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Let the cone lie on the $\hat{X}\wedge \hat{Y}$ plane (z=0) and let the $z$ axis pierce this plane at the cone's apex. If the cone's half angle is $\alpha$, then its axis of symmetry as a function of time is defined by the vector $$A(t)=\cos\alpha \left(\cos(\omega_0\,t) \hat{X} + \sin(\omega_0\,t) \hat{Y}\right)+\sin\alpha \hat{Z}$$ where \$\omega_0 = ...

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I am confused regarding the fact that when a disk is rolling on an inclined plane without slipping and similarly a solid sphere is rolling on an inclined plane without slipping then the sphere has more angular velocity. the moment of inertia of a sphere is less than the disk so it that if the moment of inertia is less then the angular velocity is more? ...

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I don't want to give away the answer immediately, since it's homework. Try to visualize one huge big wheel and one very small wheel. Try to imagine what would happen if at each instant the tangential velocity is not the same.

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