The free vectors are those that are fully specified by its magnitude, direction and sense, without being necessary to indicate a point of application or a particular origin.
Since infinite vectors can be drawn in this way, a free vector is not a single entity, but a set of parallel and identical vectors that are independent of where they are.
Figure 1. Various free vectors. Source: self made.
Let's say we have several vectors of magnitude 3 directed vertically upwards, or of magnitude 5 and inclined to the right, as in Figure 1.
Neither of these vectors is specifically applied at any point. Then any of the blue or green vectors is representative of their respective group, since their characteristics - module, direction and sense - do not change at all when they are transferred to another place in the plane.
A free vector is usually denoted in printed text by a bold, lowercase letter, for example v. Or with a lowercase letter and an arrow above it if it is handwritten text .
The advantage that free vectors have is that they can be moved through the plane or through space and maintain their properties, since any representative of the set is equally valid.
That is why in physics and mechanics they are used frequently. For example, to indicate the linear velocity of a solid that is translating it is not necessary to choose a particular point on the object. So the velocity vector behaves like a free vector.
Another example of a free vector is the pair of forces. A couple consists of two forces of equal magnitude and direction, but of opposite directions, applied at different points on a solid. The effect of a couple is not to move the object, but to cause a rotation thanks to the moment produced.
Figure 2 shows a couple of forces applied to a steering wheel. Through the forces F 1 and F 2, the torque is created that rotates the flywheel around its center and in a clockwise direction.
Figure 2. A couple of forces applied to a steering wheel gives it a clockwise turn. Source: Bielasko.
You can make some changes to the torque and still get the same rotating effect, for example increasing the force, but decreasing the distance between them. Or maintain the force and distance, but apply the torque on another pair of points on the steering wheel, that is, rotate the torque around the center.
The moment of the couple or simply couple, is a vector whose modulus is Fd and is directed perpendicular to the plane of the flywheel. In the example shown by convention the clockwise rotation has a negative direction.
Properties and characteristics
Unlike the free vector v, the vectors AB and CD are fixed (see figure 3), since they have a specified starting point and arrival point. But since they are team-lenient with each other, and in turn with the vector v, they are representative of the free vector v.
Figure 3. Free vectors, team lens vectors, and fixed vectors. Source: self made.
The main properties of free vectors are the following:
-Any vector AB (see figure 2) is, as said, representative of the free vector v.
-The module, the direction and the sense are the same in any representative of the free vector. In Figure 2, vectors AB and CD represent the free vector v and are team-lensing.
-Given a point P in space, it is always possible to find a representative of the free vector v whose origin is in P and this representative is unique. This is the most important property of free vectors and the one that makes them so versatile.
-A null free vector is denoted as 0 and is the set of all vectors that lack magnitude, direction and sense.
-If vector AB represents the free vector v, then vector BA represents the free vector - v.
-The notation V 3 will be used to designate the set of all free vectors in space and V 2 to designate all free vectors in the plane.
Solved exercises
With free vectors, the following operations can be performed:
-Sum
-Subtraction
-Multiplication of scalar by a vector
-Scalar product between two vectors.
-Cross product between two vectors
-Linear combination of vectors
And more.
-Exercise 1
A student tries to swim from one point on the bank of a river to another that is directly opposite. To achieve this, it swims directly at a speed of 6 km / h, in a perpendicular direction, however the current has a speed of 4 km / h that deflects it.
Calculate the swimmer's resultant speed and how much he is deflected by the current.
Solution
The resulting speed of the swimmer is the vector sum of his speed (with respect to the river, drawn vertically upwards) and the speed of the river (drawn from left to right), which is carried out as indicated in the figure below:
The magnitude of the resulting velocity corresponds to the hypotenuse of the right triangle shown, therefore:
v = (6 2 + 4 2) ½ km / h = 7.2 km / h
The direction can be calculated by the angle with respect to the perpendicular to the shore:
α = arctg (4/6) = 33.7º or 56.3º with respect to the shore.
Exercise 2
Find the moment of the pair of forces shown in the figure:
Solution
The moment is calculated by:
M = r x F
The units of the moment are lb-f.ft. Since the couple is in the plane of the screen, the moment is directed perpendicular to it, either outward or inward.
As the torque in the example tends to rotate the object on which it is applied (which is not shown in the figure) clockwise, this moment is considered to be pointing towards the inside of the screen and with a negative sign.
The magnitude of the moment is M = Fdsen a, where a is the angle between the force and the vector r. You have to choose a point with respect to which to calculate the moment, which is a free vector. The origin of the reference system is chosen, therefore r goes from O to the point of application of each force.
M 1 = M 2 = -Fdsen60º = -500. 20.sen 60º lb-f. ft = -8660.3 lb-f. foot
The net moment is the sum of M 1 and M 2: -17329.5 lb-f. foot.
References
- Beardon, T. 2011. An introduction to vectors. Recovered from: nrich.maths.org.
- Bedford, 2000. A. Engineering Mechanics: Statics. Addison Wesley. 38-52.
- Figueroa, D. Series: Physics for Sciences and Engineering. Volume 1. Kinematics. 31-68.
- Physical. Module 8: Vectors. Recovered from: frtl.utn.edu.ar
- Hibbeler, R. 2006. Mechanics for Engineers. Static 6th Edition. Continental Publishing Company. 15-53.
- Vector Addition Calculator. Recovered from: 1728.org
- Vectors. Recovered from: en.wikibooks.org