|A. John Mallinckrodt Professor of Physics, Cal Poly Pomona|
A Method for solving problems using Newton's Second Law
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:
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.
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.
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" forcesforces 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:
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.
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.
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!
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.
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.
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
Step 10: Simplify the resulting equations and figure out where to go from here. This is the end of "the method."
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.