Physics 40 Chapter 3: Vectors

Cartesian Coordinate System Also called rectangular coordinate system x-and y- axes intersect at the origin Points are labeled (x,y)

Polar Coordinate System Origin and reference line are noted Point is distance r from the origin in the direction of angle θ, ccw from reference line Points are labeled (r,θ)

Polar to Cartesian Coordinates Based on forming a right triangle from r and θ x = r cos θ y = r sin θ

Cartesian to Polar Coordinates r is the hypotenuse and θ an angle tanθ = y x r = x + y 2 2 θ must be ccw from positive x axis for these equations to be valid

Example 3.1 The Cartesian coordinates of a point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point. Solution: r x y 2 2 2 2 = + = + = ( 3.50 m) ( 2.50 m) 4.30 m 1 2.50 m tan ( ) = 35.5 3.50 m θ = 180 + 35.5 = 216

Vector quantities have both magnitude and direction The length of the vector represents the magnitude of the vector. The orientation represents the direction or angle of the vector. A Example: velocity of 2 km/hr, 30 degrees north of east. 2 km/hr is the magnitude, 30 degrees north or east, the direction. Scalars only have magnitude: T = 82 degrees Celsius.

Writing Vectors Text books use BOLD. Hard to write bold! The vector is written with an arrow over head and includes both magnitude and direction. Magnitude (length of vector) is written with no arrow or as an absolute value: 2km 30 A = A = magnitude = 2km A= ( 2km, 30 o NE) v A= ( A, θ )

A particle travels from A to B along the path shown by the dotted red line This is the distance traveled and is a scalar The displacement is the solid line from A to B The displacement is independent of the path taken between the two points Displacement is a vector Vector Example

Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction A = B if A = B and they point along parallel lines All of the vectors shown are equal

WARNING! Vector Algebra is Weird!! Vectors have a different rules of operation for addition, subtraction, multiplication and division than ordinary real numbers!!!! Since vectors have magnitude and direction you can not always just simply add them!!!! A & B co-linear A & B NOT colinear

When you have many vectors, just keep repeating the process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector Adding Vectors The Graphical Method

Adding Vectors Head to Tail R=A + B = B + A When two vectors are added, the sum is independent of the order of the addition. This is the commutative law of addition A + B = B + A The sum forms the diagonal of a Parallelogram!

Subtraction of Vectors A = C + ( B)

Subtracting Vectors If A B, then use A+(-B) The negative of the vector will have the same magnitude, but point in the opposite direction Two ways to draw subtraction: How do you go from A to -B?

The distance between two vectors is equal to the magnitude of the difference between them! More on this later

Parallelogram Method Addition & Subtraction R= X + Y R= X Y

R= A+ B R A B 1. Draw 1 st vector tail A at origin 2. Draw 2 nd vector B with tail at the head of the 1 st vector, A. The angle of B is measured relative to an imaginary axis attached to the tail of B. 3. The resultant is drawn from the tail of the first vector to the head of the last vector. R= A B A R B OR R B A

Adding Vectors The Graphical Method Draw vectors to scale. Draw the vectors Tip to Tail. IMPORTANT: the angle of a vector is relative its own tail! The resultant, R, is drawn from the tail of the first to the head of the last vector. Use a ruler to MEASURE the resultant length. Use a protractor to MEASURE the resultant angle.

Adding Vectors: Graphical Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find graphically (a) A + B, (b) A B, (c) B A Report all angles counterclockwise from the positive x axis. Use a scale of: 1 unit= 0.5 m 5.2m,60 o 3m,330 o 3m,150 o

Vector Components Ax = Acosθ A A y = Asinθ A θ A 2 2 x y A = A + A A = tan -1 y A ( ) x y A= A,A A= ( Aθ, ) x A

Example If the magnitude of r is 175m, find the magnitude of components x and y. x = r = r = x cosθ 175m cos50 r o x = r = r = y sinθ 175msin 50 r o x = 113m y = 134m

Vector Addition Components Method C = A+ B

Vector Components C = A+ B C tan 2 2 x y C = C + C θ C -1 y = C x C=A+B y y y C=A+B x x x

Unit Vectors A unit vector is a dimensionless vector with a magnitude of exactly 1. Unit vectors are used to specify a direction and have no other physical significance The symbols î, ĵ, and kˆ represent unit vectors They form a set of mutually perpendicular vectors (called a basis) Any vector with component A x and A y can be written in terms of unit vectors: A = A ˆi+ A ˆj + Akˆ x y z

Right Handed Coordinate System iˆ ˆj = kˆ

Vectors in 3-D The orientation of ijk doesn t matter as long as they form a right-handed system! r

Quiz Fun Which statement is true about the unit vectors i, j and k? a. Their directions are defined by a left-handed coordinate system. b. The angle between any two is 90 degrees. c. Each has a length of 1 m. d. If i is directed east and j is directed south, k points up out of the surface. e. All of the above.

