We can start by drawing a diagram of the situation.
We have a trapdoor hinged at one end, making
an angle of theta to the horizontal. The trapdoor
is uniform and heavy, so its weight acts vertically
downwards at its centre, halfway along its
length.
We are told that a force, F, acts perpendicular to the trapdoor at one end.
In order for the trapdoor to be in equilibrium,
there must be a reaction force at its pivot
point. We call this force N.
The angle N makes with the horizontal is phi. We can determine the value of F by considering the rotational
equilibrium of the system, as the moments
on the trapdoor about any point must balance.
We should take moments about the pivot point,
as this means that the unknown force, N, does
not appear in our equations. There are two
ways in which we can think about taking moments.
We can calculate the moment due to the weight
by considering the force perpendicular to
the trapdoor, and taking the length to be
half the length of the trapdoor, or we can
find the perpendicular distance, d, from the
pivot point to the weight force in terms of
the length of the trapdoor. The moment in
this case is the weight times the perpendicular distance, d.
