EGR 1101 Unit 4

1. EGR 1101 Unit 4 Two-Dimensional Vectors in Engineering (Chapter 4 of Rattan/Klingbeil text)

2. Scalars versus Vectors • A scalar is a quantity that has magnitude only. Examples: • Mass • Temperature • A vector is a quantity that has magnitude and direction, and that obeys the triangle law of addition. Examples: • Velocity • Force

3. Component Form & Polar Form • Vectors are commonly written in two different forms. • In component form, a two-dimensional (2-D) vector is expressed as the sum of an x-component and a y-component. • In polar form, a 2-D vector is expressed as having a certain magnitude in a certain direction.

4. Other Names for Component Form • Component form is sometimes called rectangular form or Cartesian form.

5. Component Form • Suppose a vector has x-component and y-component . • Then we can write the vector in component form aswhere is the unit vector in the positive-x direction and is the unit vector in the positive-y direction.

6. Polar Form • Suppose a vector has magnitude v and angle . • Then we can write the vector in polar form as= v .

7. Converting Between Component & Polar Forms • Many problems involve converting from one form to the other. This is easy if you remember your basic trig. • From polar form to component form:vx=vcos()vy = v sin() • From component form to polar form:v = vx2 +vy2  = tan-1(vy / vx)

8. This Week’s Examples • Force on a vacuum cleaner • Impedance of inductor & resistor in series • Position of a ship • Forces in static equilibrium: Hanging weight • Forces in static equilibrium: TV on a ramp

9. A New Electrical Component: The Inductor • Recall that a resistor has a resistance (R), which is measured in ohms (Ω). In diagrams, the symbol for a resistor is • An inductor has an inductive reactance (XL), also measured in ohms. In diagrams, the symbol for an inductor is

10. Impedance • Resistance (R) and inductive reactance(XL) are special cases of a quantity called impedance (Z), also measured in ohms. Impedance (Z) Resistance (R) Reactance (X) Inductive Reactance (XL) Capacitive Reactance (XC)

11. Review: Total Resistance of Resistors in Series • Recall that if two resistors are connected in series (end-to-end), total resistance is the sum of the two resistances: • Things aren’t quite this simple when a resistor and an inductor are connected in series…

12. Total Impedance • To find total impedance of a resistance and an inductive reactance in series, add them as vectors, not as scalars. • When treated as vectors, resistance always has an angle of 0, and inductive reactance always has an angle of 90.

13. Adding Vectors • Many problems involve the addition of two or more vectors. • Vectors can be added graphically or algebraically.

14. Adding Vectors Graphically • To add two vectors and graphically: • Draw the two vectors with ‘s tail placed at ‘s tip. • Then draw a third vector that extends from ‘s tail to ‘s tip. This third vector is the vector sum, which we call .

15. Adding Vectors Algebraically • To add and algebraically: • Write the vectors in component form: and • Add their x-components to get the x-component of the sum, and add their y-components to get the y-component of the sum:

16. Matrices, Vectors, and Scalars in MATLAB • In MATLAB, all quantities are treated as arrays of numbers. • A matrix has several rows and several columns. • A vector has one row and several columns, or one column and several rows. • A scalar has just one row and one column.

17. Matrices in MATLAB • Example of a 2x3 matrix (one with two rows and three columns): • To enter this in MATLAB, type:A = [7.6, 1.2, 1.5; 4.9, 3.3, 2.5]

18. Vectors in MATLAB • A row vectoris an array with just one row. • Example: v1=[7.6, 1.2, 1.5] • A column vector is an array with just one column. • Example: v2=[7.6; 1.2; 1.5]

19. Scalars in MATLAB • A scalar is treated as an array with just one row and one column. • Example: s=23.5 • Could also write this as s=[23.5]

20. Typical Use of Vectors in MATLAB • Suppose we want to plot some temperature-versus-time data. • Define vectors for time and temp, and then use plot command.

21. Matrix Multiplication versus Element-by-Element Multiplication • In MATLAB, the * operator performs matrix multiplication. • For A*B to be defined, the number of columns in A must equal the number of rows in B. • The .* operator performs element-by-element multiplication.

22. MATLAB Multiplication Example • Define two 1x3 vectors:v1 = [1, 2, 3]v2 = [4, 5, 6] • v1*v2 tries to perform matrix multiplication. An error results, since the number of columns in v1 is not equal to the number of rows in v2. • v1.*v2 performs element-by-element multiplication, giving [4, 10, 18].

23. Other Operations in MATLAB • Similar comments apply to division and exponentiation: • / performs matrix division • ./ performs element-by-element division • ^ performs matrix exponentiation • .^ performs element-by-element expntn. • Addition and subtraction are always performed element-by-element. So we don’t need special .+ and .– operators. Just use the + and – operators.

24. Static Equilibrium • The field called “statics” deals with objects in static equilibrium. For such objects, the external forces acting on the object add to zero: • Therefore (for 2 dimensions):

25. Common Types of Force • The following types of forces often arise in statics problems: • Weight • Tension • Frictional force • Normal force

26. Weight and Mass • Near the earth’s surface, an object’s weight ( ) is a vector pointing straight down. • Its magnitude (W) is equal to the object’s mass (m) times the acceleration due to gravity (g): W = mg • In metric units, g  9.81 m/s2.

27. Free-Body Diagram • For statics problem, your first step should be to draw a free-body diagram. • A free-body diagram shows the object of interest and clearly indicates all of the forces acting on that object.