1. MEASUREMENTS AND VECTORS

Units: There are 3 fundamental quantities- length, mass and time. SI (System International) Units: mks: Length = meters (m), Mass = kilograms (kg), Time = seconds (s) British Units: Inches, feet, miles, pounds Other system of units: cgs: L = centimeters (cm), M = grams (gm), T = seconds (s) 9

Conversion of units: Useful Conversion factors: 1 inch = 2.54 cm 1 m = 3.28 ft 1 mile = 5280 ft 1 mile = 1.61 km Example: convert miles per hour to meters per second:

DIMENSIONS OF PHYSICAL QUANTITIES: length(L) , mass (M), time(T) Example: Doing a problem you get the answer distance t= dv (velocity x time2) Units on left side = T Units on right side = L x L / T = L2 / T Left units and right units don’t match, so answer must be wrong!!

Scalars : Quantities having magnitude only Scalars : Quantities having magnitude only. Example: Mass, Distance, Speed. Vectors: Quantities having both magnitude and direction. Example: Displacement, Force, Velocity, Acceleration Distance : Total length covered by an object. Displacement : The shortest distance between any 2 points in a specified direction. Distance and displacement have the same units. (m, cm, km)

Unit Vectors: A Unit Vector is a vector having length 1 and no units It is used to specify a direction Unit vector u points in the direction of U Often denoted with a “hat”: u = û Useful examples are the Cartesian unit vectors [ i, j, k ] point in the direction of the x, y and z axes U û y j x i k z

Vector Notation: There are two common ways of indicating that something is a vector quantity: Boldface notation: A “Arrow” notation: A =

Vectors... The components of r are its (x,y,z) coordinates r = (rx ,ry ,rz ) = (x,y,z) Consider this in 2-D (since it’s easier to draw): rx = x = r cos  ry = y = r sin  where r = |r | (x,y) y arctan( y / x ) r  x

Magnitude of a vector: Vector addition using components: The magnitude (length) of r is found using the Pythagorean theorem: r y x The length of a vector clearly does not depend on its direction. Vector addition using components: A= (Ax i + Ay j+ Az k)

What is the resultant vector, Z, from adding A+B+C? Vector A = (1,2,3) Vector B = (4,5,0) Vector C = (2,-3,2) Z = (AXi + AYj + AZk) + (BXi + BYj + BZk) + (CXi + CYj + CZk) = (AX + BX + CX)i + (AY + BY+ CY)j + (AZ + BZ + CZ)k = (1 + 4 + 2)i + (2 + 5 - 3)j + (3+ 0 + 2)k = {7,4,5}

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.

3-4 Adding Vectors by Components If the components are perpendicular, they can be found using trigonometric functions.

Adding Vectors by Components The components are effectively one-dimensional, so they can be added arithmetically:

Adding Vectors by Components Draw a diagram; add the vectors graphically. Choose x and y axes. Resolve each vector into x and y components. Calculate each component using sines and cosines. Add the components in each direction. To find the length and direction of the vector, use: