The parallelogram method is a graphical method to add two vectors in the plane. It is frequently used to find the resultant of two forces applied to a body or of two speeds, as in the case of a swimmer who intends to cross a river perpendicularly and is deflected by the current.
To construct the parallelogram, the origins of the vectors to be added, drawn to scale, must coincide at a point.
Then auxiliary lines are drawn parallel to each vector, reaching to the end of the other, as shown in the figure above.
The sum or resultant vector, also called the net force, is the vector Fnet, which is obtained by drawing the vector that goes from the common origin of F1 Y Ftwo, to the point where the auxiliary parallel lines intersect. In the diagram of the figure these are represented by dotted lines.
The method gets its name from the figure that is formed with the addend vectors and the auxiliary lines, which is precisely a parallelogram. The main diagonal of the parallelogram is the sum vector.
It is very important to note that the order in which the addend vectors are placed does not alter the sum at all, since this operation between vectors is commutative.
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The following image shows the vectors v Y or in arbitrary units. The vector v measures 3.61 units and forms an angle of 56.3º with the horizontal, while or measures 6.32 units and an angle of 18.4º with respect to said reference line.
We are going to find its vector sum using the parallelogram method.
Choose an appropriate scale, such as the one shown in the following figure, in which the plane has been divided by a grid. The width of the square represents one (1) unit.
Since the vectors are not altered when translated, they are positioned in such a way that their origins coincide with the origin of the coordinate system (image on the left).
Now let's follow these steps:
The result can be seen in the right image, in which the resulting vector appears R.
If we want to know the magnitude of R, we can measure its length and compare it to the scale we have. And as for its direction, the horizontal axis or the vertical axis can be used as references, for example.
When using the horizontal axis or x axis, the angle that R shape with said axis is measured with the protractor and in this way we know the direction of R.
Likewise, the magnitude and direction of R can be calculated using the cosine and sine theorems, since the parallelogram formed can be divided into two congruent triangles, whose sides are the modules of the vectors or, v Y R. See worked example 1.
When the vectors are perpendicular to each other, the figure that is formed is a rectangle. The modulus of the resulting vector corresponds to the length of the diagonal, which can be easily calculated using the Pythagorean theorem.
We have the vector v, which measures 3.61 units and forms an angle of 56.3º with the horizontal, and the vector or, whose measure is 6.32 units and forms an angle of 18.4º (figure 2). Determine the modulus of the resultant vector R = or + v and the direction that the vector forms with the horizontal axis.
The parallelogram method is applied according to the steps described above, to obtain the vector R. As stated before, if the vectors are carefully drawn by following the scale and using the ruler and protractor, the magnitude and direction of R are measured directly on the drawing.
They can also be calculated directly, with the help of trigonometry and the properties of angles. When the triangle formed is not right, as in this case, the cosine theorem is applied to find the missing side.
In the triangle on the right, the sides measure u, v and R. To apply the cosine theorem it is necessary to know the angle between v Y or, that we can find with the help of the grid, appropriately positioning the angles provided by the statement.
This angle is α and is composed of:
α = (90-56.3º) + 90º + 18.4º = 142.1º
According to the cosine theorem:
Rtwo = vtwo + ortwo - 2u⋅v⋅cos α = 3.61two + 6.32two - 2 × 3.61 × 6.32 × cos 142.1º = 88.98
R = 9.43 units.
Finally, the angle between R and the horizontal axis is θ = 18.4 º + γ. The angle γ can be found using the sine theorem:
sin α / R = sin γ / u
Therefore:
sin γ = v (sin α / R) = 3.61 x (sin 142.1º / 9.43)
γ = 13.6º
θ = 18.4º + 13.6º = 32º
A swimmer is about to cross a river swimming perpendicular to the current with a constant speed of 2.0 m / s. The swimmer starts from A, however ends up at B, a point downstream, due to the current that diverted him.
If the speed of the current is 0.8 m / s and all speeds are assumed to be constant, find the speed of the swimmer as seen by an observer standing on the shore.
An observer standing on the shore would see how the swimmer is deflected according to the resulting speed VR. To find the answer we need to add vectorially the speed of the swimmer with respect to the water and the speed of the current, which we call V River:
V R = V swimmer + V River
In the figure, which is not to scale, the vectors were added to obtain V R. In this case, the Pythagorean theorem can be applied to obtain its magnitude:
VRtwo = 2.0two + 0.8two = 4.64
VR = 2.15 m / s
The direction in which the swimmer deviates from the perpendicular direction is easily calculated, noting that:
θ = arctg (2 / 0.8) = 68.2º
Then the swimmer deviates 90º - 68.2º = 27.2º from his original direction..
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