A. John Mallinckrodt   Professor of Physics, Cal Poly Pomona
Newton's
Second
Law
A Method for solving problems using Newton's Second Law
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Here is a systematic, 10-step method that will get you 90% of the way through virtually any problem that involves the use of Newton's Second law. The solution of the following sample problem will illustrate the steps of the method:

A 30 kg suitcase moves upward along a 20 degree incline under the influence of an applied horizontal force of 100 N. The kinetic coefficient of friction between the incline and the suitcase is 0.20. Find the acceleration of the suitcase.

Step 1: Construct a big schematic diagram of the physical situation. While reading and rereading the problem statement construct your diagram including every piece of information that you can extract from the statement on the diagram. Attach appropriate symbols to each important parameter in the problem whether the value of the parameter is known or not. Make straight lines straight, parallel lines parallel, perpendicular lines perpendicular, etc., to the best of your ability in order to avoid confusion later on.
Example at right. (Note how the following items extracted from the statement of the problem have been translated into specific items on the drawing: "... 30 kg suitcase ... moves upward ... 20 degree incline ... applied horizontal force of 100 N ... kinetic coefficient of friction ... is 0.20. ")

Step 2: Select a "system" to which you intend to apply Newton's Second Law. In some problems there may be more than one candidate for the "system." You may not choose the best one the first time. No problem; just choose another one and do it again.
Example: We will select the suitcase as our system because it is the thing to which many obvious forces are being applied and it is the thing whose acceleration we want to find.

Step 3: Identify all of the forces acting on "the system." Do this by drawing a dotted line around the system chosen in step 2 and identifying all physical objects that come in contact with the system. Each of these will exert a force on the system. Then look for "field" forces—forces that act without touching through the intermediary of a field of some sort. In introductory mechanics the only "field" force is the force of gravity. It is a force exerted by the earth (or some other very massive body) on the system through the intermediary of the gravitational field.
IMPORTANT! Every force on a system is exerted by some physical object outside the system. If you can't identify that object and the method of interaction (contact or field), the force DOES NOT EXIST!

The following are some commonly encountered forces and some tips on dealing with them:

• Ropes or strings These exert "tension" forces on the system. They are always directed away from the system along the direction of extension of the rope or string.

• Contacts with surfaces We generally break up the force due to contact with a surface into two components called the "normal"—meaning "perpendicular"—force and the "frictional" force. The normal force is generally a "push" type of force directed toward the system, unless the surface is sticky enabling it to exert a "pull" type of force. The frictional force is parallel to the surface, opposes motion or potential motion (i.e., a system on the verge of "slipping") and is often assumed to be related to the normal force through a "coefficient of friction." See a text for more details.

• Hinges or Pins These exert forces of arbitrary magnitude and direction as required to ensure that the point of attachment does not move.

• General pushes or pulls If a problem specifies that some object is being pushed or pulled in some direction, you may assume that the force specified is being exerted by some physical object. Don't forget to include it.

• Air resistance Air may not be visible, but it very likely does physically contact your system. Often we neglect air resistance because its effects are deemed negligible. However, if a problem specifies a certain amount of air resistance or tells you that the air resistance depends in some way on velocity or other parameters, don't forget to include it.

• Gravitational force The gravitational force—commonly called the "weight" of the system—is the only force that acts without physically contacting the system (at least until you learn about electric and magnetic forces later on.) It acts in the downward direction (by definition!) and is equal to the mass of the system times the local gravitational field strength g—commonly, but misleadingly called "the acceleration due to gravity."
Example at right. We find two objects in contact with the system—one surface and one "pusher." Thus we find a total of four forces—the normal force and the frictional force (from the surface), the push (from the pusher), and the weight (due to the only force—so far—that acts without needing to touch—gravity.)

Step 4: Draw a "free body diagram." The system may be represented by a simple circle or square; we want to focus our attention on the forces on and the resulting acceleration of the system. Draw each force with its tail at the surface of the system extending in the proper direction. Include the acceleration vector as well, but distinguish it from the force vectors by drawing it in a different looking form.
Example at right. Note that the normal force is directed perpendicular to the surface (not shown in the free body diagram), the frictional force is directed opposite the direction that the system slips with respect to the surface, the push is in its given direction, and the weight is directed "down." We show the acceleration as a different looking vector that is directed upward along the plane, but we don't know this for certain; it may be directed downward along the incline. We remind ourselves of this fact by putting a "(?)" next to the acceleration vector.

Step 5: Pick a coordinate system and determine the angles that the forces and accelerations make with the coordinate axes. It is usually "clever" to pick a coordinate system that minimizes the number of unknown vectors that will have to be broken down into components. The answers you obtain must and will be independent of your choice of coordinate system, but clever choices will yield equations that are more easily solved. You may need to do some geometrical scratch work on another sheet of paper to figure out how the angles are related to those given in the problem statement.
Example at right. We have chosen a coordinate system that requires us to determine the components of only the weight and the push—two forces about which we know a lot. Both of them lie at the angle theta (given in the problem statement as 20 degrees) from one of the axis directions.

Step 6: Write Newton's Second Law. It is the basic physical principle you are invoking; the "starting point" for your calculations. Just do it!
Example:

Step 7: Apply the basic equation to this problem. Simply write what the "sum of forces" is in this case. If the acceleration is zero, use that fact to simplify the equation too.
Example:

Step 8: Write the component equations. This is simply a matter of recognizing that every vector equation is shorthand for two (or, more generally, three) scalar equations. Simply rewrite the vector equation for each component direction with each vector quantity rewritten as the corresponding component.
Example:

Step 9: Determine what each component is in terms of the vector magnitude and trigonometric functions of the associated angles. In this step we explicitly indicate the signs of the vector components. This is also a good time to explicitly substitute "mg" for "W" if you happen to know the mass of the system
Example: Notice that the normal force is purely in the +y direction, the frictional force is purely in the -x direction, the push has a positive x-component and a negative y-component, the weight has negative x- and y-components, and the assumed acceleration is purely in the +x direction. Thus we have:

Step 10: Simplify the resulting equations and figure out where to go from here. This is the end of "the method."
Example:

Now you are on your own. What the method has done for you is to deliver a set of relationships between the magnitudes of the various forces that are required in order to satisfy Newton's second law. You now have "information" you didn't have before in the form of these equations. In some problems you may be able to get more information by applying the method to a different "system." What you do with the information will depend upon the specifics of the problem. Below we finish the sample problem.
In addition to the component equations we have the following relationship between the frictional force and the normal force:

So, solving the first component equation for a, using the relationship above for f, and substituting from the second component equation for n we get

Since a is supposed to be the magnitude of the acceleration and since magnitudes of vectors are always non-negative, this answer tells us that we chose the wrong direction for a; it must actually be directed down the incline meaning that the suitcase is slowing down. The answer is

 ©2001 by A. John Mallinckrodt ajm at csupomona.edu http://www.csupomona.edu/~ajm Many documents are provided in ".pdf" format. Click here to get the free Acrobat Reader application. The space for this page is provided by Cal Poly Pomona and is subject to its policies. Nevertheless, the opinions expressed here are my own and do not necessarily represent official policy of the University. I take full responsibility for the information presented and will appreciate being informed of errors or inaccuracies. Last modified on 5 February, 2004