Applied Maths - Mechanics 36

還不明白的薑

2009-08-11 8:23 pm

Two rigid, light rids AB, BC, each of length 2a, are smoothly jointed at B, and the rod AB is smoothly jointed at A to a fixed smooth vertical rod. The joint at B carries a particle of mass m. A small ring, also of mass m, is smoothly jointed to BC at C and can slide on the vertical rod below A. The ring rests on a smooth horizontal ledge fixed to the vertical rod at a distance 2a below A.

The system rotates about the vertical rod with constant angular velocity w. Find the force exerted by the ledge on the ring, and deduce that if the ring remains on the ledge, then aw^2 <= 3g .

回答 (1)

用戶9112

2009-08-12 4:24 am

✔ 最佳答案

Let T1 be the tension in AB and T2 the tension in BC
Since AB = BC = AC = 2a, they form an equilateral triangle.
For particle at B,
T1cos60 – T2cos60 = mg or T1 – T2 = 2mg … (1)
T1sin60 + T2sin60 = mω2r where r = 2asin30 = √3a
So √3T1 + √3T2 = 2mω2√3a
T1 + T2 = 2mω2a … (2)
(2) – (1) => T2 = mω2a – mg
At C, the particle will remain in contact with the ledge if the weight is large enough to compensate more than the vertical component of the tension in rod BC.
mg => T2cos60
mg => (mω2a – mg)/2
g + g/2 => ω2a/2
3g => ω2a

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