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Question

A small block of mass m slides along a smooth frictionless track as shown in the figure. If the block starts from rest at P at height h from the bottom, then which of the following statements are true?


A
For h=5R, the resultant force acting on the block at Q is 75mg
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B
If the force exerted by the block against the track at the top of the loop equals its weight, then h=3R
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C
For h=5R, the resultant force acting on the block at Q is 65mg
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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
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Solution

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=2mgv2=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=mv2Rv2=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

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