Concept:To successfully complete a vertical circle, the object must maintain enough velocity at the highest point so that centripetal force is entirely provided by gravity, allowing normal force or tension to drop to exactly zero.
Formula:$$ \frac{mv^2}{r} = mg + T $$
Solution:- At the minimum required velocity, the tension (or normal force) \( T \) becomes 0.
- The formula simplifies to \( \frac{mv^2}{r} = mg \).
- Mass \( m \) cancels out: \( \frac{v^2}{r} = g \).
- Solve for velocity: \( v^2 = gr \), thus \( v = \sqrt{gr} \).
Why other options are incorrect:- Option B and Option A represent velocities at other points on the circle or in different orbital scenarios.
- Option D is mathematically baseless for this specific limit boundary condition.
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