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A block follows the path shown in the figure from height $h$. If the radius of the circular path is $r$, then the relation required to complete a full circle is

1. $h<\dfrac{5r}{2}$
2. $h>\dfrac{5r}{2}$
3. $h=\dfrac{5r}{2}$
4. $h\ge\dfrac{5r}{2}$

Correct answer: $h\ge\dfrac{5r}{2}$

Solution

For a block to complete the loop, at the top the minimum condition is $N=0$, so $v_{top}^2=gr$. Using energy conservation from height $h$ to the top of the loop gives $mgh=mg(2r)+\tfrac12 mgr$, so $h=\tfrac{5r}{2}$. To complete the circle, the height must be at least this value.

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