Lecture #4: Vector Addition

Transcription

1 Lecture #4: Vector Addition ackground and Introduction i) Some phsical quantities in nature are specified b onl one number and are called scalar quantities. An eample of a scalar quantit is temperature, and another is mass. ii) Other phsical quantities in nature are specified b two or more numbers taken together. An eample, is the displacement in two dimensions which requires the magnitude as well as the direction of the motion. Force and velocit are two other eamples of vector quantities. Mostl in this lecture the displacement vector will be used as the eample of a vector. iii) It is possible to write the laws of nature without using vectors (and this was done until the late 1800's). However, the laws of nature are more simpl written and understandable using vector notation.

2 2 4.VectorAddition rev.nb Definitions: 1. Thus far we have discussed motion onl in one dimension because a) motion in one dimension occurs in nature and b) one dimensional problems are simpler to understand. 2. The concept of displacement in one dimension has some aspects of vectors. The displacement is defined D= 2-1 a) The size or magnitude of D is given b D. Eample D =5 meters means ou could either walk 5 meters in the + direction or 5 meters in the - direction. b) The sign of D is also important. Eample D = -5 meter The vector concept is a generalization of these concepts to two dimensions and three dimensions.

3 4.VectorAddition rev.nb 3 Addition of Displacement Vectors in One Dimension Suppose at first we have a displacement in the + -direction of 2 m awa from the origin as indicated below b the vector smbol A A

4 4 4.VectorAddition rev.nb Following the first displacement, suppose a second displacement of magnitude + 1 m is also in the positive direction. A This is clearl equivalent to a single displacement of magnitude 3 m in the + direction from the origin and is written as A + A + A

5 4.VectorAddition rev.nb 5 So the sum of two displacements A + is itself a displacement.

6 6 4.VectorAddition rev.nb The Order of the Displacements. Doing the displacement first and following that with the A displacement is written + A. You can convince ourself + A=A + because ou wind up in the same spot. The order of the vector addition is not important. +A A

7 4.VectorAddition rev.nb 7 The Effect of the Sign of the Displacement What happens if one of the displacements is negative? Eample = m which a displacement in the negative direction and A the same. Answer: A + = (2)m= 1 m. Graphicall ou have in this case A+ So A + in this case has its tail at the origin and tip at meters.

8 8 4.VectorAddition rev.nb Equal Vectors: The magnitude and sign (direction) of the displacement are all that is important in defining a vector. All the vectors below are considered equal because the all have length 2 meter and are in the positive direction.

9 4.VectorAddition rev.nb 9 The Graphical Procedure for Addition of Two Vectors: a) You add vector A to vector b moving vector parallel to itself until the "tail" of is at the tip of the vector A. The sum of the two vectors A + is drawn from the "tail" of A to the tip of. b) Alternativel, ou can add vector A to vector b moving vector A parallel to itself until the "tail" of A is at the tip of the vector. The sum of the two vectors + A is drawn from the "tail" of to the tip of A. c) It should be clear that A + = + A and this holds in two and three dimensions as well. d) General Propert of Vectors: If ou have a vector A having a given magnitude and direction, then ou can move vector A anwhere and as long as the magnitude and direction are the same, ou have the same vector A.

10 10 4.VectorAddition rev.nb Addition of Two Vectors NOT in the Same Direction: Suppose the vector A has length 3m in the positive -direction and the vector has length 2 and is in the positive -direction. The sum of the two vectors A+ is indicated in the diagram below: R= A + q A

11 4.VectorAddition rev.nb 11 Notice the Following: 1. The sum of two vectors A and is itself a vector since ou can walk directl from the origin to the tip of. The sum of two vectors is called the Resultant Vector R=A+. Since R is a vector, its magnitude R and direction q are specified (see item #4 below). 2. Graphical Method for Adding Vectors: Add vector A to vector b moving vector parallel to itself until the "tail" of is at the tip of the vector A. The sum of the two vectors A + is drawn from the "tail" of the first vector A to the tip of the second vector. 3. The order of addition of A and can be reversed and ou will get the same Resultant Vector R=+A. So the order two vectors are added is unimportant A + = + A. 4. The length or magnitude of the Resultant Vector R is computed using the Pthagorean Theorem that is R = H3 ml 2 + H2 ml 2. Using Mathematica we compute the numerical value in this case So the length of the Resultant Vector is R =3.6 m. 3. The direction q of the Resultant Vector is the angle q with respect to the direction indicated in the diagram above. The angle q can be computed using the Tangent function or Tan for short and in this case Tan@qD = H ê L = 2 m =0.67 The angle q is obtained from the ArcTan function since 3 m q=arctan[(/)]=arctan[0.67]=33.7 since =3 m and = 2m in the eample. If ou have Mathematica do the calculation ou get ArcTan@2 ê 3.D * p

12 12 4.VectorAddition rev.nb so the angle The factor (360/2p) converts from angular measure in radians to degrees. The are 360 in a circle an equivalentl there are 2p Radians.

13 4.VectorAddition rev.nb 13 The Graphical Method of Adding Two Vectors Not at Right Angles: (Also called the "Ruler/Protractor and Paper/Pencil Method".) Suppose there are two vectors A and as indicated below and ou wish to add them together to get A +. A

14 14 4.VectorAddition rev.nb The first thing to do is move the vector parallel to itself (so the angle does not change) until the tail of is a the tip of A. A

15 4.VectorAddition rev.nb 15 Remember this is the same vector provided the length and angle remain the same. The resultant vector A + is drawn from the tail of the first vector (that is A ) to the tip of the second vector (that is ). A + A The length of the vector A + can be measured with a ruler and the angle q measured with a protractor. The graphical method of adding vectors is useful to gain an intuition but mostl we will use another method. This method uses the components of a vector.