Adding Vectors Using Unit Vectors R = A + B ( A ˆ ˆ) ( ˆ ˆ) x Ay Bx By R = i+ j + i+ j R = + ˆi+ + ˆj ( A ) ( ) x Bx Ay By R = R ˆi+ R ˆj R x = A x + B x and R y = A y + B y x y R = R + R θ = tan 2 2 1 x y R R y x

Adding Vectors: Component Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find using components (a) A + B, (b) A B, (c) B A, (d) A 2B. Report all angles counterclockwise from the positive x axis. Use a scale of: 1 unit= 0.5 m ( ) ˆi ( ) A = 3.00m cos 30.0 + 3.00msin 30.0 A = 2.60mˆi+ 1.50mˆj B = 3.00mˆj A+ B = (2.60 + 0)m ˆi+ (1.50 + 3.00)mˆj A+ B = 2.60mˆi+ 4.50mˆj ( ) 2 2 A+ B = 2.60m + (4.50m) = 5.20m 1 4.5m θ = tan = 60 2.6m o ˆj

Adding Vectors: Component Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find using components (a) A + B, (b) A B, (c) B A, (d) A 2B. Report all angles counterclockwise from the positive x axis. Use a scale of: A= 2.60mˆi+ 1.50m ˆj B = 3.00mˆj A B = (2.60 0)m ˆi+ (1.50 3.00)mˆj A B = 2.60mˆi 1.50mˆj ( ) 2 2 A B = 2.60m + ( 1.50m) = 3.00m 1 1.5m tan = 30 2.6m 1 unit= 0.5 m ( ) ˆi ( ) A = 3.00m cos 30.0 + 3.00m sin 30.0 ˆj o o o o, θ = 360 30 = 330

Adding Vectors: Component Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find using components (a) A + B, (b) A B, (c) B A, (d) A 2B. Report all angles counterclockwise from the positive x axis. Use a scale of: 1 unit= 0.5 m A = 2.60mˆi+ 1.50mˆj B = 3.00mˆj B A= (0 2.60)m ˆi+ (3.00 1.50)mˆj B A= 2.60mˆi+ 1.50mˆj ( ) 2 2 B A = 2.60m + (1.50m) = 3.00m 1 1.5m tan = 30-2.6m o o o o, θ = 180 30 = 150

Adding Vectors: Component Method Each of the displacement vectors A and B shown has a magnitude of 3.00 m. Find using components (a) A + B, (b) A B, (c) B A, (d) A 2B. Report all angles counterclockwise from the positive x axis. Use a scale of: 1 unit= 0.5 m A = 2.60mˆi+ 1.50mˆj 2B = 6.00mˆj A 2 B = (2.60 0)m ˆi+ (1.50 6.00)mˆj A 2B = 2.60mˆi 4.50mˆj ( ) 2 2 A 2B = 2.60m + ( 4.50m) = 5.2m 1 4.5m tan = 60 2.6m o o o o, θ = 360 60 = 300

The distance between two vectors is equal to the magnitude of the difference between them! Ch3. Two vectors in the xy plane have Cartesian coordinates (2.00, 4.00) m and ( 3.00, 3.00) m. Draw the vectors, write them with ijk vectors and find the distance between the points.

Mule Problem The helicopter view shows two people pulling on a stubborn mule The system is in equilibrium. Write the force vectors in ijk form for both people and find the net force they pull with. Then find the force, as an ijk vector and also magnitude and direction that the mule pulls back with. Forces are measured in units of newtons (N). 1 ( ) Nˆ ( ) F = 120cos 60.0 i+ 120sin 60.0 Nˆj = (60.0ˆi+ 104 ˆj)N F = 80.0cos 75.0 i+ 80.0sin 75.0 Nˆj 2 = ( 20.7ˆi+ 77.3 ˆj)N ( ) Nˆ ( ) 3 1 2 ( ˆ ˆ) F = F + F = 39.3i+ 181 j N ( ˆ ˆ) F = F= 39.3 i 181 j N 1 181 θ = tan = 77.8 39.3 SW F 2 2 = 39.3 + 181 = 185 N 1 181 θ = tan = 77.8 39.3

Problem Three displacement vectors of a croquet ball are shown in Figure P3.49, where A = 20.0 units, B = 40.0 units, and C = 30.0 units. Find (a) the resultant in unit vector notation and (b) the magnitude and direction of the resultant displacement. R x R y = 40.0cos 45.0 + 30.0cos 45.0 = 49.5 = 40.0sin 45.0 30.0sin 45.0 + 20.0 = 27.1 R = 49.5ˆi+ 27.1ˆj R ( ) ( ) 2 2 = 49.5 + 27.1 = 56.4 1 27.1 θ = tan = 28.7 49.5

4 Problem A person going for a walk follows the path shown. The total trip consists of four straightline paths. At the end of the walk, what is the personʹs resultant displacement measured from the starting point? d1 = 100mˆi d ˆ 2 = 300mj d ˆ ˆ ˆ ˆ 3 = 150cos 30.0 mi 150sin 30.0 mj= 130mi 75.0mj d = 200cos 60.0 mˆi+ 200sin 60.0 mˆj= 100mˆi+ 173mˆj ( ) ( ) ( ) ( ) 1 2 3 4 ˆ ˆ ( ) R = d + d + d + d = 130i 202 j m R ( ) ( ) 2 2 = 130 + 202 = 240 m 1 202 φ = tan 57.2 130 = θ = 180 + φ = 237

Next Lab! Force Table! R = A+ B 0 = A+ B+(- R ) B R A The Balancing Force! -R