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Teacher's Guide

Vectors: Motion and Forces in Tw o Dimen sions - Lesson 3

Forces in Two Dim ensions

Addition ofForces | Resolution ofForces | Equilibrium and Statics | Net Force Problems RevisitedInclined Planes | Double Trouble in 2 Dimensions

Addition of Forces

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 affect offorces 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.

A force board (or force table) is a common physics lab apparatus that has three (or more) chains or

cables attached to a center ring. The chains or cables exert forces upon the center ring in three

different directions. Typically the experimenter adjusts the d irection of the three forces, makes

measurements of the amount of force in each direction, and determines the vectorsum of three

forces. Forces perpendicular to the plane of the force board are typically ignored in the analysis.

Suppose that a force board or a force table is used such that there are three forces acting upon an

object. (The object is the ring in the center of the force board or force table.) In this situation, two of the forces are acting in

two-dimensions. A top viewof these three forces could be represented by the following diagram.

The goal of a force analysis is to determine the net force and the corresponding acceleration. The net force is the vectorsum 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 vectorsum 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 vectorsum ) 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 vectorsum 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

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