Gravitational potential in rotational system

[jackdamiels]

Homework Statement

The system is balancing rod on its tip. The coordinate system is polar, and the problem is one dimensional with angle $$\theta$$. Only forcee on this rod is gravitational force. The problem is finding potential in therms of moment of inertia I. My question is it possible to express gravitational potential similar to this: $$U(\theta) = k I \cos\theta$$

Homework Equations

Moment of inertia of rod about his and $$\frac{1}{3}ml^2$$
gravitational potential on the centar of mass : $$1/2mgl\cos\theta$$

The Attempt at a Solution

my attempt was integrating $$U(\theta)=-\int I \alpha d\theta$$




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[jbsteele]

1] U(\theta)=\int I \alpha d\theta [2] \alpha= -g\sin\theta [3] U(\theta)=-\int I g\sin\theta d\theta [4] U(\theta)=-Ig \int \sin\theta d\theta [5] U(\theta)=-Ig(-\cos\theta )+C [6] U(\theta)=-Ig \cos\theta + C