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Question

A body is moving in a vertical circle such that the velocities of body at different points are critical. The ratio of velocities of body at angular displacements 600 and 1200 from the lowest point is:
  1. 5:2
  2. 3:2
  3. 3:1
  4. 2:1

A
5:2
B
3:1
C
2:1
D
3:2
Solution
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Given that velocities at different points are critical
so velocity at top point (T)
VT=gR...(1)
From the diagram
in right triangle OMA
MA=OP=OPcosθ
At point A, θ=60°
HA=OGOPHA=RRcos60°=R2...(2)
in right triangle OMB
MB=QP=QPcos(180θ)
At point B, θ=120°
HB=OG+OQHB=R+Rcos60°=3R2...(4)
Using energy conservation equation at point T and A
12mVT2+mg(2R)=12mVA2+mgH1...(2)
putting the value of eqn (1), eqn (2) in eqn (4)
12m(gR)2+mg(2R)=12mVA2+mgR2
VA=2gR
Using energy conservation equation at point T and B
12mVT2+mg(2R)=12mVB2+mgH2...(2)
putting the value of eqn (1), eqn (3) in eqn (4)
12m(gR)2+mg(2R)=12mVB2+mg3R2
VB=2gR
Ratio of VA and VB=2:1
So the correct option is (D)


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