16 J. B. CHERRIMAN : THE MOTION OF A 
2. Particular case. 
A uniform chain is in motion on a fixed smooth vertical curve, the ends being free. 
and no impressed force except gravity ac‘ing. 
Taking the axis of y vertical, and positive ordinates measured downwards, 
dv _ , dy 
mds ren ds .6 a + dT. 
Integrating over the whole length / of the chain 
dv 
Z— = à — 
te AC er) 
where 7, 7, are the ordinates of the upper and lower ends of the chain. 
To find the tension at any point, 
dv 
dT = mds Frm dy 

—m ds Ÿ (Ye — Y) — mg dy 
and therefore 

L=mg (Y2— Y) ; —mgy + const, 
where s is the distance measured from some fixed point of the curve to the point where 
the tension is required. Also at the free end 
0 — mg Ye— Y) 7 — M 9 Y + const, 
and therefore 
mg G—y) 

T= mg (Y2— y) 
= G—9 + 6—s) 1) 
m 1 
= “A (nh + Yale —y I) 
where /,, /, are the lengths of the segments into which the point in question divides the 
length 7 of the chain. Hence also the tension at the middle point of the chain is 
expressed by 
mg} a+) ay i. 
Cor. The preceding results can also be obtained by the direct application of the 
principle of vis viva; for, 
i ml. v? = Zm ds fg dy 
— mg /* y ds 
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