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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
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Kinematics & Dynamics in 2 & 3 Dimensions; Vectors
First, a reviewof some
Math Topicsin Ch. 1.Then, some
Physics Topicsin 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 kmNote!
DR < D1 + D2
(scalar addition)
D1 = 10 km E, D2 = 5 km N.Resultant = 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
DR = 11.2 kmNote!
DR < D1 + D2
• 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 RMeasure 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.
• 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 as2 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
Mathematically, we can move vectors around (preserving their magnitudes & directions)
A common error!
Parallelogram Method
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 VectorsTo 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 tofind the length of R & theangle β.
Answers:Length = 48.2 km
β = 38.9º
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