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Question

Figure shows a loop track of radius r. A box starts sliding from a platform at a distance (h) above the top of the loop and goes around the loop without falling off the track. Find the minimum value of h for a successful looping. Friction is negligible at all surfaces.


A
2r
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B
r
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C
r2
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D
3r2
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Solution

The correct option is C r2
Let the box be of mass (m)
Let point B be taken as a reference point,
Applying law of conservation of energy between A and B
mg(h+2r)=12mv2
v=2g(h+2r)
To complete the loop, the box must have a speed of at least 5gr at the bottom, otherwise it will loose contact and start following the projectile parabolic trajectory. Thus we have
2g(h+2r)5gr
h+2r52r
hr2

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