Graphical Analytical Component Method Vector Addition

Math sTUFF What does an ordered pair mean in math? Ex:(2,3)

Vectors Quantities having both magnitude and direction Magnitude: How much (think of it as the length of the line) Direction: Which way is it pointing? Can be represented by an arrow-tipped line segment Examples: Velocity Acceleration Displacement Force

Question: Compare the two vectors. What makes them different

Answer Direction The magnitude or length is exactly the same

Vector Terminology Two or more vectors acting on the same point are said to be concurrent vectors. The sum of 2 or more vectors is called the resultant (R). A single vector that can replace concurrent vectors Any vector can be described as having both x and y components in a coordinate system. The process of breaking a single vector into its x and y components is called vector resolution.

More Vector Terminology Vectors are said to be in equilibrium if their sum is equal to zero. A single vector that can be added to others to produce equilibrium is call the equilibrant (E). Equal to the resultant in magnitude but opposite in direction. E + R = 0 E = - R E = 5 N R = 5 N at 180 ° at 0°

What is the resultant of the following vectors? E= 10 N at 0 degrees R = 20 N at 0 degrees

Answer 30 N at 0 degrees

Question 20 N at 45 degrees 10 N at 225 degrees Do not get freaked out by the angles, Think about it for a second.

Answer 10 N at 45 degrees

Using the Graphical Method of Vector Addition: Vectors are drawn to scale and the resultant is determined using a ruler and protractor. Vectors are added by drawing the tail of the second vector at the head of the first (tip to tail method). The order of addition does not matter. The resultant is always drawn from the tail of the first to the head of the last vector.

Example Problem A 50 N force at 0° acts concurrently with a 20 N force at 90°. R   R and  are equal on each diagram.

Adding vectors!

Question: Add these two vectors together b

Answer b a R= a+b

Motion Applications Perpendicular vectors act independently of one another. In problems requesting information about motion in a certain direction, choose the vector with the same direction.

Example Problem: Motion in 2 Dimensions A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s. What is the resultant velocity of the boat?

A boat heads east at 8. 00 m/s across a river flowing north at 5 A boat heads east at 8.00 m/s across a river flowing north at 5.00 m/s. 5.00 m/s N 8.00 m/s E River width

What is the resultant velocity of the boat? Draw to scale and measure. 5.00 m/s N 8.00 m/s E R = 9.43 m/s at 32°

Advantages and Disadvantages of the Graphical Method Can add any number of vectors at once Uses simple tools No mathematical equations needed Must be correctly draw to scale and at appropriate angles Subject to human error Time consuming

Solving Vectors Using the Analytical Method A rough sketch of the vectors is drawn. The resultant is determined using: Algebra Trigonometry Geometry

Quick Review Right Triangle c is the hypotenuse B c2 = a2 + b2 c sin = o/h cos = a/h tan = o/a a A + B + C = 180° B = 180° – (A + 90°) C A b tan A = a/b tan B = b/a

These Laws Work for Any Triangle. A + B + C = 180° C Law of sines: a = b = c sin A sin B sin C b a B A c Law of cosines: c2 = a2 + b2 –2abCos C

Use the Law of: Sines when you know: Cosines when you know: 2 angles and an opposite side 2 sides and an opposite angle Cosines when you know: 2 sides and the angle between them

For right triangles: Draw a tip to tail sketch first. To determine the magnitude of the resultant Use the Pythagorean theorem. To determine the direction Use the tangent function.

To add more than two vectors: Find the resultant for the first two vectors. Add the resultant to vector 3 and find the new resultant. Repeat as necessary.

Advantages and Disadvantages of the Analytical Method Does not require drawing to scale. More precise answers are calculated. Works for any type of triangle if appropriate laws are used. Can only add 2 vectors at a time. Must know many mathematical formulas. Can be quite time consuming.

Solving Vector Problems using the Component Method Each vector is replaced by 2 perpendicular vectors called components. Add the x-components and the y-components to find the x- and y-components of the resultant. Use the Pythagorean theorem and the tangent function to find the magnitude and direction of the resultant.

Vector Resolution y = h sin  x = h cos  h y  x + ++ - +-

Components of Force: y x

Question What are the components of the following force 25N @ 12 degrees North of West

Answer West is 180 degrees to 12 degrees north of west is 168 degrees The X component is -24.45N The Y component is 5.20N You can confirm you answer –X and +Y would be found in the second quadrant on a graph so this answer makes sense

Example: x y + 0.09 + 6.74 6 N at 135° 5 N at 30° 5 cos 30° = +4.33 5 sin 30° = +2.5 6 cos 45 ° = - 4.24 6 sin 45 ° = + 4.24 + 0.09 + 6.74 5 N at 30° R = (0.09)2 + (6.74)2 = 6.74 N  = arctan 6.74/0.09 = 89.2°

Tangent Function The tangent function has 2 places that it is not defined (you get an error on your calculator) 90 degrees and 270 degrees The x and y components tell you the angle range Angle Range Operation 0 to 90 Nothing special needed 90 to 180 Add 180 degrees 180 to 270 270 to 360 Add 360 degrees

Question: Critical THinking My X component was negative and my y component was negative as well. My calculator told me that my answer was 22 degrees. What is my true angle?

Answer My evidence: Negative X Negative Y We are in quadrant three (between 180 degrees and 270) I got 22 degrees, so I must take 180+22 to get 202 degree as my angle! Using my tangent rules

Solve the following problem using the component method. 10 km at 30 6 km at 120

Adding Vectors To find the magnitude: pythagorean theorum X component Y component 6 km @ 120 degrees 6 km * cos(120) = 6 km * sin(120) = 10 km @ 30 degrees 10 km * cos (30) = 10 km * sin(30) = Add them together To find the magnitude: pythagorean theorum To find the direction: 1. Take into account if either X or y is + or – 2. Use any trig function SOH CAH TOA to find angle

Critical Thinking Question 2 I get a positive x and a negative y component when I add them together. What degree range is my angle in?

Answer X is positive so that can only mean either quadrant 1 or 4 Y is negative so that means you have to get quadrant 4 as your answer 270 to 360 degrees

Notes Make sure that all angles are measured from the x axis (0 degrees) Report both the magnitude and the direction otherwise the vector is wrong! Keep track of signs, They give you a clue to where the angle of the vector actually is.

Adding two vectors Find the resultant magnitude: X component Y component 12 N @ 135 degrees 15 N @ 200 degrees Resultant X and Y Find the resultant magnitude: Find the resultant direction: