Forces

Princess Sophie E:

A ladder rests with its foot on smooth horizontal ground and its upper end against a smooth vertical wall.
The ladder is uniformed and weighs 300N and inclined to the wall at an angle x. What horizontal force must be applied to the foot of the ladder to prevent it from slipping if
a) x=30degrees
b) x=arctan(3/4)

Hey Princess Sophie E!

To keep the ladder in place, the resultant force at its bottom must be in the direction of the ladder.
The vertical force (the "normal" force) at the bottom is equal to the weight W = 300 N.
And suppose we call the horizontal force F.
Then that means that the ratio between the horizontal force F and the vertical force N must be equal to the ratio of the distance of the foot to the wall (call that d) and the height of the top (call that h).

It so happens that tan(x) is equal to the ratio of d and h.
That is:

tan(x) = d/h

So this has to be equal to the ratio of F and W.

tan(x) = d/h = F/W

Can you then find F if you know that x = 30 degrees?
And also if x=arctan(4/3)?