Situation 1: Suppose I have two blocks on a table placed side by side in contact with each other. Do they have a contact force acting at the interface? How can we find the magnitude of such a force?
Situation 2: If I'm holding a pen between two fingers of my hand such that its centre of mass and my two fingertips are collinear, what is the contact force between the pen and the finger above it? What about the one below?
It seems that you are interested in a Newtonian/classical interpretation.
You can apply the equilibrium equation. The basic principle is that if an object is at rest then the sum of the forces applied on it is zero.
So isolate one of the objects with mass $m$ on the table. The forces on it are :
* the weight $\vec P=\vec g\cdot m$ where $\vec g$ is Earth's gravitational acceleration ($\approx 9.8 m/s^2$)
* the force $\vec T$ the table is applying on the object, and
* the force the other object is applaying $\vec F$
Since $m$ is at rest it must hold that
$\vec F+\vec T+\vec g\cdot m=0 $.
If we suppose that the two objects are cubic or anyway in contact with vertical plane surfaces then we know the force $F$ is horizontal and we can divide in two scalar equations :
$t=g\cdot m$
$F=0$
This actually tells us that the total force the bodies are exchanging is zero but it doesn't say how it could be distributed on the surface.
For the second problem, other than the weight we have already seen we have two forces which are respectively applied by the first and second finger, so all we can say is that $\vec F_1+\vec F_2=-m\cdot\vec g$.
We cannot determine the two forces without further information. This is pretty intuitive since you could hold the pen lightly or stronger still keeping it firm and in the same exact position.
There is a force of gravitational attraction between all objects which have mass. So even in the absence of external forces, two objects will move closer until the contact force between them balances the gravitational attraction. For very small objects the attractive force could have some other origin, such as the Casimir force or the van der Waals force.
For human-sized objects these forces are extremely weak and are usually ignored in Newtonian Mechanics. If two such objects were placed in contact without any constraint keeping them together, then the contact force would separate them because there is no force to oppose it.
One possible constraint is friction. Two blocks could be squeezed together, placed on a rough surface, then released. If the contact force is less than the limit of static friction they will remain in contact. Given 2 such blocks in contact, we cannot determine the size of the contact force between them unless we know their history. We could only do so by measurement. If the surface on which the blocks are place is frictionless then there is no constraint so the blocks would separate until the contact force is zero.
The 2nd situation is no different from 2 blocks placed one on top of the other on a horizontal surface. The contact forces are easily calculated from the masses of the blocks and the requirement that the forces on them are balanced since the blocks are not accelerating.
