With what minimum speed v must a small ball should be pushed...

With what minimum speed $v$ must a small ball should be pushed inside a smooth vertical tube from a height $h$ so that it may reach the top of the tube? Radius of the tube is $R$. (Assume radius of cross-section of tube is negligible in comparison to $R$.) (A) $\sqrt{2 g(h+2 R)}$ (B) $\frac{5}{2} R$ (C) $\sqrt{g(5 R-2 h)}$ (D) $\sqrt{2 g(2 R-h)}$

With what minimum speed $v$ must a small ball should be pushed inside a smooth vertical tube from a height $h$ so that it may reach the top of the tube? Radius of the tube is $R$. (Assume radius of cross-section of tube is negligible in comparison to $R$.)
(A) $\sqrt{2 g(h+2 R)}$
(B) $\frac{5}{2} R$
(C) $\sqrt{g(5 R-2 h)}$
(D) $\sqrt{2 g(2 R-h)}$

Key Concepts

Critical Condition at the Top of a Vertical Circle

When the object reaches the top of the circular path in a vertical setting, the minimum speed required is determined by the necessity that the gravitational force alone is just enough to supply the required centripetal force (i.e., the normal force becomes zero). This critical condition leads to a minimum speed of ?(gR) at the top, which is crucial in analyzing and ensuring the object can continue along the circular path.

Conservation of Mechanical Energy

This concept involves equating the total mechanical energy (kinetic plus potential energy) at one point in the motion to that at another, assuming there are no non-conservative forces doing work. In problems involving motion in a vertical tube or loop, the energy imparted at one point (for example, by giving an initial speed) is converted into gravitational potential energy as the object rises, and the remaining kinetic energy must be sufficient to maintain the motion over the curved path.

Centripetal Force and Circular Motion

An object moving in a curved path, such as a circular tube, requires a centripetal force that is directed toward the center of the circle. This force is provided by the net effect of the applied forces (such as gravity and the normal force). The requirement for centripetal force is often expressed as v²/R, where v is the speed of the object and R is the radius of the circle.

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