Rotating Pendulum: Is This Eqn Correct?

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The discussion revolves around verifying the correctness of a specific equation related to a rotating pendulum. Participants request the equation to be presented horizontally for clarity and seek explanations for each term in the equation to assess its validity. The use of LaTeX for formatting equations is encouraged to enhance readability. Ultimately, the original poster confirms they have found the answer independently but appreciates the assistance offered. The conversation highlights the importance of clear communication and proper formatting in mathematical discussions.
Prabs3257
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Homework Statement
Rod of mass M and length 2R can rotate about an Axis passing through O in vertical plane a disc of mass M and radiusR/2 is hinged to the other end P of the rod and can freely rotate about p when the disc is at the lowest point both rod and disc has angular velocity w if rod rotates by maximum angle 60 degree with downward vertical find w in terms of r and g (all hinges are smooth)
Relevant Equations
Conservation of energy
IMG_20190823_215039.JPG
is this eqn correct
 

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Please turn the equation so that it can be read horizontally. Also, please explain what each term in the equation is supposed to represent. Then perhaps we will be able to tell you if it is correct and if it isn't, we will be able to point out where you went wrong. Writing the equations in LaTeX is encouraged and will be of great help.
 
Leave it i got the answer but thanks for your time 😊
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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