16 16 4.VectorAddition rev.nb Rectangular Components of a Vector: Consider the vector C indicated in the diagram below C q You can get vector C b adding on vector C in the direction and having length 1 m to another vector C in the direction and having a length 3 m. C C q C

17 4.VectorAddition rev.nb 17 in other words, C=C +C. C and C are called the rectangular components of the vector C. In the eample above the length C =1 and the length C =3.

18 18 4.VectorAddition rev.nb Unit Vectors: ` and ỳ The unit vector ` is a vector of length one meter in the direction. The reason it is called a "unit" vector is because the length of ` is one. Similarl, the unit vector ỳ is a vector of length one meter in the direction. The unit vectors ` and ỳ are drawn in the diagram below. ỳ ` Using the unit vector ` ou can write C = C ` where C is the length C of the vector C. In the eample above C =1 and C = 1 `. Similarl, using the unit vector ỳ ou can write C =C ỳ where C is the length C of the vector C. In the eample above C =3 and C =3 ỳ. Also ou can write, C=C `+C ỳ =1 ` ỳ in this eample. NOTATION: Some books use ì and j` instead of ` and ỳ for the units vectors.

19 4.VectorAddition rev.nb 19 Addition of Two Vectors Using Unit Vectors Consider the addition of two vectors A and not perpendicular to each other A This time let us add the two vectors A + using the unit vector concept. First write vectors A and using the unit vector notation and get A=2 `+ 0 ỳ and =1 ` ỳ. Net add the - components of vectors A and and the components of vectors A and to obtain A + = J2 ` + 0 ỳ M + J1 ` + 2 ỳ M = ` J2 L + H 0 + 2L ỳ = 3 ` + 2 ỳ so the resultant vector is A + =3 ` + 2 ỳ. The length of the resultant vector (or sum) is A + = = 13 = 3.6 The angle q that the vector makes with respect to the ais is given b Tan[q]=(2/3)=0.67 so q=34.

20 20 4.VectorAddition rev.nb Another Eample: Adding Two Vectors A and First add the two vectors A+ using the graphical (or ruler and protractor) method A Move the vector parallel to itself until the tail of is at the tip of A obtaining A

21 4.VectorAddition rev.nb 21 The vector sum A+ goes from the tail of A to the tip of A A+ So from the diagram we see that the vector sum A+ has an -component of and the component of.

22 22 4.VectorAddition rev.nb You can get the same answer from the vector sum + A b first moving the vector A parallel to itself until the tail of A is at the tip of obtaining A A The vector sum + A goes from the tail of to the tip of A + A A

23 4.VectorAddition rev.nb 23

24 24 4.VectorAddition rev.nb The two vectors A and above can also be added using components and unit vectors. First write A and in terms of their components obtaining get A=2 `+ 1 ỳ and = ` - 2 ỳ. Net add the -components of vectors A and and the components of vectors A and to obtain A + = J2 ` + 1 ỳ M + J ` - 2 ỳ M = ` J2 L + H 1-2L ỳ = ` - 1 ỳ so the resultant vector is A + = ` - 1 ỳ. The length of the resultant vector (or sum) is A + = HL 2 + HL 2 = 2 = 1.41 The angle q that the vector makes with respect to the ais is given b Tan[q]=()/()=1 so q=45.

25 4.VectorAddition rev.nb 25 General Procedure for Adding Two Vectors using Components Here a "cookbook" procedure is given for adding two vectors and obtaining the resultant vector, R = A Resolve the vector A into its components A and A so that A= A + A. 2. Resolve the vector into its components and so that = Add the -component of A to the -components of to obtain the -component of the resultant R so R =A Add the -component of A to the -components of to obtain the -component of the resultant R so R =A Add the -component of R to the -components of R to obtain the resultant R so R=R +R. Technicall ou are now done but ou might also be asked to 6. Component the magnitude of the resultant R = R 2 + R 2 and/or 7. The angle q of the resultant using q=arctan[/].

26 26 4.VectorAddition rev.nb Vectors in Other Dimensions One Dimension: A vector in one dimension is just one number but usuall a vector in one dimension is called a scalar. It has a magnitude (length) and direction (the sign) but usuall this is consider just one number which can have positive and negative values. Three Dimensions: A vector A in three dimension has three components: A, A, and A z. There are three unit vectors in three dimensions namel `, ỳ, and z` and A can be written as A =A `+A ỳ+a z z` in three dimensions. We will seldom discuss motion in three dimensions because two dimensions has most of the complications of three dimensional problem. Also, it is more difficult to draw three dimensional figures.

27 4.VectorAddition rev.nb 27 Some Trigonometric Functions You Need to Know: Review Trigonometric functions are tabulated relations that allow ou to use the concept of similar triangles easil in phsics. Similar triangles are two triangles of different size but have the same three angles. The right triangle below has a 90 angle b definition. An angle of 90 angle is the same as an angle of p/2 Radians. The other two angles of the triangle add to a+q=90. The side h is opposite the 90 angle and is known as the hpotenuse. a b q 90 Î a The Tangent function (or Tan for short) given b Tan@qD = b a = opposite side adacent side where b is the side opposite the q angle and a is the side adjacent or net to the q angle. The Sine (or Sin for short) function is given b Sin@qD = b h = opposite side hpotenuse and the Cosine (or Cos for short) function is given b Cos@qD = a h = adacent side hpotenuse

28 28 4.VectorAddition rev.nb Also the Pthagorean Theorem is the relation h 2 = a 2 + b 2 The inverse trignometric functions are call ArcTan, ArcSin, and ArcCos. The are equivalent to the InverseTangent, InverseSine, and InverseCos functions. On some calculators these inverse functions are written Tan, Sin, and Cos respectivel. Rodne L. Varle (2010).