Book against a wall and forces If you take a book with mass of 1kg and push it against the wall. With how much force do you have to push the book so it does not fall?
The problem is I know how to calculate this problem, you say $F_{friction}$=$F_{gravitational}$ and $F_{wall / normal}$ = $F_{human / push}$. 
The problem is solvable if you say that $F_{gravitational}$ = $F_{normal / wall}$, but why is this true? 
How would you calculate this problem if you didn't know that $F_{normal}$ = $F_{gravitational}$, how would you prove this statement? 
For me the problem is that $F_{normal}$ = $F_{gravitational}$ * $\cos(\alpha)$, but cos(angle) is 0, I don't understand the relationship between $F_{gravitational}$ and $F_{normal}$ in this scenario.
 A: What you need is a relationship between the frictional force and the normal force.  
The Wikipedia article on friction has $$F_\mathrm{f} \leq \mu F_\mathrm{n}$$ where $\mu$ is the coefficient of friction.  You want to find the minimum normal force necessary, i.e. when this inequality becomes an equality.
A: I think it would help if you rotate the figure by 90degrees. Instead of seeing it as what force to push the book with, make the book horizontal. Like this:-


So, now you push the book downwards, the force due to gravity will act leftwards, and the fictional force will be rightwards.
Now the question becomes simpler. From the figure, assume that $F_{g}$ is a force acting on the book to the left. $F_{f}$ is the frictional force and $F_{push}$ is the force you apply(which will act analogous to gravity in actual horizontal systems). The question now converts to what should the frictional coefficient be so the book doesn't move. So we equate $F_{f}$=$F_{g}$.
Since, 

$F_{g}$ = mg   (where 'm' is mass of the book and 'g' is the gravitational constant) 
Hence,
$F_{f}$ = mg
or, since in $F_{f}$ is just friction coefficient times your applied force 
u$F_{push}$ = mg  (where $F_{push}$ is the force you apply)

therefore, $F_{push}$= mg/u .

So as you can see its not a matter of proving the statement $F_{frictional}$=$F_{gravitational}$ , but in order to satisfy the condition that the book doesn't move, $F_{frictional}$=$F_{gravitational}$, becomes a condition which must be satisfied(else the book would move).
Edit:: I tried to add an image to explain the figure, but can't do so because of low reputation points.
Edit:: Got enough reps to add images now, hope this clears it up.
