GATE ME 2022 Set 1 — Question 32
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Theory of Machines & Vibrations → Free Vibration → Undamped Free Vibration
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Question
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 and outer radius are such that , and the acceleration due to gravity is . If the time period of small amplitude simple harmonic motion is given by , where is the ratio of circumference to diameter of a circle, then ________ (round off to 2 decimal places).
Correct answer
2.62 to 2.7
Solution
The time period of a physical pendulum is given by , where is the mass moment of inertia about the pivot point , is the mass, is the acceleration due to gravity, and is the distance from the pivot to the center of gravity .
1.Mass moment of inertia of an annular disc about its center : .
2.Given , substituting this: .
3.Using the parallel axis theorem, the moment of inertia about pivot (at distance from ): .
4.The distance from pivot to center is .
5.Substituting into the time period formula: .
6.Comparing with the given form , we find .
7.Rounding to two decimal places, .
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