Chapter 3 Kinematics in Two or Three Dimensions; Vectors
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Transcript of Chapter 3 Kinematics in Two or Three Dimensions; Vectors
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Chapter 3
Kinematics in Two or Three Dimensions; Vectors
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Homework
Tuesday:
Read Chapter 4
2014: Do P. 77 1,2,4,5,11,12,13,14
2013: Do p. 77 1, 2, 4, 5, 18
2012: Do p. 77 1, 2, 4, 5, 11, 12, 13, 14, 17
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How do we calculate the motion of this skier in two
dimensions?
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How do we calculate the motion of this skier in two dimensions?
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How do we calculate the tension in these ropes?
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3-1 Vectors and Scalars
A vector has magnitude as well as direction.
Some vector quantities: displacement, velocity, force, momentum
A scalar has only a magnitude.
Some scalar quantities: mass, time, temperature
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3-2 Addition of Vectors—Graphical Methods
For vectors in one dimension, simple addition and subtraction are all that is needed.
You do need to be careful about the signs, as the figure indicates.
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3-2 Addition of Vectors—Graphical MethodsIf the motion is in two dimensions, the situation is somewhat more complicated.
Here, the actual travel paths are at right angles to one another; we can find the displacement by
using the Pythagorean Theorem.
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3-2 Addition of Vectors—Graphical Methods
Adding the vectors in the opposite order gives the same result:
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3-2 Addition of Vectors—Graphical Methods
Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.
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3-2 Addition of Vectors—Graphical Methods
The parallelogram method may also be used; here again the vectors must be tail-to-tip.
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3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar
In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction.
Then we add the negative vector.
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3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar
A vector can be multiplied by a scalar c; the result is a vector c that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.
V
V
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Adding & Subtracting Vectors
• Vectors can be added or subtracted from each other graphically.
• Each vector is represented by an arrow with a length that is proportional to the magnitude of the vector.
• Each vector has a direction associated with it.• When two or more vectors are added or
subtracted, the answer is called the resultant.• A resultant that is equal in magnitude and
opposite in direction is also known as an equilibrant.
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Adding Vectors using the Pythagorean Theorem
3 m4 m+ =
3 m
4 m
5 m
If the vectors occur such that they are perpendicular to one another, the Pythagorean theorem may be used to determine the resultant.
A2 + B2 = C2
(4m)2 + (3m)2 = (5m)2
When adding vectors, place the tail of the second vector at the tip of the first vector.
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+
Adding & Subtracting Vectors
3 m4 m-
7 m=
If the vectors occur in a single dimension, just add or subtract them.
• When adding vectors, place the tail of the second vector at the tip of the first vector.
• When subtracting vectors, invert the second one before placing its tail at the tip of the first vector.
4 m 3 m 7 m+ =
7 m
7 m
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Law of Cosines
If the angle between the two vectors is more or less than 90º, then the Law of Cosines can be used to determine the resultant vector.
C2 = A2 + B2 – 2ABCos
C2 = (7m)2 + (5m)2 – 2(7m)(5m)Cos 80º
C = 7.9 m
7 m 5 m
+ =
5 m7 m
= 80º
C
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Example 1:
P
P
P
P
The vector shown to the right represents two forces acting concurrently on an object at point P. Which pair of vectors best represents the resultant vector? P
Resultant
(a)
(d)(c)
(b)
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How to Solve:
P
P
1. Add vectors by placing them tip to tail.
P
or
2. Draw the resultant.
P
Resultant
This method is also known as the Parallelogram Method.
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How to Solve:
P
Resultant
This method is also known as the Parallelogram Method.
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3-4 Adding Vectors by Components
Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.
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3-4 Adding Vectors by Components
If the components are perpendicular, they can be found using trigonometric functions.
Remember:
soh
cah
toa
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3-4 Adding Vectors by Components
The components are effectively one-dimensional, so they can be added arithmetically.
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3-4 Adding Vectors by Components
Adding vectors:
1. Draw a diagram; add the vectors graphically.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using sines and cosines.
5. Add the components in each direction.
6. To find the length and direction of the vector, use:
and .
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3-4 Adding Vectors by Components
Example 3-2: Mail carrier’s displacement.
A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0° south of east for 47.0 km. What is her displacement from the post office?
