.) 
at \aj cos0y \g JJ e, \cos fl — p/ ’ 
/a\ « /*®> / « — cos 9 \ , 
••• ( 4 -) 
where t represents the time of the body’s passing from the inclination to zero. 
* Poisson, Dynamique, 2“^ partie, 565 ; Poncelet, Mecanique Industrielle ; Moseley, Mechanical Principles 
of Engineering, Art. 129. 
