VIEWS: 9 PAGES: 28 CATEGORY: College POSTED ON: 10/25/2012
Topic 1 (Cont.) Vectors Lecture Outline • Vectors and Scalars • Presentation of Vectors • Addition and Subtraction of vector • Component of Vector 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 Presentation of Vectors • On a diagram, each vector is represented by an arrow • Arrow pointing in the direction of the vector • Length of arrow is proportional to the magnitude of the vector • Symbol V or V • Magnitude of the vector: V or V Addition of Vectors—Graphical Methods 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. 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. Adding the vectors in the opposite order gives the same result: Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method. The parallelogram method may also be used; here again the vectors must be tail-to-tip. Subtraction of Vectors In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction. Then we add the negative vector. Multiplication of a Vector by a Scalar A vector V can be multiplied by a scalar c; the result is a vector c V that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction. ConcepTest 3.1a Vectors I 1) same magnitude, but can be in any If two vectors are given direction such that A + B = 0, what 2) same magnitude, but must be in the same can you say about the direction magnitude and direction 3) different magnitudes, but must be in the same direction of vectors A and B? 4) same magnitude, but must be in opposite directions 5) different magnitudes, but must be in opposite directions ConcepTest 3.1a Vectors I 1) same magnitude, but can be in any If two vectors are given direction such that A + B = 0, what 2) same magnitude, but must be in the same can you say about the direction magnitude and direction 3) different magnitudes, but must be in the same direction of vectors A and B? 4) same magnitude, but must be in opposite directions 5) different magnitudes, but must be in opposite directions The magnitudes must be the same, but one vector must be pointing in the opposite direction of the other in order for the sum to come out to zero. You can prove this with the tip-to-tail method. ConcepTest 3.1b Vectors II Given that A + B = C, and 1) they are perpendicular to each other that lAl 2 + lBl 2 = lCl 2, 2) they are parallel and in the same direction how are vectors A and B 3) they are parallel but in the opposite oriented with respect to direction each other? 4) they are at 45° to each other 5) they can be at any angle to each other ConcepTest 3.1b Vectors II Given that A + B = C, and 1) they are perpendicular to each other that lAl 2 + lBl 2 = lCl 2, 2) they are parallel and in the same direction how are vectors A and B 3) they are parallel but in the opposite oriented with respect to direction each other? 4) they are at 45° to each other 5) they can be at any angle to each other Note that the magnitudes of the vectors satisfy the Pythagorean Theorem. This suggests that they form a right triangle, with vector C as the hypotenuse. Thus, A and B are the legs of the right triangle and are therefore perpendicular. ConcepTest 3.1c Vectors III Given that A + B = C, 1) they are perpendicular to each other and that lAl + lBl = 2) they are parallel and in the same direction lCl , how are vectors 3) they are parallel but in the opposite direction A and B oriented with 4) they are at 45° to each other respect to each other? 5) they can be at any angle to each other ConcepTest 3.1c Vectors III Given that A + B = C, 1) they are perpendicular to each other and that lAl + lBl = 2) they are parallel and in the same direction lCl , how are vectors 3) they are parallel but in the opposite direction A and B oriented with 4) they are at 45° to each other respect to each other? 5) they can be at any angle to each other The only time vector magnitudes will simply add together is when the direction does not have to be taken into account (i.e., the direction is the same for both vectors). In that case, there is no angle between them to worry about, so vectors A and B must be pointing in the same direction. Adding Vectors by Components Any vector can be expressed as the sum of two other vectors, which are called its components. The process of finding the component is known as resolving the vector into its component. Because x and y axis is perpendicular, they can be calculate using trigonometric functions. The components are effectively one-dimensional, so they can be added arithmetically. Adding vectors: 1. Draw a diagram 2. Choose x and y axes. 3. Resolve each vector into x and y components. 4. Calculate each component using sines and cosines. 5. Add the components in each direction. 6. To find the length and direction of the vector, use: and . Example 3-2: Mail carrier’s displacement. A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office? Example 3-3: Three short trips. An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45°) for 440 km; and the third leg is at 53° south of west, for 550 km, as shown. What is the plane’s total displacement? ConcepTest 3.2 Vector Components I 1) it doubles If each component of a 2) it increases, but by less than double vector is doubled, what 3) it does not change happens to the angle of 4) it is reduced by half that vector? 5) it decreases, but not as much as half ConcepTest 3.2 Vector Components I 1) it doubles If each component of a 2) it increases, but by less than double vector is doubled, what 3) it does not change happens to the angle of 4) it is reduced by half that vector? 5) it decreases, but not as much as half The magnitude of the vector clearly doubles if each of its components is doubled. But the angle of the vector is given by tan q = 2y/2x, which is the same as tan q = y/x (the original angle). Follow-up: If you double one component and not the other, how would the angle change? ConcepTest 3.3 Vector Addition You are adding vectors of length 1) 0 20 and 40 units. What is the only 2) 18 possible resultant magnitude that 3) 37 you can obtain out of the 4) 64 following choices? 5) 100 ConcepTest 3.3 Vector Addition You are adding vectors of length 1) 0 20 and 40 units. What is the only 2) 18 possible resultant magnitude that 3) 37 you can obtain out of the 4) 64 following choices? 5) 100 The minimum resultant occurs when the vectors are opposite, giving 20 units. The maximum resultant occurs when the vectors are aligned, giving 60 units. Anything in between is also possible for angles between 0° and 180°.