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3-4 Adding Vectors by Components
Example 3-3: Three short trips.
An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45°) for 440 km; and the third leg is at 53° south of west, for 550 km, as shown. What is the plane’s total displacement?
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Vector vs. Scalar
The resultant will always be less than or equal to the scalar value.
670 m
270 m
770 m
868 m
dTotal = 1,710 md = 868 m
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Homework
2014: Read and provide notes on 4.2 Do p. 78 27, 28, 29, 30, 31
2013: Do p. 78 21, 23, 27, 28, 29, 30
2012: Do p. 78 21, 23, 27, 28, 29, 30, 31
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d = 23 km
dx = d cos dx = (23 km)(cos 30°)dx = 19.9 km
dy = d sin dy = (23 km)(sin 30°)dy = 11.5 km
Example 2:
• A bus travels 23 km on a straight road that is 30° North of East. What are the component vectors for its displacement?
x
d
dx
dy
= 30°
y
East
North
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Algebraic Addition
• In the event that there is more than one vector, the x-components can be added together, as can the y-components to determine the resultant vector.
y
x
a
b
c
ay
ax cx
cy
bx
by
Rx = ax + bx + cx
Ry = ay + by + cy
R = Rx + Ry
R
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Properties of Vectors
• A vector can be moved anywhere in a plane as long as the magnitude and direction are not changed.
• Two vectors are equal if they have the same magnitude and direction.
• Vectors are concurrent when they act on a point simultaneously.
• A vector multiplied by a scalar will result in a vector with the same direction.
P
F = ma
vector vectorscalar
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Properties of Vectors (cont.)
• Two or more vectors can be added together to form a resultant. The resultant is a single vector that replaces the other vectors.
• The maximum value for a resultant vector occurs when the angle between them is 0°.
• The minimum value for a resultant vector occurs when the angle between the two vectors is 180°.
• The equilibrant is a vector with the same magnitude but opposite in direction to the resultant vector.
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Properties of Vectors (cont.)
4 m 3 m 7 m+ =
3 m4 m+
1 m=
180°
R-R
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Key Ideas
• Vector: Magnitude and Direction• Scalar: Magnitude only• When drawing vectors:
– Scale them for magnitude.– Maintain the proper direction.
• Vectors can be analyzed graphically or by using coordinates.
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Add the following by tip/tail, parallelogram, and trig. Find the resultant and the equilibrant:
• 1) Add the scaled vectors tip/tail • 2) Draw the resultant vector• 3) Add the vectors in the other
order to make a parallelogram.• 4) Calculate the resultant using
trig• 5) Find the equilibrant
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Add the following by tip/tail, parallelogram, and trig. Find the resultant and the equilibrant:
– Example 1:• 10 m/sec at 0 degrees• 5 m/sec at 180 degrees
– Example 2:• 5 m/sec at 30 degrees• 3 m/sec at 60 degrees
– Example 3:• 4 m/sec at 45 degrees• 4 m/sec at 135 degrees
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Add the following by tip/tail, parallelogram, and trig. Find the resultant and the equilibrant:
– Example 4:• 10 m/sec at 10 degrees• 5 m/sec at 20 degrees
– Example 5:• 5 m/sec at 40 degrees• 3 m/sec at 220 degrees
– Example 6:• 4 m/sec at 315 degrees• 4 m/sec at 260 degrees
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Draw these and find the following by trig:
– Example 7:• 5 m/sec at 270 degrees• 10m/sec at 60 degrees• 15m/sec at 120 degrees• Then graph the resultant and equilibrant
– Example 8:• 5 m/sec at 0 degrees• 5 m/sec at 135 degrees• 10 m/sec at 270 degrees• Then graph the resultant and equilibrant
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Draw these and find the following by trig:
– Example 9:• 5 m/sec at 40 degrees• 10m/sec at 50 degrees• 15m/sec at 60 degrees• Then graph the resultant and equilibrant
– Example 10:• 5 m/sec at 60 degrees• 5 m/sec at 120 degrees• 10 m/sec at 270 degrees• Then graph the resultant and equilibrant
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Now do the following on the map:
– Start at RCK• Go 5 cm North• Go 10 cm at 10 degrees N of W• Go 20 cm at -80 degrees• Go 5 cm at 190 degrees• Go 18.5cm at 50 degrees (N of E)• Where are you?