

 When breaking vectors into components: For the horizontal component,
vectors pointing to the left are
negative while vectors pointing to the
right are positive.
 The vertical component of the vector are positive
when pointing up and
negative when pointing down.
 All objects are assume to be ideals, masses are points without shape
(even when diagrams show shapes), strings are massless (if not indicated
otherwise) and stretchless (no rubber band kind of properties), and pulley
are massless and frictionless. Later on this notes, same of these
restrictions will be lifted.


The idea of the FBD is to replace the effect of objects in each other for the
forces representing their actions. Therefore reducing to one object the diagram
associated with the FBD resulting from a complex situation.


 Read the problem carefully. Make sure that you understand the given data
and the asked information.
 Select the order in which you are going to work the problem.
 Start by solving from the object to which less unknowns are
associated with.
 Also, start solving for objects for which there is less number of
forces applied.
 Draw the free body diagram for the object.
 Break every force and acceleration into their x and y
components.
 The axis should be selected such that the minimal number of vectors
need to be break down into components.
 If possible, select the xaxis along the horizontal component
and the yaxis along the vertical component.
 From the diagram, obtain the net force
acting on the selected object for each vector component, one for xaxis
and another for yaxis components.
 As a direct application of Newton's second law, the previous net force is also
related to the acceleration of the object
for each individual component.
 Notice that results from step 5 and step 6 are obtained from two
independent analysis. However, these results must be the same. Therefore,
equating these two independent results for the net force an equation can be
obtained.
 Repeat steps 3 to 7 as need it until all the unknown can be solve from
the different equations.


In the figure, two weights are hanging from the ceiling.
The first weight is 200 N, w_{1} = 200 N, and the second
weight is 300N, w_{2} = 300N. Find the tensions T_{1}
and T_{2}. 

The tension T_{2} is

a) 
500 N 


b) 
100 N 

c) 
200 N 
N 
d) 
300 N


e) 
None of the above. 
and the tension T_{1} is
N 
a) 
500 N 


b) 
100 N 

c) 
200 N 

d) 
300 N


e) 
None of the above. 

Solution:
This example is sufficiently simple to be solved without
calculations. Thus, it is a good example to start understanding the
method for solving problems involving Newton's laws of motion.
Following step two described above, it is simpler to solve for the
bottom weight first, w_{2}. The corresponding FBD is
(Notice that the effect of the rope is replaced by the tension. This
tension pulls up the object) 
Free Body Diagram Object 2

In this problem, all vectors are along the vertical
component. Therefore, it is not necessary to break the vectors into
components.
Vertical Direction
Using step 5, the net force is the result of the vector addition
between the tension of the rope (vector pointing up; then, positive) and
the weight of the object (vector pointing down; then, negative).
Following step 6, the net force is related to the
acceleration of the block. But the acceleration of the block is zero as
indicated in the diagram. Therefore,
.
Equating these two relations for the net force acting
on object 2, the following equation is obtained
This equation can be immediately solved for T_{2}


Vertical Direction
In this case, the net force is the result of the vector addition
between the tension of the rope (vector pointing up; then, positive) and
the vectors pointing down (vectors pointing down; then, negatives); the
weight of the object and the tension of the bottom rope. Notice that
T_{2} pulls on object one downward, for a review on tension
go to the
Tension
section of these notes. Therefore, the net force is
Applying Newton's second law to the first weight,
Equating these two relations, the final equation is obtained

