The correct option is D If the block should not fall off at the top of the circular track, it must be released from a minimum height of h=5R2
(i) Using energy conservation between P and Q, taking Q as a reference,
UP+KEP=UQ+KEQ
⇒0+mg(4R)=12mv2
⇒4gR=v22
At point Q,
N=mv2R=m×8gRR=8mg
Thus, Resultant force at Q
FQ=√N2+(mg)2
FQ=√64m2g2+m2g2=√65 mg
(ii) For block to exert force on track equal to weight,
Normal force exerted by track at the top = mg
Equating forces at topmost point :
mv2R=mg+N=2mg⇒v2=2Rg
Considering point P to be at height h above the ground and applying Law of conservation of energy between P and the topmost point of circular loop :
KEP+UP=KEtop+Utop
0+mg(h)=12m(2Rg)+mg(2R)
Therefore, required height, h=3R
(iii) If the block just falls off at the top of the loop, then normal reaction at the top N=0
At the top:
N+mg=mv2R⇒v2=Rg
Applying conservation of energy between point P and the top of the loop:
mghmin+0=mg(2R)+12mv2
⇒mghmin=mg(2R)+mgR2=5mgR2
⇒hmin=5R2