Chapter 1-Part II Vectors Chapter Opener. Caption: This snowboarder flying through the air shows an example of motion in two dimensions. In the absence of air resistance, the path would be a perfect parabola. The gold arrow represents the downward acceleration of gravity, g. Galileo analyzed the motion of objects in 2 dimensions under the action of gravity near the Earth’s surface (now called “projectile motion”) into its horizontal and vertical components. We will discuss how to manipulate vectors and how to add them. Besides analyzing projectile motion, we will also see how to work with relative velocity.

Vectors and Scalars A vector has magnitude as well as direction. Some vector quantities: displacement, velocity, force, momentum A scalar has only a magnitude. Some scalar quantities: mass, time, temperature Figure 3-1. Caption: Car traveling on a road, slowing down to round the curve. The green arrows represent the velocity vector at each position.

Scalar vs. Vector Scalar Quantity A mathematical expression possessing only magnitude characterized by a positive or negative number

Scalar vs. Vector Vector Physical quantity that requires both a magnitude and a direction for its complete description. Must be added using Vector Operations

Examples: Mass, Volume Force, Velocity SCALARS AND VECTORS Scalars Vectors Examples: Mass, Volume Force, Velocity Characteristics: It has a magnitude It has a magnitude (positive or negative) and direction Addition rule: Simple arithmetic Various Methods Special Notation: None Bold font, a line, an arrow Statics:The Next Generation (2nd Ed.) Mehta, Danielson, & Berg Lecture Notes for Sections 2.1,2.2,2.4

Properties of Vectors A vector can be moved anywhere in a plane as long as the magnitude and direction are not changed. Two vectors are equal if they have the same magnitude and direction. Vectors are concurrent when they act on a point simultaneously. A vector multiplied by a scalar will result in a vector with the same direction and a larger magnitude. P F = ma vector scalar vector

Properties of Vectors (cont.) Two or more vectors can be added together to form a resultant. The resultant is a single vector that replaces the other vectors. The maximum value for a resultant vector occurs when the angle between them is 0°. The minimum value for a resultant vector occurs when the angle between the two vectors is 180°. The equilibrant is a vector with the same magnitude but opposite in direction to the resultant vector.

Components of a Vector A Vector Magnitude Vector Designation 500 N Head Tail Vector Designation 500 N

Adding Vectors Vectors can be added to each other graphically. Each vector is represented by an arrow with a length that is proportional to the magnitude of the vector. Each vector has a direction associated with it.

Using the Graphical Method of Vector Addition Vectors are drawn to scale and the resultant is determined using a ruler and protractor. Vectors are added by drawing the tail of the second vector at the head of the first (tip to tail method). The order of addition does not matter. The resultant is always drawn from the tail of the first to the head of the last vector.

Adding Vectors If the vectors occur in a single dimension, just add them. + = 4 m 7 m 3 m 7 m When adding vectors, place the tail of the second vector at the tip of the first vector.

Addition of Vectors—Graphical Methods You do need to be careful about the signs, as the figure indicates. Figure 3-2. Caption: Combining vectors in one dimension.

Addition of Vectors—Graphical Methods 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. Figure 3-3. Caption: A person walks 10.0 km east and then 5.0 km north. These two displacements are represented by the vectors D1 and D2, which are shown as arrows. The resultant displacement vector, DR, which is the vector sum of D1 and D2, is also shown. Measurement on the graph with ruler and protractor shows that DR has a magnitude of 11.2 km and points at an angle θ = 27° north of east.

Addition of Vectors—Graphical Methods Adding the vectors in the opposite order gives the same result: Figure 3-4. Caption: If the vectors are added in reverse order, the resultant is the same. (Compare to Fig. 3–3.)

Addition of Vectors—Graphical Methods Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method. Figure 3-5. Caption: The resultant of three vectors: vR = v1 + v2 + v3.

Addition of Vectors—Graphical Methods The parallelogram method may also be used; here again the vectors must be tail-to-tip. Figure 3-6. Caption: Vector addition by two different methods, (a) and (b). Part (c) is incorrect.

Parallelogram Law By drawing construction lines parallel to the vectors, the resultant vector goes from the point of origin to the intersection of the construction lines

Adding Vectors using the Pythagorean Theorem If the vectors occur such that they are perpendicular to one another, the Pythagorean theorem may be used to determine the resultant. = 3 m 4 m 5 m + 4 m 3 m A2 + B2 = C2 (4m)2 + (3m)2 = (5m)2 When adding vectors, place the tail of the second vector at the tip of the first vector.

Law of Cosines If the angle between the two vectors is more or less than 90º, then the Law of Cosines can be used to determine the resultant vector. 7 m 5 m + =   = 80º C C2 = A2 + B2 – 2ABCos  C2 = (7m)2 + (5m)2 – 2(7m)(5m)Cos 80º C = 7.9 m

Multiplication of a Vector by a Scalar A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction. Figure 3-9. Caption: Multiplying a vector v by a scalar c gives a vector whose magnitude is c times greater and in the same direction as v (or opposite direction if c is negative).

Example: P Resultant The vector shown to the right represents two forces acting concurrently on an object at point P. Which pair of vectors best represents the resultant vector? P P (a) (d) (c) (b) P P

Adding Vectors by Components Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other. Figure 3-10. Caption: Resolving a vector V into its components along an arbitrarily chosen set of x and y axes. The components, once found, themselves represent the vector. That is, the components contain as much information as the vector itself.

Adding Vectors by Components Remember: soh cah toa If the components are perpendicular, they can be found using trigonometric functions. Figure 3-11. Caption: Finding the components of a vector using trigonometric functions.