Post on 29-Dec-2015
transcript
VectorMathematics
Physics 1
Physical Quantities
A scalar quantity is expressed in terms of magnitude (amount) only.
Common examples include time, mass, volume, and temperature.
Physical Quantities
A vector quantity is expressed in terms of both magnitude and direction.
Common examples include velocity, weight (force), and acceleration.
Representing Vectors
Vector quantities can be graphically represented using arrows.– magnitude = length of the arrow– direction = arrowhead
Vectors
All vectors have a head and a tail.
Vector Addition
Vector quantities are added graphically by placing them head-to-tail.
Head-to-Tail Method
1. Draw the first COMPONENT vector with the proper length and orientation.
2. Draw the second COMPONENT vector with the proper length and orientation starting from the head of the first component vector.
Head-to-Tail Method
3. The RESULTANT (sum) vector is drawn starting at the tail of the first component vector and terminating at the head of the second component vector.
4. Measure the length and orientation of the resultant vector.
South
East
Resultant
To add vectors, move tail to head and then draw resultant from original start to final point.
Resultant is (sqrt(2)) 45◦ south
of East
South
East
Resultant
Vector addition is ‘commutative’ (can add vectors in either order)
Resultant is (sqrt(2)) 45◦ south
of East
South
East
Resultant
Vector addition is ‘commutative’ (can add vectors in either order)
Resultant is (sqrt(2)) 45◦ south
of East
South
East
Resultant
Co-linear vectors make a longer (or shorter) vector
Resultant is 3 magnitude South
Co-linear vectors make a longer (or shorter) vector
Resultant is 3 magnitude South
Nor
thN
orth
EastEast
Can add multiple vectors.Just draw ‘head to tail’ for each vector
Resultant is magnitude
45◦ North of East 2
Nor
thSo
uth
EastEast
Adding vectors is commutative.
Resultant is magnitude
45◦ North of East
Nor
th
East
Nor
th
East
22
Nor
th
South
East
WestResultant=0
Equal but opposite vectors cancel each other out
Resultant is 0.
Vector Addition – same directionA + B = R
B
A
A B
R = A + B
Vector Addition
• Example: What is the resultant vector of an object if it moved 5 m east, 5 m south, 5 m west and 5 m north?
Vector Addition – Opposite direction(Vector Subtraction) .
A + (-B) = RA
B
-B
A
-B A + (-B) = R
Vectors• The sum of two or more vectors is called the
resultant.
Practice
Vector Simulator http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html
Polar Vectors
Every vector has a magnitude and direction
direction anglemagnitude
Right Triangles
SOH CAH TOA
Vector Resolution
Every vector quantity can be resolved into perpendicular components.
Rectilinear (component) form of vector:
yx
A
Ax
Ay
Vector Resolution
Vector A has been resolved into two perpendicular components, Ax (horizontal component) and Ay (vertical component).
Vector Resolution
If these two components were added together, the resultant would be equal to vector A.
A
Ax
Ay
Vector Resolution
When resolving a vector graphically, first construct the horizontal component (Ax). Then construct the vertical component (Ay).
Using right triangle trigonometry, expressions for determining the magnitude of each component can be derived.
Vector Resolution
Horizontal Component (Ax)
A
Axcos
cosAAx
A
Ax
Ay
Vector Resolution
Vertical Component (Ay)
sin yA
A
sinyA A
A
Ax
Ay
Drawing Directions
EX: 30° S of W– Start at west axis and move south 30 °– Degree is the angle between south and west
N
S
EW
Vector Resolution
Use the sign conventions for the x-y coordinate system to determine the direction of each component.
(+,+)(-,+)
(-,-) (+,-)
N
E
S
W
Component Method
1. Resolve all vectors into horizontal and vertical components.
2. Find the sum of all horizontal components. Express as SX.
3. Find the sum of all vertical components. Express as SY.
Component Method
4. Construct a vector diagram using the component sums. The resultant of this sum is vector A + B.
5. Find the magnitude of the resultant vector A + B using the Pythagorean Theorem.
6. Find the direction of the resultant vector A + B using the tangent of an angle q.
Adding “Oblique” Vectors
Head to tail method works, but makes it very difficult to ‘understand’ the resultant vector
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
Using Calculator For Vectors
Can use the “Angle” button on TI-84 calculator to do vector mathematics
Using Calculator for Vectors