Ashley Abid Nicole Bogdan Vectors. Vectors and Scalars A vector quantity is a quantity that is fully...
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Transcript of Ashley Abid Nicole Bogdan Vectors. Vectors and Scalars A vector quantity is a quantity that is fully...
Ashley AbidNicole Bogdan
Vectors
Vectors and Scalars
A vector quantity is a quantity that is fully described by both magnitude and direction.
Scalars are quantities that are fully described by a magnitude (or numerical value) alone.
Examples of Vectors Examples of Scalars
Displacement (distance in a direction)
Distance (m)
Velocity (distance over time) Temperature (C or F)
Acceleration (velocity over time)
Energy (J)
Force (newtons * acceleration) Time (s)
Drawing Vectors
All vectors can be represented as arrows.
Tail Head
Magnitude of Vectors in One Dimension
• Vectors acting in the same direction produce the greatest magnitude force
• Vectors acting in opposite directions produce the
smallest magnitude force
• At 0 degrees, magnitudeis greatestAt 180 degrees, magnitudeis the smallest
What is displacement?Displacement is an object's overall change in position. It takes direction into account.
If a person walks around the perimeter of the diagram, the total distance traveled would be4m + 2 m + 4 m + 2 m = 12m
However, the total displacement is calculated as4 m East + 2 m South + 4 m West + 2 m North = 0 m.East and West cancel one another out, just like North and South.
Vector Fundamentals in Two DimensionsVectors can be added together to form a resultant
vector. The vectors added together are called component
vectors. They are represented with compass directions on the x and y axis.
Resultant --> <-- components
Adding Vectors and Calculating Resultants
The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together.
There are two methods to calculate resultants:
Head to TailTail to Tail (Parallelogram Method)
Head to Tail Method
1. Place the two vectors next to each other so that the head of one vector is touching the tail of the other vector.
2. Draw the resultant vector by connecting the remaining head and tail.
V1 + V2 = RV1
V2
Resultant Vector (R)
Head to Tail Method
Often, vectors must be rearranged for the head to tail method. The angles must remain the same.
The Pythagorean TheoremIf two vectors are perpendicular to each
other, you can solve for their resultant using the
Parallelogram Method
1. Draw the two components 2. Extend parallel lines to each of with their tails touching. the components so that
their lines meet
3. The resultant is the diagonal extended from one corner to the next.
V1
V2
R
Trigonometry Review for Parallelogram Method
The hypotenuse represents the resultant force. The adjacent and opposite represent the components.
Finding Horizontal and Vertical Components to a Vector
The vertical and horizontal components make a triangle and so we can use sine and cosine to calculate a missing component.
The formulas Rx =R cosθ and Ry = R sinθ
are used.
Ver
tica
l C
ompon
ent
R
y
Horizontal Component
Rx
θ
Finding an Equilibrant
• Equilibrium is any situation where the net force acting on an object is zero.
• It is called equilibrium because all the forces acting on the object equal out and cancel each other.
• This third force that would do the cancelling out is called the equilibrant.
• The equilibrant is a vector that is the exact same size as the resultant would be, but the equilibrant points in exactly the opposite direction.
• For this reason, an equilibrant touches the other vectors head-to-tail like any other vector being added.
Drawing Equilibrants
The equilibrant is the exact same size as the resultant would be, but the equilibrant points in exactly the opposite direction
(short animation)http://www.stmary.ws/highschool/physics/home/animations3/forces/resultants_90_degrees.swf
Review Question #1
Review Question #1 Solved
Ry = R sinθRy = 300 N * sin(60)=(2) 260 N
Review Question #2
Review Question #2 Solved(3)
Use the parallelogrammethod to find the missing component.
Review Question #3
Review Question #3 SolvedResultant:(F1^2 + F2^2) = R^2R = 14 N North East
Equilibrant is equal inmagnitude butopposite in direction
Equilibrant = 14 N South West (1)
R = 14 N
Review Question #4
Review Question #4 Solved(1) decreases
At 0 degrees, bothobjects work in the same direction. Their
magnitude is added.
At 90 degrees, the pythagorean theorem is used
6 N ^2 + 8 N ^2 = 10 N ^214 N > 10 N.
6 N + 8 N
14 N
Review Question #5
Question #5 Solved
(1) Law of TrianglesThe sum of any two sides of a triangle
cannot be smaller than the third side1 N + 3 N < 5 N