556 THE PRESSURE OF THE CYLINDER ON THE POINT ON WHICH IT ROLLS. 
The pressure of the cylinder on its point of contact with the plane on which it rolls. 
Let A' be the point where the point A 
of the cylinder was in contact with the 
plane. 
Let A'N=a?, NG=y. 
— X= horizontal pressure on M in di- 
rection A'M. 
Y=vertical pressure on M in direc- 
tion MC. 
Since the centre of gravity G moves as 
lected there, all the impressed forces were applied to it, we have, by the principle of 
d’ALEMBERT, 
/ 
W^_ 
9 
9 dt^ 
=Y-W 
But since CA—a, CG=h, MCA=^, 
x=a^—h sin 
y — a—hcos 
dx , , 
= {a — h COS 0) 
dt 
dy , . M 
dt* 
d'^x , . 
^=Asin 
\Jt) +{<‘-hco%e)-^ 
d'^y , 
df- 
=h cos^ 
sin 6 
df 
Assume 
.•. by equation (29.) 
= M f€l\^ 
\dt) 
\dt^J 
= -N, 
d^x 
-^=MA sin 0 — N(a— A cos 0) 
d^y 
cos sin 0 ; 
by equation (28.), 
X 
= M sin cos j 
W/if,, ... 1 
(28.) 
(29.) 
( 30 .) 
