7 junio, 2024

Free body diagram: what it is, how to do it, examples, exercise

What is a free body diagram?

A Free-Body diagramisolated body diagram or force diagram, is a scheme where the forces acting on a body are represented by arrows.

It is necessary to make sure to include in the diagram all the forces that act on the object, and since it is a vector quantity, the arrow is in charge of indicating its direction and its sense, while the length of the same provides an idea of ​​the module or intensity.

In figure 1 we have an example of a free body diagram that we are going to analyze.

The situation is as follows: a traffic light hanging at rest from some cables (figure 1a). Two forces act on it, one is the one exerted by the Earth, which is weight. In the diagram it is denoted as Fg and acts vertically downward.

The other force is the tension in the vertical rope, called you3 and that goes vertically upwards, holding the traffic light and preventing it from falling to the ground.

When a problem has more than one object, then it is necessary to draw a diagram for each one separately.

The knot between the inclined ropes and the rope that holds the traffic light is considered a point object and its free body diagram is shown in Figure 1c. Note that for the knot, the tension you3 is directed downwards.

It is important to note that the free body diagram should not show the forces that the object exerts on other bodies, but only those that act on it.

free body diagram examples

The free body diagram allows the application of Newton’s laws and with them determine the state of motion or rest of the object on which the forces act. In the case of the traffic light shown, we can determine the value of the tensions in the cables that hold the traffic light, knowing its weight.

Once these data are known, suitable cables are selected to hang from the traffic light and to fulfill its function without collapsing.

Free body diagrams are used to describe various everyday situations, such as these:

A person pulling a trunk or container

It is very common for people to have to move heavy objects such as the container of the figure. To do this they must exert a force F on the container, which in this example is horizontal and to the right, which is the direction of movement.

But this is not the only force that acts on it, there is also the normal one no, exerted by the flat surface of the dolly. And finally there is the weight of it: Fg, directed vertically downwards.

The normal is a force that arises whenever two surfaces are in contact and is always perpendicular to the exerting surface. In this case, the dolly exerts a normal on the container.

A block sliding down an inclined plane

Some desks have the table slightly inclined to make it more comfortable to take notes and read. It also has a pencil holder slot, but we have all put the pencil on the table outside of the slot and seen how it slides on the table.

What forces act on the pencil?

The same ones that act on the block shown in the following free body diagram:

the normal FN is the force that the table top exerts on the pencil or the resting block. Unlike the previous example, the normal is not vertical, but inclined. Remember that the normal is the force that the table exerts on the block and is perpendicular to it. Since the table is tilted, the normal one too.

as always the weight Fg is vertical, regardless of the inclination of the system.

And finally we have a new force acting, which is kinetic friction Ffr between the table and the pencil or block. Friction is also a contact force, but unlike normal, it is a tangential (parallel) force to the surface. Also note that it is always directed in the opposite direction of motion.

Atwood’s machine

Atwood’s machine is a simple machine consisting of a light, frictionless pulley on the rail, through which a light, inextensible rope passes.

Two objects of masses m1 and m2 are hung from it. When one of the objects goes up, the other goes down, as shown in figure 4a:

Since there are two objects, a free-body diagram is made for each one separately. For both objects there are only two forces: the tension in the string you and the respective weights.

In the figure, each weight is expressed directly as the product of the mass times the acceleration. For its part, the tension is always directed vertically along the taut string.

solved exercise

Apply Newton’s laws to determine the acceleration with which the masses of the Atwood machine shown in the previous section move.

Solution

Newton’s second law states that the sum of forces is equal to the product of mass times acceleration.

The sign convention in each mass can be different, so we are going to take the direction of movement as positive, as the graph indicates, the first mass goes up and the second goes down.

In some problems, the statement does not provide information, so the signs must be assigned arbitrarily and if the result of the acceleration is negative, then the system of masses moves in the opposite direction to what was initially assumed.

-For dough 1 (rise):

T – m1g = m1a

-For dough 2 (low):

-T + m2g = m2a

Both equations form a system of linear equations of two unknowns, as the voltage appears with a different sign in each equation, we simply add them term by term and the voltage cancels:

m2g – m1g = m1a + m2a

a = m2g – m1g / (m1 + m2)

References

Bauer, W. 2011. Physics for Engineering and Science. Volume 1. Mc Graw Hill.
Giancoli, D. 2006. Physics: Principles with Applications. 6th. Ed Prentice Hall.
Serway, R., Vulle, C. 2011. College Physics. 9na Ed. Cengage Learning.
Tipler, P. (2006) Physics for Science and Technology. 5th Ed. Volume 1. Editorial Reverté.
Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. mcgrawhill

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