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Conventions for solving Dynamic Problems

  1. When breaking vectors into components: For the horizontal component, vectors pointing to the left are negative while vectors pointing to the right are positive.
  2. The vertical component of the vector are positive when pointing up and negative when pointing down.
  3. All objects are assume to be ideals, masses are points without shape (even when diagrams show shapes), strings are massless (if not indicated otherwise) and stretchless (no rubber band kind of properties), and pulley are massless and frictionless. Later on this notes, same of these restrictions will be lifted.

 

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Introduction to Free Body Diagrams (FBD).

The idea of the FBD is to replace the effect of objects in each other for the forces representing their actions. Therefore reducing to one object the diagram associated with the FBD resulting from a complex situation.

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Steps for solving problems involving Newton's equations.

  1. Read the problem carefully. Make sure that you understand the given data and the asked information.
  2. Select the order in which you are going to work the problem.
    1. Start by solving from the object to which less unknowns are associated with.
    2. Also, start solving for objects for which there is less number of forces applied.
  3. Draw the free body diagram for the object.
  4. Break every force and acceleration into their x and y components.
    1. The axis should be selected such that the minimal number of vectors need to be break down into components.
    2. If possible, select the x-axis along the horizontal component and the y-axis along the vertical component.
  5. From the diagram, obtain the net force acting on the selected object for each vector component, one for x-axis and another for y-axis components.
  6. As a direct application of Newton's second law, the previous net force is also related to the acceleration of the object for each individual component.
  7. Notice that results from step 5 and step 6 are obtained from two independent analysis. However, these results must be the same. Therefore, equating these two independent results for the net force an equation can be obtained.
  8. Repeat steps 3 to 7 as need it until all the unknown can be solve from the different equations.

 

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Weights hanging from Ceiling

In the figure, two weights are hanging from the ceiling. The first weight is 200 N, w1 = 200 N, and the second weight is 300N, w2 = 300N. Find the tensions T1 and T2.

The tension T2 is

 

a)

 500 N

 

b)

 100 N

 

c)

 200 N

N

d)

 300 N

 

e)

None of the above.

and the tension T1 is

N

a)

 500 N

 

b)

 100 N

 

c)

 200 N

 

d)

 300 N

 

e)

None of the above.

Solution:

This example is sufficiently simple to be solved without calculations. Thus, it is a good example to start understanding the method for solving problems involving Newton's laws of motion.

Following step two described above, it is simpler to solve for the bottom weight first, w2. The corresponding FBD is (Notice that the effect of the rope is replaced by the tension. This tension pulls up the object)

Free Body Diagram Object 2

In this problem, all vectors are along the vertical component. Therefore, it is not necessary to break the vectors into components.

Vertical Direction

Using step 5, the net force is the result of the vector addition between the tension of the rope (vector pointing up; then, positive) and the weight of the object (vector pointing down; then, negative).

Following step 6, the net force is related to the acceleration of the block. But the acceleration of the block is zero as indicated in the diagram. Therefore, .

Equating these two relations for the net force acting on object 2, the following equation is obtained

This equation can be immediately solved for T2

Vertical Direction

In this case, the net force is the result of the vector addition between the tension of the rope (vector pointing up; then, positive) and the vectors pointing down (vectors pointing down; then, negatives); the weight of the object and the tension of the bottom rope. Notice that T2 pulls on object one downward, for a review on tension go to the Tension section of these notes. Therefore, the net force is

Applying Newton's second law to the first weight,

Equating these two relations, the final equation is obtained

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com