A small bead ‘B’ of mass m is free to slide on a fixed smooth vertical wire , as indicated in the
diagram. One end of a light elastic string, of unstreched length a and force constant 2mg/a is attached to B. The string passes through a smooth fixed ring R and the other end of the string is attached to the fixed-point A, AR being horizontal. The point O on the wire is at same horizontal level as R and AR = RO = a.
(i) In the equilibrium position, find OB.
(ii) The bead B is raised to a point C of the wire above O, where OC = a and is released from rest. Find the speed of the bead as it passes O and find the greatest depth below O of the bead in the subsequent motion.
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) OB = a/2
(ii) v = 
, d = 2a
Sol. (i)
mg = T cos θ
mg = 
× 
x = 
(ii)
K 
+ mga = 
K (2a – a) 2 + 
mv 2 
(2a
2 – a 2 ) + mga = 
mv 2 ⇒ 
= v
K 
+ mg a = 
K 
– mg y
K2a
2 + mg a = 
K (a 2 + y 2 ) – mgy
2a 2 + mga = 
a 2 + 
y 2 – mg y
3 mg a – mg a = 
– mg y
2 mga = 
– mg y
2a 2 = y 2 – ay
⇒ y 2 – ay – 2a 2 = 0
y 2 + ay – 2ay – 2a 2
⇒ y (a + y) = 2a (y + a)
⇒ y = 2a
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