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Guide 5-1a. Solving 1-Dimensional Net Force Problems

Net force problems are the second major type of problem we'll study. (DVATs were the first.)

Here's the encapsulated net force method for 1-dimensional problems
  1. Draw a picture of the situation and indicate the directions of the acceleration and velocity.

  2. Choose the directions for +x and +y.

  3. List the givens and the goal.

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

  5. Write net force equation.

  6. Apply Newton's Second Law.

  7. Solve the problem algebraically.

  8. Substitute values and units and calculate a numerical answer.

  9. Check your answer.

Details are given below. A step-by-step example is provided at the bottom.

Step 1

Draw a picture of the situation.

On your picture, draw and label a vector for each of the acceleration and velocity.

Step 2

Choose the directions for +x and +y.

An inconvenient choice for the positive direction 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 3

List the givens and the goal.

List all given values, including units. The direction that you select for +x (and +y for a 2-dimensional problem) will determine the signs of any given values of position, velocity, and acceleration. If there is more than one object of interest in a problem, 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

Isolate the object of interest and draw a force diagram.

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.

Step 5

Write the net force equation for each coordinate direction.

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 6

Apply Newton's 2nd Law.

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

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 values and units and calculate 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.)

Examples

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



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