Chapter 4: How do we describe Vectors, Force and Motion? Objectives 4 To note that energy is often associated with matter in motion and that motion is controlled by forces. 4 To learn how to represent forces and motion by vectors. 4 To learn how to obtain the combined effect of two or more vectors acting upon the same point of a body 4 To understand how a vector can act in directions other than its own. 1Physics is Life

4-1 Change, Motion, and Force Practically all of the changes we see in the word about us are the result of motion. Examples are: Motion is controlled or changed by means of force. Examples are: 2Physics is Life u Day and night caused by rotation of earth on its axis u The winds and their effects are caused by motion of air. u When a car stalls, it must be pushed to get it going. u When the engine is running, it is the engine which pushes the car forward. u In order to make the car slow down ot to srope altogether, a force must be applied to the brakes.

4-2 Displacement is a Vector Motion generally involves a change of position of the object being moved. A change of position is called a displacment. We make a distinction between distance and displacement. Displacement (blue line) is how far the object is from its starting point, regardless of how it got there.. Distance traveled (dashed line) is measured along the actual path. 3Physics is Life

4 4-2 Displacement is a Vector

A study of motion will involve the introduction of a variety of quantities which are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories - vectors and scalars. A vector quantity is a quantity which is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity which is fully described by its magnitude.vectors and scalars Quantities such as displacements are called vectors. A vector is characterized by the fact that it has both magnitude and direction. 5Physics is Life

4-3 Velocity and Force are Vectors Two important vectors related to the study of motion are velocity and force. Velocity is a vector whose magnitude is the speed of the body and whose direction is the direction of the motion of the body. For example: If an airplane is traveling westward, its velocity is stated as 500 km/h westward. It is evident that a force is a vector, since the effect a force has on a body depends not only on the size of the force but also on the direction in which it acts. 6Physics is Life

4-4 Representing a Vector A vector is represented by an arrow drawn to some selected scale. The length of the arrow shows the magnitude of the vector. The direction of the arrowhead shows the direction of the vector. 7Physics is Life

8 Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram which make it an appropriately drawn vector diagram.vector diagrams a scale is clearly listed a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail. the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North) 4-4 Representing a Vector

4-5 Resultant of Two Vectors 4-6 Vectors acting in the Same Direction 4-7 Vectors Acting in Opposite Directions For vectors in one dimension, simple addition and subtraction are all that is needed. You do need to be careful about the signs, as the figure indicates. 9Physics is Life

Resultant of Two Vectors 4-6 Vectors acting in the Same Direction 4-7 Vectors Acting in Opposite Directions

Vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more north than east. Thus, there is a clear need for some form of a convention for identifying the direction of a vector which is not due East, due West, due South, or due North. There are a variety of conventions for describing the direction of any vector. The two conventions which will be discussed and used in this unit are described below 4-8 Vectors acting in Any Direction

1. The direction of a vector is often expressed as an angle of rotation of the vector about its "tail" from either east, west, north, or south. For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction)tail 2. The direction of a vector is often expressed as an counterclockwise angle of rotation of the vector about its "tail" from due East. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit.tail

Physics is Life Vectors acting in Any Direction Two illustrations of the second convention (discussed in last slide) for identifying the direction of a vector are shown below. This is a link where you can practice your vector directions: e.cfm e.cfm

4-9 General Method of Finding a Resultant If the motion is in two dimensions, the situation is somewhat more complicated. Here, the actual travel paths are at right angles to one another; we can find the displacement by using the Pythagorean Theorem. 14Physics is Life

4-11 Effect of the angle Between two vectors on the Resultant 4-12 Resultant of Three or More Vectors As the angle between two vectors increases, their resultant decreases (See Figure 4-8, page. 55). To find how we find the resultant of three or more vectors. We can add graphically by using the “tail-to-tip” method. 15Physics is Life

Another example of the use of the head-to-tail method is illustrated below. The problem involves the addition of three vectors: 20 m, 45 deg m, 300 deg m, 210 deg. SCALE: 1 cm = 5 m The head-to-tail method is employed as described above and the resultant is determined (drawn in red). Its magnitude and direction is labeled on the diagram. SCALE: 1 cm = 5 m 16Physics is Life

4-13 Demonstrating that Forces Combine Vectorially 4-14 Forces in Equilibrium 4-15 The Equilibrant The simplest way to neutralize a force acting on any point of a body is to apply an equal and opposite force to that point. Forces that cancel out each other’s effect are said to be in equilibrium. The single force that can neutralize the effects of two other forces acting on a given point of a body is called their equilibrant. (Figure 4-12, pg. 58) 17Physics is Life

Physcis is Life Demonstrating that Forces Combine Vectorially 4-14 Forces in Equilibrium 4-15 The Equilibrant (10.0 N) 2 + (15.0 N) 2 = (Resultant) 2 Resultant = 18.0 N Tan  =opp/adj  = tan -1 (10/15) = 36.7°

4-17 Relationships in Triangles The direction of a resultant vector can often be determined by use of trigonometric functions. Most students recall the meaning of the useful mnemonic SOH CAH TOA from their course in trigonometry. SOH CAH TOA is a mnemonic which helps one remember the meaning of the three common trigonometric functions - sine, cosine, and tangent functions. These three functions relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. 19Physics is Life

4-18 Single Vectors May be Broken into Components 4-19 Resolving a Vector into Components 4-20 Resolving Vectors into Perpendicular Components We have been adding or combining two or more vectors to find their resultant. The vectors combined in this way are called components 20Physics is Life

If the components are perpendicular, they can be found using trigonometric functions. 21Physics is Life 4-18 Single Vectors May be Broken into Components 4-19 Resolving a Vector into Components 4-20 Resolving Vectors into Perpendicular Components

Phsyics is Life Single Vectors May be Broken into Components 4-19 Resolving a Vector into Components 4-20 Resolving Vectors into Perpendicular Components MIT LECTURE SUMMARY ON VECTORS: or%20resolution&um=1&ie=UTF-8&sa=N&hl=en&tab=wv#q=vector+&hl=en&emb=0 or%20resolution&um=1&ie=UTF-8&sa=N&hl=en&tab=wv#q=vector+&hl=en&emb=0