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Let us consider that a particle of mass m is executing a vertical circular motion,centering the point O as shown in figure and having been attached to an in-extensible string of length R.
Let P represents an arbitrary position of the particle , where the string makes an angle theta with the vertical. If v be the velocity of the particle at this position then considering forces on the particle we can write
For position P
(mv^2)/R=T-mgcostheta........[1]
For position L (lowest bottom position where theta=0)
(mv_"bot"^2)/R=T_"bot"-mgcos0^@,
where v_"bot" is the velocity of the particle at L
=>(mv_"bot"^2)/R=T_"bot"-mg...........[2]
For position H (highest top position where theta=180^@)
(mv_"top"^2)/R=T_"top"-mgcos180^@,
where v_"top" is the velocity of the particle at H
=>(mv_"top"^2)/R =T_"top"+mg...........[3]
Considering conservation of energy at L and H we can write
KE" at H"
= KE" at L" -"gain of PE due to lift of height 2R (L to H) "
1/2mv_"top"^2=1/2mv_"bot"^2-2mgR
=>v_"top"^2=v_"bot"^2-4gR.......[4]
The particle will complete the circle , if the string doesn't slack at the highest point when theta =180^@ There must be centripetal force to make this happen. The minimum centripetal force required can be had by putting T_"top"~~0 in equation [3].
(m(v_"top"^2)_"min")/R =0+mg
=>(v_"top")_min=sqrt(gR).......[5]
Inserting this value in equation [4] we can get the minimum required velocity of the particle at position L
=>v_"top"^2=v_"bot"^2-4gR
=>(v_"top"^2)_ min=(v_"bot"^2)_"min"-4gR
=>(v_"bot"^2)_"min"=(sqrt(gR))^2+4gR
=>(v_"bot")_"min"=sqrt(5gR)
