Vector addition, subtraction Fundamentals of 2-D vector addition, subtraction.
-
Upload
brice-tyler -
Category
Documents
-
view
258 -
download
2
Transcript of Vector addition, subtraction Fundamentals of 2-D vector addition, subtraction.
2 Dimensional Motion, Forces
• Combining displacements or forces in 2 dimensions is not as straightforward as we have previously done– Not just simple addition or subtraction
• The direction for motion in 2 dimensions cannot be described by a simple plus or minus
Description
Boy is walking 25 meters in the North East Direction
Displacement value is =
25m [NE] or 25m (45°)
2D values
• Always written in the formMagnitude, direction
• Several ways to write the direction
[ ] use some combination of North, South, East West and the angle of direction
A single angle measured from the horizontal axis pointing right (Polar coordinates)
Vectors
• Arrows that visually represent velocity, force, displacement, and acceleration
• Indicates direction and size
• Writing vectors: 54N [E]
Step Back a bit…
• Displacement, Velocity, Acceleration, Forces… can all be represented by vectors
• We have already used vectors in FBD
Direction using N,S,W,and E
• Use brackets
• First letter is the cardinal direction line used as the start of the measurement of the angle
• Number the value of the angle measured
• Last letter is the cardinal direction towards which you measure (never on the same dimension)
Coming up with the direction
Started with the south direction
Moving towards the east direction
34°
[S 34 E]
Polar Coordinates
• Angle is from 0 to 360 degrees
• 0 and 360° point due east (to the right)
• Measured going counterclockwise
Problems
Angle Descriptors:
[W 75 S]
[ S 15 W]
(255°)60°
75°Angle descriptors:
[W 60 N]
[N 30 E]
(120°)
Problem
• Describe the direction of the following vector. Come up with at least 4 correct values
13°
Resultant Vector
• The vector value obtained when 2 or more vectors are combined (added or subtracted)
One Dimensional vector combinations
• 5 m [N] + 7 m [N] = ___________
• 5 m [N] + 7 m [S] = ___________
• 5 m [N] – 7 m [N] = ___________
• 5 m [N] – 7 m [S] = ___________
Combining Vector values in a single dimension
• Add if going the same direction
• Subtract if going opposite directions
Basics
• Combine all forces in the same dimensions
• Draw the vectors to form 2 sides of a right triangle (head to tail).
• Does not matter which one you will start with. Will determine angle write-up
• Use Pythagorean theorem to solve for side
• Use inverse trig to find angle
Combination of 2 vectors not in the same dimension
• 5 m [N] + 7 m [E] = _____________
5 m [N]
7 m [E]
The resultant vector
Use Pythagorean theorem to solve for the 3rd side,
and
tan-1 to find the angle
Answer
• To find the side:
52 + 72 = resultant2 = 8.9
To find the angle tan-1 (7/5) = 54.5°
Vector description: 8.9 m [ N 54.5 E]
Answer
• First, combine 53 m/s [S] and 127 m/s [N]
Subtract small from large since they go in opposite directions
127 – 53 = 74 m/s [N]
(use direction of larger value)
Answer, continued
• Use Pythagorean theorem to solve for the hypotenuse
• Use tan-1 (inverse tangent) to solve for the angle
Final answer : 99.8 m/s [W 47.8 N]
Class Problem
• A 110 N force and a 55 N force both act on an object at point P. The 55 N force acts at 0%. What is the magnitude and direction of the resultant force?
Point p55 N
110 N
Combining vectors in 2-dimensions
• Resultant vector: The vector that represents the sum of 2 or more vector values
• Drawing the resultant: Draw one vector. From the end of the first, draw the second.
• The resultant is the arrow drawn from the beginning of the first vector to the end of the last vector
Adding vectors in 2 dimensions
• If dimensions are perpendicular:
• Draw given vectors head to tail
• The resultant is drawn from the beginning of the first vector to the end of the last vector.
However
• If an overall value is being sought… answer will be 2-D
• Overall velocity
• Overall Displacement
• Overall Net Force
Overall Velocity
• If the overall velocity of the boat is wanted, then the velocity of the boat and the water must be combined to form the resultant vector
• We next figure out how to solve for the resultant
Steps
• Use Pythagorean theorem or trig to find the length of the hypotenuse (resultant vector)
• Use inverse trig to find the angle of inclination
• The angle to be solved for is between the initial side drawn and the hypotenuse!