What is the distance of the point, on the rod from its end A, that has zero instantaneous acceleration at the moment of releasing the weight? Mark the ratio of this distance to the length of the rod.

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TestHub · Accepted Answer

Let the COM of the rod accelerate with a0 and the rod rotate with angular acceleration α: $T \frac{ℓ}{2} = \frac{m ℓ ^{2} α}{12} \Rightarrow T = \frac{m ℓ α}{6}; T = m a_{0} \Rightarrow a_{0} = \frac{ℓ α}{6}$

Acceleration of end A: $a_{A} = a_{0} + \frac{ℓ α}{2} = 4 a_{0}$

Point P at distance y from centre with zero acceleration: $a_{P} = a_{0} - α y = 0 \Rightarrow y = \frac{a _{0}}{α} = \frac{ℓ}{6}$

Distance from end A: $d = \frac{ℓ}{6} + \frac{ℓ}{2} = \frac{2 ℓ}{3} \Rightarrow \frac{d}{ℓ} = \frac{2}{3} \approx 0.67$

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