Guide 5-1a. Solving Net Force Problems

Here's the encapsulated net force method.
  1. List the givens and the goal.

  2. Draw a picture of the situation and indicate the directions of the acceleration and velocity.

  3. Isolate the object of interest and draw a force diagram.

  4. Choose the positive direction in the direction of the acceleration.

  5. Resolve the forces into components.

  6. Write net force equations and apply Newton's Second Law to each coordinate direction.

  7. Solve the problem algebraically.

  8. Substitute and solve for a numerical answer.

  9. Check your answer.

Details are given below.

Step 1. List the givens and the goal.


Step 2. Draw a picture of the situation and indicate the directions of the acceleration and velocity.


Step 3. Isolate the object of interest and sketch the forces.

Title your force diagram with the name of the object on which the forces are acting. Use a point to represent the object and vectors to represent the forces acting on the object of interest.  Label the forces using standard symbols:  N for normal, T for tension, f for friction, and mg or W for weight. We will see in Chapter 6 that surface friction forces need to be subscripted with s or k depending on the type of friction.

If there is more than one object of interest in a problem, then draw and label a force diagram for each object.  Use subscripts to clearly distinguish between the objects. Mnemonic subscripts are recommended. For example, if you had two objects side-by-side on a table, label the masses mL and mR for left and right. This makes it easier to read your work and for you to avoid confusing the objects.


Step 4. Choose a convenient coordinate system.

An inconvenient choice can make the algebra much more complicated than it need be or may lead you to an incorrect solution.  Generally follow this rule:  Select one of the positive axes to be in the direction of the acceleration.  This way, the acceleration will be positive and will have one component that is 0.  In some problems--sliding blocks on inclined planes, for example--this will mean that your axes won't be horizontal and vertical.  Instead, they'll be parallel and perpendicular to the plane.


Step 5. Resolve the forces into components.

Many net force problems are 2-dimensional.  So you'll need to determine the components of the forces.  Remember to do this symbolically.  As always, numbers aren't substituted until the last step.


Step 6. Apply Newton's Second Law to each coordinate direction.

Newton's Second Law may be stated in compact form as .  The net force and the acceleration are both vectors.  Be sure to use the 'net' subscript to distinguish the net force from the generic force symbol.  In component form:

In writing the Fnet equations, treat the individual force symbols as representing magnitudes (positive numbers) and explicitly indicate direction by placing a + or - sign in front of the force symbol. For example, for an object resting on a horizontal table with the positive direction defined to be up, the net force equation in the vertical direction is Fnet,y = N - W. In this equation, both N and W represent positive numbers. The - sign indicates that the direction of the weight is down.


Step 7. Solve the problem algebraically.

Setting up the problem as described above is the physics.  The rest is mostly algebra.  Having written your net force equations, solve them for the unknown(s).  Apply any specific conditions, constraints, or assumptions that are needed to solve the problem. Examples include equality of tension forces exerted by the same string on different objects, frictionless surfaces, massless and inextensible (unstretchable) strings, and massless and frictionless pulleys.  State such conditions and make it clear how you apply them. The result of this step will be a symbolic equation for the unknown.


Step 8. Substitute and solve for a numerical answer.

If you're given numerical values, substitute values with units into the result of Step 7 and reduce.


Step 9. Check your answer.

As always, check that the signs, units, and values in your final answer make sense.

Also, check that your algebraic result reduces to what you expect in special cases. (Examples of this will be provided later.)


Here's an example of a 1-dimensional problem,

and

here's an example of a 2-dimensional problem.