Strategy for Drawing Free-Body Diagrams and Solving Newton’s Law Problems
(either zero or non-zero acceleration)
1. Isolate every relevant body of a system and draw one free-body diagram for each body.
(Definition: A system is a collection of bodies under consideration.) This helps you to avoid mixing
up different forces relevant to various bodies of the system. In the most complicated cases, we will
consider a system consisting of 3 or 4 bodies.
2. For each free-body diagram, identify each and every force acting on this body under consideration.
(a) Identify internal forces: Exam and identify other bodies within the system that interact with
this body that is under consideration in this free-body diagram. When there is interaction, there
exists a pair of action-reaction forces acting on these two bodies (one on each body), according
to Newton’s third Law. Such forces can also be called internal forces. Identify pairs of such
internal forces. For example, you may have a system consisting of two boxes in contact with
each other. You normally need to consider a pair of normal forces between them and a pair of
friction forces between them (although under certain conditions, these forces can be zero).
I always use the same symbol to represent a pair of action-reaction forces acting on two
interacting body. On the free-body diagram, they are drawn in opposite directions.
(b) There are also external forces exerted on the bodies within the system, by bodies outside the
system. Since their corresponding reaction forces act on bodies outside the system, you do not
need to consider these reaction forces. For the two-box system example given above, each box
has its weight, which is the gravity force provided by the earth. The earth is not a part of the
system under consideration, therefore, in this case, the weight is an external force.
3. Draw and label each and every relevant force vector on a free-body diagram. Orient each force
vector according to its direction. Sometimes, if you have a sense of the relative magnitudes of the
forces, then, draw the lengths of the force vectors accordingly. What I mean is, if you know that
Force #1 is larger than Force #2, then you will draw a longer vector for Force #1. Assign a symbol for
each force, known or unknown.
As my preferred approach, I always identify one point on a body, typically the center of mass of the
body, from which all force vectors are drawn. As you will see in the future when we study the
rotational motion of rigid bodies, I will modify this approach.
4. Based on the directions of the forces, select the most convenient directions for the coordinate axes.
Why? Because the next step is to decompose all the vectors into their x-components and ycomponents. It is true that you can assign the coordinate system in any way you want. However, if
you orient the coordinate axes wisely, you can simplify the work of calculating the force
components (for example, reducing the number of components you will get), and, as a result, the
Newton’s Law equations you will set up later will be simpler (contain fewer terms and symbols, and
easier to solve). I will illustrate this point during our lectures.
5. Decompose all the force vectors into their x-axis and y-axis components in the selected coordinate
system. Pay special attention to the directions of the vectors to determine if the components are
positive or negative.
6. Now, you can set up a pair of equations for each body:
F
ix
 ma x
(along the x-axis)
 ma y
(along the y-axis)
i
F
iy
i
where m is the mass of the body and the summation is over the index i which is used to label the
forces acting on the body under consideration in the corresponding free-body diagram.
These two equations are the same as the following two for a case in which there are 4 forces (i = 1,
2, 3, or 4) acting on this body:
F1x + F2x + F3x + F4x = max
(along the x-axis)
F1y + F2y + F3y + F4y = may
(along the y-axis)
The acceleration can be either zero or non-zero.
7. If your system contains more than one relevant body, you need to draw one free-body diagram for
each body. For each body, you need to formulate one set of equations described above. These sets
of equations constructed for all the relevant bodies in the system can be used to solve for unknown
force(s) and/or accelerations.
8. There are steps you may consider that can help reducing the number of symbols needed to label
forces and accelerations. This also makes the equations simpler, since the number of variables in
the equations is reduced. As an obvious example, I use the same symbol to represent a pair of
action-reaction force acting on a pair of interaction bodies. As another example, if two bodies in a
system are connected by a non-stretchable rope, then, often, the accelerations of these two bodies
have equal magnitude (but the directions may be different). In this case, you may use the same
symbol to represent the magnitude of the accelerations for these two bodies. But, you should pay
attention to the directions of the accelerations, since the components of the accelerations depend
on the direction of the acceleration. I will illustration this point during our lectures.