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## Homework Statement

A rod of length L and mass m which is free to rotate round it's edge is held, initially, horizontally.

What are: the angular velocity and acceleration at angle θ

The force at the nail (the axis)

## Homework Equations

Torque and moment of inertia: ##M=I\alpha##

## The Attempt at a Solution

Conservation of energy:

$$mg\frac{L}{2}\cos\theta=\frac{1}{3}mL^2\omega\;\rightarrow \omega=\frac{3}{2}gl\cos\theta$$

$$M=I\alpha:\; mg\frac{L}{2}\sin\theta=\frac{1}{3}mL^2\alpha\;\rightarrow \alpha=\frac{3g}{2L}\sin\theta$$

The linear, tangential acceleration ##a_t=\alpha\frac{L}{2}##

I take forces in the tangential direction, F is the reaction:

$$mg\frac{L}{2}\sin\theta-F=m\cdot \frac{3g}{2L}\sin\theta \frac{L}{2}\;\rightarrow\; F=\frac{1}{4}mg\sin\theta$$

But i am not sure.

The reverse of the axial component of gravity ##mg\cos\theta## is the second component of the reaction in the nail.