When we look at force in a two-dimensional world, especially for objects that aren't moving, there are some important math tools that help us understand what's going on. These tools show us how forces act on things and help us calculate the resulting forces that affect how things stay still.

Free Body Diagrams (FBDs)
One key tool is the Free Body Diagram. An FBD is like a picture that shows all the forces acting on a single object. It helps us see normal forces and frictional forces clearly.

When an object, like a block, is resting on a surface, the normal force ($F_N$) pushes straight up from the surface. The frictional force ($F_f$) works along the surface, trying to stop the object from moving. Using FBDs, students can carefully look at each force, label them, and this makes later calculations easier.

Equilibrium Equations
If an object is not moving, the total of all forces and moments acting on it is zero. This leads us to the equilibrium equations:

• Force Equilibrium:
  • For left and right (the x-direction): $\sum F_x = 0$
  • For up and down (the y-direction): $\sum F_y = 0$

These equations help us find unknown forces like the normal force and frictional force. For example, if there's a block on a flat surface and someone pushes it at an angle, we can break that push into two parts—up and sideways—to find out the normal force and the frictional force that tries to stop the block from sliding.

Frictional Force Calculations
We can figure out the frictional force with the equation: $F_f \leq \mu F_N$

Here, $\mu$ means the coefficient of friction. This shows how friction depends on the normal force. It’s important to know how much force we can use before the object starts to move.

Calculating Normal Forces
When dealing with slopes or ramps, we can find the normal force using some basic math rules. For example, if you place a weight $W$ on a slope at an angle $\theta$, the normal force can be calculated like this: $F_N = W \cos(\theta)$

In this case, the weight is the force due to gravity pulling down on the object, and as the slope becomes steeper, the normal force gets smaller.

Static Friction vs. Kinetic Friction
It’s also important to know the difference between static friction and kinetic friction. Static friction is what holds an object in place before it starts to move. Kinetic friction kicks in when the object is sliding. Usually, the coefficient of static friction ($\mu_s$) is higher than that of kinetic friction ($\mu_k$), which affects how we calculate the forces in each case.

Mathematical Relationships
There are even more relationships we can see based on specific problems, especially with angles and forces. The formulas showing the basic components of forces are: $F_x = F \cos(\theta)$ $F_y = F \sin(\theta)$

Conclusion
To wrap it up, when we study friction and normal forces in two dimensions, we use tools like Free Body Diagrams, equilibrium equations, and friction calculations. Understanding how these forces relate to each other helps us solve real-life problems with stationary systems. By learning these concepts, students can better understand and manage the complexity of forces acting on objects.