In Unit 2 we studied the use of Newton's second law and free-body diagrams to determine the net force and acceleration of objects. In that unit, the forces acting upon objects were always directed in one dimension. There may have been both horizontal and vertical forces acting upon objects; yet there were never individual forces that were directed both horizontally and vertically. Furthermore, when a free-body diagram analysis was performed, the net force was either horizontal or vertical; the net force (and corresponding acceleration) was never both horizontal and vertical. Now times have changed and you are ready for situations involving forces in two dimensions. In this unit, we will examine the effect of forces acting at angles to the horizontal, such that the force has an influence in two dimensions - horizontally and vertically. For such situations, Newton's second law applies as it always did for situations involving one-dimensional net forces. However, to use Newton's laws, common vector operations such as vector addition and vector resolution will have to be applied. In this part of Lesson 3, the rules for adding vectors will be reviewed and applied to the addition of force vectors.
Methods of adding vectors were discussed earlier in Lesson 1 of this unit. During that discussion, the head to tail method of vector addition was introduced as a useful method of adding vectors that are not at right angles to each other. Now we will see how that method applies to situations involving the addition of force vectors.
The goal of a force analysis is to determine the net force and the corresponding acceleration. The net force is the vector sum of all the forces. That is, the net force is the resultant of all the forces; it is the result of adding all the forces together as vectors. For the situation of the three forces on the force board, the net force is the sum of force vectors A + B + C.
One method of determining the vector sum of these three forces (i.e., the net force") is to employ the method of head-to-tail addition. In this method, an accurately drawn scaled diagram is used and each individual vector is drawn to scale. Where the head of one vector ends, the tail of the next vector begins. Once all vectors are added, the resultant (i.e., the vector sum) can be determined by drawing a vector from the tail of the first vector to the head of the last vector. This procedure is shown below. The three vectors are added using the head-to-tail method. Incidentally, the vector sum of the three vectors is 0 Newton - the three vectors add up to 0 Newton. The last vector ends where the first vector began such that there is no resultant vector.
The purpose of adding force vectors is to determine the net force acting upon an object. In the above case, the net force (vector sum of all the forces) is 0 Newton. This would be expected for the situation since the object (the ring in the center of the force table) is at rest and staying at rest. We would say that the object is at equilibrium. Any object upon which all the forces are balanced (Fnet = 0 N) is said to be at equilibrium.
In addition to knowing graphical methods of adding the forces acting upon an object, it is also important to have a conceptual grasp of the principles of adding forces. Let's begin by considering the addition of two forces, both having a magnitude of 10 Newton. Suppose the question is posed:
How would you answer such a question? Would you quickly conclude 20 Newton, thinking that two force vectors can be added like any two numerical quantities? Would you pause for a moment and think that the quantities to be added are vectors (force vectors) and the addition of vectors follow a different set of rules than the addition of scalars? Would you pause for a moment, pondering the possible ways of adding 10 Newton and 10 Newton and conclude, "it depends upon their direction?" In fact, 10 Newton + 10 Newton could give almost any resultant, provided that it has a magnitude between 0 Newton and 20 Newton. Study the diagram below in which 10 Newton and 10 Newton are added to give a variety of answers; each answer is dependent upon the direction of the two vectors that are to be added. For this example, the minimum magnitude for the resultant is 0 Newton (occurring when 10 N and 10 N are in the opposite direction); and the maximum magnitude for the resultant is 20 N (occurring when 10 N and 10 N are in the same direction).
1. Barb Dwyer recently submitted her vector addition homework assignment. As seen below, Barb added two vectors and drew the resultant. However, Barb Dwyer failed to label the resultant on the diagram. For each case, that is the resultant (A, B, or C)? Explain.
4. Billie Budten and Mia Neezhirt are having an intense argument at the lunch table. They are adding two force vectors together to determine the resultant force. The magnitude of the two forces are 3 N and 4 N. Billie is arguing that the sum of the two forces is 7 N. Mia argues that the two forces add together to equal 5 N. Who is right? Explain.
Both Billie and Mia could be right. Yet with the lack of information about the direction of the two vectors, it is impossible to tell who is right. The only conclusion that we can make is that the sum of the two vectors is no greater than 7 N (if the two vectors were directed in the same direction) and no smaller than 1 N (if the two vectors were directed in opposite directions).
Vectors are commonly depicted as an arrow with an attached angle. This depiction is due to the fact that vectors are quantities with both magnitude and direction. The arrow represents its magnitude while its angle represents its direction. Some of the most common vectors that can be observed in our daily lives are acceleration, force, velocity, and displacement.
Success at this question demands an understanding of the head-to-tail method of vector addition (see Physics Rules section above). Analyze each diagram to see if the vectors are added using the head-to-tail principle. Check to insure that the resultant is drawn in the indicated direction. Rule out any choices that either fail to follow the head-to-tail approach or represent the resultant improperly.
Often times, the key to success on a physics question is a good game plan - an effective strategy. The best strategy for this question is to get out a writing utensil and some scratch paper and to add the two vectors.
Two or more vectors can be added using the head-to-tail vector addition method (see Physics Rules section below). The vectors are added such that the tail of one vector starts at the head of the previously drawn vector. Thus, a head leads into a tail (or a tail follows a head). However the resultant is the one vector in the diagram which is drawn from the tail of the first to the head of the last vector. In other words, the resultant vector has a tail which starts at the tail of a vector and a head which ends at the head of another vector. When you see a vector which has these characteristics, then you know it is the resultant.
Two or more vectors can be added using the head-to-tail vector addition method (see Physics Rules section above). The vectors are added such that the tail of one vector starts at the head of the previously drawn vector. Thus, a head leads into a tail (or a tail follows a head). However the resultant is the one vector in the diagram which is drawn from the tail of the first to the head of the last vector. In other words, the resultant vector has a tail which starts at the tail of the first vector and a head which ends at the head of the last vector. Once you have found the resultant, you can identify the starting vector and trace the order in which the three vectors were added.
When a vector diagram is used to add three vectors using the head-to-tail method, the resultant is drawn with a specific direction and orientation. The proper direction for drawing the resultant is ...
Let a and b be any two vectors. Let O be any point. Let us assume that OA and OB are two line segments such that OA = a and AB = b. Join O and B. Then OB is defined as the sum of the vectors a and b.
i.e. The sum of two co-initial vectors OA and OC is given by OB where OB is the diagonal of the parallelogram OABC having OA and OC as the adjacent sides.
Vectors are really accurate approximations of values with direction. They are often used to model motion of all types. From you walking down your street to pickup a gallon of milk to an astronaut blasting off in a rocket. There are many times when we need to model all of the forces that surround the motion of something. Such as a car moving on a road. The force of gravity exerts a resulting vector of drag of friction on the car. So, while the car can have a vector of its own to describe the motion another vector can be drawn to describe the frictional forces that are exerted on the car. You can then determine the overall net force by adding those two vectors together. This worksheet series shows students how to use vector addition to find the end result of two vectors which in motion is usually a measure of net force. 2b1af7f3a8