Kinematics & Dynamics in 2 & 3 Dimensions; Vectors Math Topics

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Kinematics & Dynamics in 2 & 3
Dimensions; Vectors
First, a review
of some
Math Topics
in Ch. 1.
Then, some
Physics Topics
in Ch. 4!
Vectors: Some Topics in Ch. 1, Section 7
General Discussion.
Vector  A quantity with magnitude & direction.
Scalar  A quantity with magnitude only.
• Here, we’ll mainly deal with Displacement &
Velocity. But, our discussion is valid for any vector!
• The Ch. 1 vector review has a lot of math!
It requires a detailed knowledge of trigonometry!
Problem Solving
• A diagram or sketch is helpful & vital!
• I don’t see how it is possible to solve a
vector problem without a diagram!
Coordinate Systems
Rectangular (Cartesian)
Coordinates
• “Standard” coordinate axes.
• A point in the plane is (x,y)
• If its convenient, we could
reverse + & -
- ,+
+,+
-,-
+,-
A “Standard Set” of
xy Coordinate Axes
Vector & Scalar Quantities
• Vector  Quantity with magnitude &
direction.
• Scalar  Quantity with magnitude only.
Equality of Two Vectors
•Consider 2 vectors, A & B
A = B means A & B have
the same magnitude & direction.
Vector Addition, Graphical Method
• Addition of Scalars:
We use “Normal” arithmetic!
• Addition of Vectors: Not so simple!
• Vectors in the same direction:
– We can also use simple arithmetic
• Example 1: Suppose we travel 8 km East on day
1 & 6 km East on day 2.
Displacement = 8 km + 6 km = 14 km East
• Example 2: Suppose we travel 8 km East on day
1 & 6 km West on day 2.
Displacement = 8 km - 6 km = 2 km East
“Resultant” = Displacement
Adding Vectors in the
Same Direction:
Graphical Method of Vector Addition
• For 2 vectors NOT along the gsame
line, adding is more complicated:
• Example: D1 = 10 km East
D2 = 5 km North.
What is the resultant (final) displacement?
• 2 Methods of Vector Addition:
–Graphical (2 methods of this also!)
–Analytical (TRIGONOMETRY)
Graphical Method of Adding Vectors
“Recipe”
• Draw the first vector.
• Draw the second vector starting
at the tip of the first vector
• Continue to draw vectors “tip-to-tail”
• The sum is drawn from the tail of the
first vector to the tip of the last vector
Example:
• Example: 2 vectors NOT along the same line. Figure!
D1 = 10 km E, D2 = 5 km N.
Resultant = DR = D1 + D2 = ?
• In this special case ONLY, D1 is perpendicular to D2.
• So, we can use the Pythagorean Theorem.
DR = 11.2 km
Note!
DR < D1 + D2
(scalar addition)
D1 = 10 km E, D2 = 5 km N.
Resultant = DR = D1 + D2 = ?
DR = 11.2 km
Note!
DR < D1 + D2
The Graphical Method of Addition
• Plot the vectors to scale, as in the figure.
• Then measure DR & θ.
Results in DR = 11.2 km, θ = 27º N of E
• This example illustrates general rules of
graphical addition, which is also called the
“Tail to Tip” Method.
• Consider R = A + B (See figure!).
Graphical Addition Recipe
1. Draw A & B to scale.
2. Place the tail of B at the
tip of A
3. Draw an arrow from the
tail of A to the tip of B
4. This arrow is the Resultant R
Measure its length & the angle with the x-axis.
Order Isn’t Important!
Adding vectors in the opposite order gives the same
result: In the example in the figure,
DR = D1 + D2 = D2 + D1
Graphical Method of
Vector Addition
• Adding (3 or more) vectors:
V = V1 + V2 + V3
• Even if the vectors are not at right angles, they
can be added graphically using the tail-to-tip
method.
Parallelogram Method
• A 2nd Graphical Method of Adding Vectors
(equivalent to the tail-to-tip method, of course!)
V = V1 + V2
1. Draw V1 & V2 to scale from a common origin.
2. Construct a parallelogram using V1 & V2 as
2 of the 4 sides.
3. Resultant V = Diagonal of the
Parallelogram from a Common Origin
(measure length & the angle it makes with the x axis)
See Figure Next Page!
Parallelogram Method
A common error!
Mathematically, we can move vectors around
(preserving their magnitudes & directions)
Subtraction of Vectors
• First, Define The Negative of a Vector:
- V  vector with the same magnitude (size) as
V but with opposite direction.
Math: V + (- V)  0
• Then add the negative vector.
• For 2 vectors, V1 & V2:
Subtracting Vectors
 To subtract one vector from another, add the
first vector to the negative of the 2nd vector,
as in the figure below:
Multiplication by a Scalar
• A vector V can be multiplied by a scalar c
V' = cV
V'  vector with magnitude cV & same direction as V.
• If c is negative, the resultant is in the
opposite direction.
Example
• Consider a 2 part car trip:
• Displacement A = 20 km due North.
• Displacement B = 35 km 60º West of North.
• Find (graphically) resultant displacement vector R
(magnitude & direction). R = A + B. See figure below.
Use ruler & protractor to
find the length of R & the
angle β.
Answers:
Length = 48.2 km
β = 38.9º
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