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for the whole mass may be divided into two parts — for the 
portion that is still coiled up the angular velocity will be 
the same for each particle, and equal to the angular velocity 
of the body about the axis. For this portion then we shall 
have the expression 
d /M 
dt) 
For the unwound portion 
the angular velocity about the axis is not the same for each 
particle. At any instant y is the same for each particle, 
also ^ is the same for each particle. The equation of 
moments for this part will take the form 
d dx f cx /T)2 2X \ 
dtdt i^+V)) 
The geometrical equations will be the same as before. The 
positive direction of the axis of x is taken vertically down- 
wards. Suppose w the attached weight to have n times the 
mass of the tape, then the equation of motion may be 
written 
d dx f ny 3 cx( R 2 4- By 2 ) 
dt dt\ 2 + 4 y 
= cxgy + ymrBry 
(Hoc • 
The coefficient of — may also be written 
1 /2t r 2 R 4 -cV\ 
4A tt ; 
Writing, for brevity, this in the form F(&), the equation of 
motion becomes 
~F(z) + =cxgy + gmr’B? 
The solution of this equation is 
(S) =(^f{fM™ + n*WW(x)d X +c } 
If we suppose the motion to start from rest, the constant 
will be 0. Performing the integration, the result will be 
dx y_ /8y_ /of~ nx 3 ^ 
at x 2 \f ttR 2 V 
