Kinematics Vectors and Scalars Scalars A scalar quantity

Kinematics Vectors and Scalars

Scalars �A scalar quantity is a quantity that has magnitude only and has no direction in space Examples of Scalar Quantities: � Length � Area � Volume � Time � Mass

Vectors �A vector quantity is a quantity that has both magnitude and a direction in space Examples of Vector Quantities: � Displacement � Velocity � Acceleration � Force

Vector � https: //youtu. be/b. OIe 0 DIMb. I 8

Vector Diagrams � Vector diagrams are shown using an arrow � The length of the arrow represents its magnitude � The direction of the arrow shows its direction

Resultant of Two Vectors � The resultant is the sum or the combined effect of two vector quantities Vectors in the same direction: 6 N 6 m 4 m 4 N = 10 m Vectors in opposite directions: 6 m/s 10 m/s = 4 m/s 6 N 10 N = 4 N

The Parallelogram Law � � � When two vectors are joined tail to tail Complete the parallelogram The resultant is found by drawing the diagonal The Triangle Law � � When two vectors are joined head to tail Draw the resultant vector by completing the triangle

Problem: Resultant of 2 Vectors Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? Solution: � � Complete the parallelogram (rectangle) The diagonal of the parallelogram ac represents the resultant force The magnitude of the resultant is found using Pythagoras’ Theorem on the triangle abc a 5 N � b � θ 12 N 13 12 d N 5 c Resultant displacement is 13 N 67º with the 5 N force

Problem: Resultant of 3 Vectors Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below. Solution: � Find the resultant of the two 5 N forces first (do right angles first) 5 � So, Resultant = 10 N – 7. 07 N = 2. 93 N in the direction of the 10 N force 7. 07 5 N a N 93 N 2. � Now find the resultant of the 10 N and 7. 07 N forces The 2 forces are in a straight line (45º + 135º = 180º) and in opposite directions 10 � N d 90º θ 45º 135º 5 N c 5 b

Recap � What is a scalar quantity? � Give 2 examples � What is a vector quantity? � Give 2 examples � How are vectors represented? � What is the resultant of 2 vector quantities? � What is the triangle law? � What is the parallelogram law?

Identifying Direction A common way of identifying direction is by reference to East, North, West, and South. (Locate points below. ) Length = 40 m N 40 m, 50 o N of E W 60 o 50 o 60 o E 40 m, 60 o N of W 40 m, 60 o W of S S 40 m, 60 o S of E

Identifying Direction Write the angles shown below by using references to east, south, west, north. N W 45 o E 50 o S 0 S of 50 Click to. Esee the Answers. . . W N E S 450 W of N

Vectors and Polar Coordinates Polar coordinates (R, q) are an excellent way to express vectors. Consider the vector 40 m, 500 N of E, for example. 90 o 180 o 270 o 90 o 40 m R 180 o 50 o 0 o R is the magnitude and q is the direction. 270 o q 0 o

Vectors and Polar Coordinates Polar coordinates (R, q) are given for each of four possible quadrants: 90 o q (R, ) = 40 m, 50 o 120 o 210 o 180 o 60 o 50 o 60 o 0 o q (R, ) = 40 m, 120 o q (R, ) = 40 m, 210 o 3000 270 o q (R, ) = 40 m, 300 o

Rectangular Coordinates y (-2, +3) (+3, +2) + (-1, -3) + x Reference is made to x and y axes, with + and - numbers to indicate position in space. Right, up = (+, +) Left, down = (-, -) (x, y) = (? , ? ) (+4, -3)

Trigonometry Review � Application Trigonometry y = R sin q R y of Trigonometry to Vectors x = R cos q q x R 2 = x 2 + y 2

Example 1: Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30 o. The height h is opposite 300 and the known adjacent side is 90 m. h 300 h = (90 m) tan 30 o 90 m h = 57. 7 m

Example 2: A person walks 400 m in a direction of 30 o N of E. How far is the displacement east and how far north? N N R q x y 400 m E 30 o y=? x=? E The x-component (E) is ADJ: x = R cos q The y-component (N) is OPP: y = R sin q

Example 2 (Cont. ): A 400 -m walk in a direction of 30 o N of E. How far is the displacement east and how far north? N 400 m 30 o x=? Note: x is the side adjacent to angle 300 y=? E x = (400 m) cos 30 o = +346 m, E ADJ = HYP x Cos 300 x = R cos q The x-component is: Rx = +346 m

Example 2 (Cont. ): A 400 -m walk in a direction of 30 o N of E. How far is the displacement east and how far north? N 400 m 30 o x=? Note: y is the side opposite to angle 300 y=? E y = (400 m) sin 30 o = + 200 m, N OPP = HYP x Sin 300 y = R sin q The y-component is: Ry = +200 m

Example 2 (Cont. ): A 400 -m walk in a direction of 30 o N of E. How far is the displacement east and how far north? N 400 m 30 o Rx = Ry = +200 m E The x- and y- components are each + in the first quadrant +346 m Solution: The person is displaced 346 m east and 200 m north of the original position.