The problem you've provided involves a rod that can rotate frictionlessly around a horizontal axis $A$. The goal is to determine the forces exerted by the rod on the axis $A$ in terms of both magnitude and direction when the rod has rotated 45° from its initial horizontal position. The key to solving this problem involves using principles of rotational dynamics and mechanics.

Key Steps for the Solution:

1. Moment of Inertia Calculation: Since we are asked to calculate a relevant moment of inertia, we assume the rod has a uniform mass distribution. For a rod rotating about an axis through one end (point $A$), the moment of inertia is: $I = \frac{1}{3} m L^2$ where $m$ is the mass of the rod and $L$ is the length of the rod.

2. Conservation of Energy: The rod is released from rest in a horizontal position. At this position, it has potential energy due to gravity. As it rotates, this potential energy converts into rotational kinetic energy. At an angle of 45°, both the kinetic energy and the remaining potential energy need to be considered. The total mechanical energy can be written as: $m g \left( \frac{L}{2} \right) = \frac{1}{2} I \omega^2 + m g \left( \frac{L}{2} \sin 45^\circ \right)$ where $\omega$ is the angular velocity of the rod at 45°.

3. Forces at the Axis $A$: To find the forces on the rod at point $A$, we need to compute both the radial and tangential forces. The radial force is related to the centripetal acceleration $a_c = \omega^2 L/2$, and the tangential force relates to the torque and angular acceleration.

Let me now compute and simplify these expressions to provide you with the exact values of the forces and directions.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can still apply the key principles mentioned earlier to manually solve the problem. If you'd like more details or have any further questions, feel free to ask!

Here are 5 related questions to expand your understanding:

1. How is the moment of inertia of a rod about an axis through its end derived?
2. How does the conservation of energy apply to rotational motion?
3. What is the difference between radial and tangential forces in rotational systems?
4. How do we calculate angular velocity using energy principles?
5. How does the angle of rotation affect the forces acting on a rotating object?

Tip: Always remember that forces in rotational motion can be broken down into components (radial and tangential), making it easier to analyze their effects.