418 On the rolling Pendulum. 
molecules of the pendulum. The same formula will be more 
conveniently written thus, j 
x § ddx 
Sdm.) aes 
Now, if 7 be the distance of dm from the axis of the cylin- 
der, we have 
bx + ce by t —Sdm.gix =0. 
x=r7 cos >: 
also, by the rolling of the cylinder, it is manifest that c¢ is the 
distance of its axis from the fixt vertical plane; wherefore, 
¥y = 7 sing — co: consequently, 
dSa=8o xX —rsing 
Sy = 36 x (r cos d — Cc) 
die die dg? 
Tae ape Mas TS 
ddy _ dd@ dg? 
dt? dt? dt? 
and the preceding equation will therefore become 
pe eee (fr2dm — 2c cos >. frdm + cm) 
r cos } 
r sino: 
(r cos¢ —c) — 
dt? 
dg? pn 
+c rey, 4 dm 
+ gsing. frdm. 
Now, by the nature of the centre of gravity, 
Srdm=ma: 
and if we put mk* for the momentum of inertia of an axis pass- 
ing through the centre of gravity parallel to the cylinder, then 
Sr>dm = m (k? + a’) 
Hence, if we leave out m and put 
ht = k* + (a—c)*; 
the foregoing equation will become 
ddQ § 
O=sR Ut 2.ac(1— cos ¢)} ope 
1o2 S 
77a SIN G+ Gasin gy. 
In the case of very sma!] vibrations, we may reject the terms 
above the first order; and then we get, 
dado Pian rs * 
oa ig sind = 0), 
Now this equation, which is true of a body of any figure, belongs 
4 he 5 
to a simple pendulum of the length —s and hence if + denote 
the time of a complete oscillation of the rolling pendulum in a_ 
very small are, we shall have 
pi) We he 9 
Petal iy a ch ali? 
§ pycale 
a= 7% : 
g m being 
