Physics Chapter 6 Vectors 6-1

Introduction:

If two strong people hold on to the ends of a light rope several meters long and both pull until it is straight it would be reasonable to assume that they could keep the rope straight if I placed a light mass in the center of the rope.

Lets give it a try. What happened and why did it happen? You will discover the answer to this question, as you understand the vector nature of forces. (Two Straight against 1V)

6.1 GRAPHICAL METHOD OF VECTOR ADDITION

A vector quantity can be represented by an arrow-tipped line segment. The length of the line, drawn to scale, represents the magnitude of the quantity. The direction of the arrow indicates the direction of the quantity. This arrow-tipped line segment represents a vector. Just as we can represent a vector graphically, we can add vectors graphically. They also can be represented in printed materials in boldface type, A, B.

Vector Addition in One Dimension

Suppose a child walks 200 m east, pauses, and then continues 400 m east. To find the total displacement, or change in position of the child, we must add the two vector quantities.

In Figure 6-1 a,A and B, drawn to scale, are vectors representing the two segments of the child's walk. The vectors are added by placing the tail of one vector at the head of the other vector. It is very important that neither the direction nor the length of either vector is changed during the process. A third vector is then drawn connecting the tail of the first vector to the head of the second vector. This third vector represents the sum of the first two vectors. It is called the resultant of A and B. The resultant is always drawn from the tail of the first vector to the head of the last vector.

When vectors are added, the order of addition does not matter. The tail of A could have been placed at the head of B. Figure 6-1 b shows that the same vector sum would result.

To find the magnitude of the resultant, R, measure its length using the same scale used to draw A and B. In this situation, the total change in position is 200 m east + 400 m east = 600 m east.

The two vectors can have different directions, Figure 6-1 c. If the child had turned around after moving 200 m east and walked 400 m west, the change of position would have been 200 m east + 400 m west or 200 m west.

Note that in both cases, the vectors are added head to tail, and the directions of the original vectors are not changed.

Vectors can be used to determine the resultant of forces as in the example below. The football team pulled with a force of 1500N to the left and the tennis team pulled with a force of 1200N to the right. What is the resultant force?

Vector Addition in Two Dimensions

So far we have looked at motion in only one dimension. Vectors can also represent motion in two dimensions. In Figure 6-2,A and B represent the two displacements of a student who walked first 95 m east (00)and then 55 m north (900).

The vectors are added by placing the tail of one vector at the head of the other vector. Always draw the east (00) vector first. The resultant of A and B is drawn from the tail of the first vector to the head of the second vector. To find the magnitude of the resultant, R, measure its length using the same scale used to draw A and B. Its direction can be found with a protractor. The direction is expressed as an angle measured counterclockwise from the horizontal. In Figure 6-2, the resultant displacement is 110 m at 300 . We will use 0 to 360 for direction.

Force vectors are added in the same way as position or velocity vectors. In Figure 6-3, a force, A, of 45 N and a force, B, of 65 N are exerted on an object at point P.

Force A acts in the direction of 600, force B acts at 00. The resultant, R, is the sum of the two forces. Vectors representing forces A and B are drawn to scale. R is found by moving A without changing its direction or length until the tail of A is located at the head of B. The resultant is drawn from the tail of the first vector, B, to the head of the second vector, A. As before, the magnitude of R is determined using the same scale used for A and B. The angle is again found with a protractor. In this case, R is 96 N acting in a direction of 240. A single force of 96 N acting in a direction of 240 will have exactly the same effect as two forces, 45 N at 600and 65 N at 00, acting at the same time.

Addition of Several Vectors

Often, more than two forces act at the same time on the same object. To find the resultant of three or more vectors, follow the same procedure you used to add two vectors. Just be sure to place the vectors head-to-tail. The order of addition is not important – unless you want to read from the horizontal as we do – so although R is the same in both cases – it is more consistent to draw the 00 vector first. In Figure 6-4a, the three forces, A, B, and C, are acting on point P. In Figures 6-4b and 6-4c, the vectors are added graphically. Note that the resultant is the same in both sketches although two different orders of addition are used. Remember, when placing vectors head-to-tail, the direction and length of each vector must not be changed.

Do practice problems 6-1

Independence of Vector Quantities

Perpendicular vector quantities are independent of one another. A motor boat heads east 0 0 at 8.0 m/s across a river that flows north 900 at 5.0 m/s. Starting from the West Bank, the boat will travel 8.0 m east in one second. In the same second, it also travels 5.0 m north. The velocity north does not change the velocity east. Neither does the velocity east change the velocity north. These two perpendicular velocities are independent of each other. Perpendicular vector quantities can be treated independently of one another.

In figure 6-5, the two velocities of the boat are represented by vectors. When these vectors are added, the resultant velocity, VR, is 9.4 m/s at 320. You can also think of the boat as traveling, in each second, east 8.0 m and north 5.0 m at the same time. Both statements have the same meaning.

Suppose that the river is 80 meters wide. Because the boat’s velocity is 8 m/s east, it will take the boat 10 seconds to cross the river. During this 10 seconds, the boat will also be carried 50 meters downstream. In no way does the downstream velocity change the velocity of the boat across the river, unless the boat is set on a heading against the stream.

Physics Chapter 6 Vectors 6-1

Do practice problems 6-2

Do Concept Review 6-1