GATE ME 2016 Set 1 — Question 47
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Theory of Machines & Vibrations → Free Vibration → Undamped Free Vibration
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Question
A solid disc with radius is connected to a spring at a point above the center of the disc. The other end of the spring is fixed to the vertical wall. The disc is free to roll without slipping on the ground. The mass of the disc is and the spring constant is . The polar moment of inertia for the disc about its centre is .
The natural frequency of this system in rad/s is given by
Correct answer
(A) √((2 K (a+d)²)/(3 M a²))
Solution
Using the energy method, let be the angular displacement of the disc. The linear displacement of the center is . The displacement of the point where the spring is attached is .Total Kinetic Energy ():
Total Potential Energy ():
By conservation of energy, :
The natural frequency is .
Total Potential Energy ():
By conservation of energy, :
The natural frequency is .
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