Trigonometry + Mechanics

Consider ( Ring + block + string + Earth ) as a system

Interesting thing of this question is that we don't need to calculate it's velocity ! Because change in potential energy of system is zero , So By Cons. of energy, change in kinetic energy of system is zero and since system is constrained So velocity of both bodies must be zero !

Mathematically :

Since decrease in Potential energy of ring :

$\Delta PE\quad =\quad -mgL\tan { ({ 37 }^{ 0 }) } \quad =\quad -\quad \cfrac { 3gL }{ 4 }$ .

Increase in PE of block :

$\Delta PE\quad =\quad +MgL(\sec { ({ 37 }^{ 0 }) } \quad -\quad 1)\quad =\quad +\cfrac { 3gL }{ 4 } \quad$ ..

So ${ \Delta PE }_{ net }\quad =\quad 0\quad =\quad { \Delta KE }_{ net }\quad \quad$ .

$\boxed { { V }_{ ring }\quad =\quad 0 }$ .

Q.E.D

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Note : Constrained motion relation between ring and block is ;

${ V }_{ block }\quad =\quad { V }_{ ring }\sin { ({ 37 }^{ 0 }) }$ .

you r right buddy actually i have gone through all the calculation and the result came the same..