A rod is arranged at an angle of 30° from the horizontal. Attached to the rod with two strings is the mass m, as shown. The rod is rotated, maintaining its direction in space, so that m travels in a circular path. The strings are of equal length , and make angles of 60° with the rod as shown. Take the length of the strings as 2.4m. Calculate the minimum value of the tangential speed (in m/s) of the mass such that the string with tension T2 does not become slack when the mass is directly above the rod.

Engineering

Physics

Circular Motion in Vertical Plane

Free Body Diagram

Vertical Circular Motion

Question

A rod is arranged at an angle of 30° from the horizontal. Attached to the rod with two strings is the mass m, as shown. The rod is rotated, maintaining its direction in space, so that m travels in a circular path. The strings are of equal length , and make angles of 60° with the rod as shown. Take the length of the strings as 2.4m. Calculate the minimum value of the tangential speed (in m/s) of the mass such that the string with tension T₂ does not become slack when the mass is directly above the rod.

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r = cos 30° = $\frac{2 \sqrt{3}}{2}$
T₂ cos 30° + T₁ cos 30° + mg cos 30° = $\frac{{mv}^{2}}{r}$
T₂ sin 30° + mg sin 30° = T₁ sin 30°
Just slack ⇒ T₂ = 0° ⇒ T₁ = mg
2mg cos 30° × L cos 30° = mv²
v² = 2 × $\frac{3}{4}$ L × 10
v² = $\frac{3}{2}$ × 10 × 2.4 = 36 ⇒ v = 6m/s