A rigid uniform annular disc is pivoted on a knife edge A in a uniform gravitational field as shown, such that it can execute small amplitude simple harmonic motion in the plane of the figure without slip at the pivot point. The inner radius r and outer radius 𝑅 are such that r 2 = R 2 /2, and the acceleration due to gravity is g. If the time period of small amplitude simple harmonic motion is given by T = \beta\pi\sqrt{R\/ g} where π is the ratio of circumference to diameter of a circle, then β = ________ (round off to 2 decimal places).

GATE Mechanical Engineering (ME) 2022 Shift 1 Solved Paper

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A rigid uniform annular disc is pivoted on a knife edge A in a uniform gravitational field as shown, such that it can execute small amplitude simple harmonic motion in the plane of the figure without slip at the pivot point. The inner radius r and outer radius 𝑅 are such that r² = R²/2, and the acceleration due to gravity is g. If the time period of small amplitude simple harmonic motion is given by

T = βπ{√{R\/ g}}

where π is the ratio of circumference to diameter of a circle, then β = ________ (round off to 2 decimal places).

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