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In the image below, the system is in equilibrium. Some force is applied at the bottom of the vertical pole and is being opposed by some force at the top by the horizontal pole... But I feel as though the opposing force wouldn't be equal to the applied force because there is a distance between the two points of action. How would this distance affect the force required to oppose the initial force? And how could I get an expression for the opposing force in terms of the initial force, F?

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If you do a free body diagram you will see that a moment needs to be added in order to balance the forces and moments. The forces will be equal and opposite and the moment will depend on the distance between the forces as well as their magnitude.

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$$ M = F d$$

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