THE COMPLETE GUIDE TO PERFECT FO

1) Draw a picture for the relevant objects in the problem. The picture should include the object, the situation, the direction of velocity (labeled), and the direction of the acceleration (labeled).

2) Draw a separate force diagram for each mass, with the mass represented as a point. Forces should be labeled as arrows on the force diagram with mnemonic symbols. If there is more than one mass, you MUST use mnemonic subscripts appropriate to that mass. We've only encountered the following five forces so far:

symbol	force	direction
mg	*Weight* (gravitational force of earth on an object, a *non-contact* force)	on Earth, directed toward the center of the Earth (later we will see that gravitational forces act along the line joining the centers of the two objects involved)
N	*Normal* (subscripts can be used for the object that applies the force, particularly if there is more than one normal force in the problem)	perpendicular to the plane of the contact surface; always a "push" force
T	*Tension*	along the string, cord, etc.; always a "pull " force; the tension in a given string is everywhere the same because the string is massless!)
*f_k*	*Kinetic friction* ( f_k = u_kN)	opposite to the direction of the relative motion
*f_s*	*Static friction* (f_s< u_sN)	opposite to direction of the relative motion that would have occurred had there been no friction

a) If the acceleration of the object is 0, make sure that there are opposing forces both horizontally and vertically (or parallel to the plane and perpendicular to the plane, if the problem is on an inclined plane) so that the net force could conceivably be zero.

b) If the acceleration is not zero, make sure that there is a net force that exists in the direction of the labeled acceleration.

c) If the object is moving in a circle, make sure that there is a net force toward the center of the circle.

3) Define positive in the direction of the acceleration on the force diagram. If the problem is two-dimensional, make sure that you have defined positive in two different (but mutually perpendicular) directions.

If there are two different masses in the problem, make sure that each one's positive direction is in the direction of its own acceleration even if that means that different directions are positive for different masses in the problem.

4) Decide on the two mutually-perpendicular directions for which you will write the F_net equations:

A) If the mass is accelerating, you MUST choose the acceleration direction as one of your two directions for writing the F_net equations; the other direction (in which the acceleration is zero) must be perpendicular to the acceleration direction.

Example 1: if an object is accelerating down along an inclined plane (a skier, for example), you must define + down along the inclined plane, and you must write the F_net equation by summing forces along along the incline. The other F_net equation must be written for the direction perpendicular to the first direction, i.e., perpendicular to the incline.

Example 2: if an object is moving in a circle on an inclined plane, (e.g., a race car on a banked speedway), you must choose to define + toward the center of the circle (i.e., horizontally), since the acceleration of the car is in that direction, and write the F_net equation for this direction. The other F_net equation must be written for the direction perpendicular to the first, i.e., for the vertical direction.

B) If the mass is not accelerating, you may define + any way you wish. It is usually easiest to choose 2 mutually perpendicular directions such that you have to break up the fewest forces into components.

5) If any forces are not exactly parallel to or perpendicular to the directions chosen for the F_net equations, break them up into their two components. Remember that the side opposite q is (hypotenuse) sin q and the side adjacent to q is (hypotenuse) cos q .

6) Write the F_net equations for each direction and for each mass. This means to add up the forces (noting which forces are + and which are -, using the direction defined to be +) along each direction, and then set F_net = ma. Label F_net 's and accelerations with H and V or ^ and || . Make sure that the F_net equations contain the identical letters/symbols that are used on your force diagram. Note that each F_net equation will have three parts: (a) it will start with F_net or S F (with subscripts appropriate to the direction or the mass involved), (b) a middle part that contains the symbols (N, mg, T, f_k, f_s ) for all the forces on the force diagram, and (c) end with ma (again, with subscripts appropriate to the direction or the mass involved). There may also be a fourth part, if the acceleration has the special values of either 0 or v²/r (if in circular motion).

7) Isolate the unknown: i.e., solve for the unknown in terms of the knowns, without using numbers.

8) Think about and try to understand your answer physically. For example, if you find that the allowable acceleration in a problem is a<m_sg, we first interpret " m_sg " as the maximum acceleration possible, and then try to understand in word, not formula, language why this maximum acceleration should depend on the coefficient of static friction and the strength of the gravitational field.

9) Substitute in numbers and units; check solutions (if problem is odd) in the back of the text.