Equilibriums, Inclined, Pulleys, and Trains

 Equilibrium
An equilibrium is when a system is not moving at all. Since this is true this must mean that all the forces are perfectly balanced and the sum of the forces on one side aren't greater than the forces acting against it. When dealing with these questions though it is always important to list ones assumptions.
Assumptions:
-there is no friction
-the results of the x and y components are 0
-F=ma


Inclined

In a inclined plane a new force is introduced which is friction. Friction can be split into static and kinetic friction for when the object is not moving and moving respectively. When solving these questions one must tilt their head to think in a new direction, since while the force of gravity is dragging the object down in the same direction, friction, the normal force, and perhaps acceleration are in a totally different direction. It is also interesting to note that the coefficient of friction is the tan of the degrees between the plane and the ground.
Assumptions
-positive axes in direction of acceleration
-normal force perpendicular to the plane
-no air resistance
-no movement in the y direction


Pulleys
A pulley is simply a system where a weight or an object is attached to a string which is on an axle and support the rope holding the pulley up or another weight connect to the same string. Tension is introduced in this system which is energy stored in the rope. Since the objects attached to the same rope, their tension is the same.
Assumptions:
-No friction or weight in the rope and pulley
-positive axes in the direction of acceleration
-no air resistance or wind
-acceleration is equal
-tension is equal
-two free body diagrams


Trains
Finally we are at trains. A train is simply a system where several objects are connect by rope and a force is acting on one end of it causing it to move. The other objects are dragged with it due to tension. A simple way to solve it is to draw it first as one big free body diagram with their combined mass since the tensions cancel each other out. This way one can quickly find the acceleration of the system and won't have to substitute one equation into another other and other again.
Assumptions
-acceleration is equal
-no air resistance
-same number of fbds as number of masses
-position axis in direction of acceleration
-weightless ropes/cables