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Mjk 2 denoting the instantaneous moment of inertia. From 
(1) and (3) we have the following equation : — 
+ ) - M ^ sina w 
Also we have the following geometrical equation, since the 
space x is described by a circle of variable radius : — dx - ydd . 
Also the mass in contact with the plane will be bdcx where c 
denotes thickness, b and d breadth and density. If we 
regard the rolling portion as a circular cylinder its mass 
will be bd^y 2 , supposing it to have the same density as the 
unrolled portion. Let M be the whole mass, and B, initial 
radius, thus M = 7rR 2 frd Since the mass is constant we 
obtain the equation ^y 2 = ttB, 2 - c x, since y and x are the 
coordinates of the centre of gravity, this equation gives a 
y 2 
parabola as its locus. Also Jc 2 = y. 
j^sina. 
Hence equation (4) may be written 
d 2 x c / dx \ 2 2 
d#-^\dt) = 
By integration we obtain 
(ST = { A - | jsina(irE 2 - cxf } 
A denoting the constant. If this be determined by the sup- 
position that the mass starts from rest, the equation may 
be written as follows 
cx V 
7CR 2 J 
S) 2= l! sina,rR2 { 1 -( 1 “® 2 } 
1- 
cx 
ttRT 2 
or if l denotes the length of the tape 
© 2 4 3in ©-( 1_ f) s } 
Hence as x approaches to the value l the velocity increases 
indefinitely. The whole tension at the fixed point at any 
time will be 
