Forces at Key Points

R TOP T + mg → centre BOTTOM T − mg = $mv^{2}$/R SIDE v = $\sqrt{3gR}$ Min Speeds (string) Top: $\sqrt{gR}$ Side: $\sqrt{3gR}$ Bottom: $\sqrt{5gR}$ Tb − Tt = 6mg Min Speeds (rod) Top: 0 Bottom: $\sqrt{4gR}$ = 2$\sqrt{gR}$ v (tangential)

String vs. Rod — Key Distinction

A string can only pull (tensile force). Minimum tension = 0 at the top ⟹ v_top ≥ $\sqrt{gR}$.

A rigid rod can push (compressive) or pull. At the top with v = 0: rod pushes inward, providing centripetal force. So v_top,min = 0.

Tension Difference Formula

$T_{\text{bottom}} - T_{\text{top}} = 6mg \quad \text{(independent of v, m, R)}$

This is a powerful check formula — always valid for any speed (not just minimum).