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Question

A pencil of hexagonal cross-section of side a is placed on a rough surface as shown.

The platform starts performing S.H.M. perpendicular to length of the pencil in horizontal plane with angular frequency ω. Considering there is sufficient friction between pencil and platform such that there is no slipping between them, the maximum amplitude of oscillation such the pencil does not topple is gαω2. Then the value of α is (Answer upto two digits after the decimal point)

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Solution

The acceleration of the platform is Aω2, where A is its amplitude.

Taking frame of reference as the platform, there will be a pseudo force of mω2A on the block acting at its COM as shown in the figure.
By balancing the forces, we get,
f=mω2A;N=mg
By balancing the torque about COM, we get,
f3a2=Na2
mω2Aa32 = mga2A = g3ω2
α = 3

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