Graphical Mechanics (Part 3)

A ring of mass mm is connected with a spring of stiffness kk whose other end is fixed\text{fixed} as shown. Ring is constrained to move along wedge\text{wedge} shaped y=x24\displaystyle y = \frac{x^2}{4}. The system is released from rest\text{rest}. Find its Time Period tt in seconds\text{seconds}.

If t=ab×π where a,bN and gcd(a,b)=1\displaystyle t = \sqrt{\frac{a}{b}} × \pi \text{ where } a, b \in \N \text{ and } \gcd(a, b) = 1, enter answer as a+ba + b.

Details and Assumptions

  • All surfaces are smooth\color{#3D99F6}{\text{smooth}}

  • Take acceleration due to gravity g=10m/s2\color{#3D99F6}{g = 10m/s^2}

  • Spring is ideal\color{#3D99F6}{\text{ideal}} and is initially in its natural state\color{#3D99F6}{\text{natural state}}

  • Stiffness of spring k=mg N/m\color{#3D99F6}{k = mg \text{ N/m}}

  • Initially, ring is at (2,1)\color{#3D99F6}{(-2, 1)} and other end of spring is fixed at (0,1)\color{#3D99F6}{(0, 1)}

Inspiration Aniket Sanghi

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