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cheeky_virus (10)

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Sir, my name is Cyril Johnson and I am from Jabalpur.I am greatly thankful to elessar_iitkgp  and biki(303) for helping me in the previous kinematics problem.Sir, I am new to this site and I dont know how to rate the Genius like you all.So, kindly excuse me.I will learn it soon.
Sir,now i have a different question for you.
The following question is of circular motion+work ,force and energy.
a 2Kg block is fixed at one end of a 3 mt long string(radius of the exibited motion) and released from horizontal position.find the minimum velocity required to keep the circular motion going.
Sir, is this practically proven?Please kindly explain me the physics behind it.
    
swatygupta (689)

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Sure i am that this day we are the masters of our fate, that the task set before us is not above our strengths; that its pangs and toils are not beyond our endurance. As long as we have faith in our own cause and an unconquerable will to win, victory will not be denied us.
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elessar_iitkgp (2317)

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Thanx for the appreciation.
Let the velocity with which the particle is projected be v0 downwards and the velocity of the particle at the topmost point of the circle be v.
System: Particle + Earth
There are no external forces on the system that do work on the particle. Hence the total mechanical energy is conserved.
Now, assuming the horizontal position to be the zero PE level,
(1/2)mv02 = (1/2)mv2 + mgL
where L is the length of the string.
v2 =  v02 - 2gL ------------------(1)
Now, for the string to cross the topmost point,
v2 0
v02 - 2gL 0
v0 2gL --------------(2)

At the highest point,
Now, at the topmost point,
T + mg = mv2 /L
T = mv2 /L - mg
T = mv02 /L - 3mg  (Using (1))
For the particle to cross the highest point
T 0
mv02 /L - 3mg 0
v0 3gL  ---------------------(3)

From (1) and (3),
v0 3gL
v0min = 3gL = 310 m/s

Good work biki.



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biki (1628)

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thanks .... frnd.... for the appraisal....

let me try this.
taking the lowest position of the body as the reference level of potential energy, the body is thus at a height of (radius) m from the ref. level.
So using energy conservation .... the total energy at this point must be equal to the K.E. at the lowest position (as P.E. at lowest position =0 ).
Let v0 = velocity at lowest position for which the body completes circular motion.
let T = tension in the string at highest position.
The limiting condition for completing circular motion in case the body is attached to a string is that the string does not become slack at the highest point.
in case of rods... the centrifugal force is not reqd. at highest position as the rod will never become slack. So in case of rods , we take velocity at top position as zero and just equate energy conservation principle between initial and final state (without considering any other things) 
Now, coming back to the given problem, the minimum condition for this to happen is that all forces in opposite direction become equal at highest point.
let v = velocity at top position.
So mg + T = mv2/r ________(1)
Now using conservation of mech. energy at top and bottom positions....
(1/2)mv02 = (1/2)mv2 + mg(2r)
or mv2 = mv02 - 4mgr _____(2)
using (2) in (1)...
mg+ T = (mv02 - 4mgr) /r
or mv02 = 5mgr + Tr.
Now the v0 will be minimum if T is minimum.
Now Tmin =0
So mv02 = 5mgr
or v0 =  5gr.
So v0 =  (5 x 9.8 x 3) = 73 m/s
Now let v / = velocity at initial horizontal position (at height = r from ref. level)
So again by conservation of mech. energy...
(1/2) m.(v / )2 + mgr = (1/2) m.v02
Putting values of m,g,r and v0 in (3)..
v / = 9.39 m/s
 

salman khan
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