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Chapter 3. Vectors. Vectors and Scalars , Addition of vectors Subtraction of vectors. Physics deals with many quantities that have both Magnitude Direction VECTORS !!!!!. y. . x. r. Scalar. A scalar quantity is a quantity that has magnitude only and has no direction in space. - PowerPoint PPT Presentation
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CHAPTER 3
1
Vectors
2
Vectors and Scalars,Addition of vectorsSubtraction of vectors
x
y
r
Physics deals with many quantities that have both
MagnitudeDirection
VECTORS !!!!!
3
A scalar quantity is a quantity that has magnitude only and has no direction in space
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Examples of Scalar Quantities:
Length Area Volume Time Mass
Scalar
A vector quantity is a quantity that has both magnitude and a direction in space
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Examples of Vector Quantities: Displacement Velocity Acceleration Force
Vector
A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.
Two vectors are equal if they have the same direction and magnitude (length).
Blue and orange vectors have same magnitude but different direction.
Blue and green vectors have same direction but different magnitude.
Blue and purple vectors have same magnitude and direction so they are equal.
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Examples A = 20 m/s at 35° NE B = 120 lb at 60° SE
C = 5.8 mph/s west
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Example
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•The direction of the vector is 55° North of East
•The magnitude of the vector is 2.3.
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Try Again
Direction:
Magnitude:
43° East of South
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Try Again
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It is also possible to describe this vector's direction as 47 South of East.
Why?
Vector Addition vectors may be added graphically or analytically
Triangle (Head-to-Tail) Method1. Draw the first vector with the proper length and orientation.
2. Draw the second vector with the proper length and orientation originating from the head of the first vector.3. The resultant vector is the vector originating at the tail of the first vector and terminating at the head of the second vector.
4. Measure the length and orientation angle of the resultant. 11
Adding vectors in same direction:Example: Travel 8 km East on day 1, 6 km
East on day 2. Displacement = 8 km + 6 km = 14 km East Example: Travel 8 km East on day 1, 6 km
West on day 2. Displacement = 8 km - 6 km = 2 km East
“Resultant” = Displacement
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13
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Adding more than two vectors graphically
Subtraction of Vectors -graphically
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bad
ba c whereas)b(ad
Parallelogram (Tail-to-Tail) Method1. Draw both vectors with proper length and orientation originating from the same point.2. Complete a parallelogram using the two vectors as two of the sides.3. Draw the resultant vector as the diagonal originating from the tails.4. Measure the length and angle of the resultant vector.
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Components of Force:
x
y
Resolving a Vector Into Components
+x
+y
A
Ax
Ayq
The horizontal, or x-component, of A is found by Ax = A cos q.
The vertical, ory-component, of A is found by Ay = A sin qBy the Pythagorean Theorem, Ax
2 + Ay2 = A2
Every vector can be resolved using these formulas, such that A is the magnitude of A, andq is the angle the vector makes with the x-axis.
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Analytical Method of Vector Addition1. Find the x- and y-components of each vector.
Ax = A cos q Ay = A sin q Bx = B cos q By = B sin q Cx = C cos q Cy = C sin q
2. Sum the x-components. This is the x-component of the resultant.
Rx
3. Sum the y-components. This is the y-component of the resultant.
Ry
4. Use the Pythagorean Theorem to find the magnitude of the resultant vector.Rx
2 + Ry2 = R2
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5. Find the reference angle by taking the inverse tangent of the absolute value of the y-component divided by the x-component.
q = Tan-1 Ry/Rx
6. Use the “signs” of Rx and Ry to determine the quadrant. NE
(+,+)NW
(-,+)
SW(-,-)
SE(-,+)
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Step 1 Sketch the given vector with the tail
located at the origin of an x-y coordinate system. (Ex. 25 m at an angle of 36º)
25 m
36º
Step 2 Draw a line segment from the tip of the
vector perpendicular to the x-axis
25 m
Notice, you now have a right triangle with a known hypotenuse and known angle measurements
36º
Step 3 Replace the perpendicular sides of the
right triangle with vectors drawn tip – to - tail
25 m
Step 4 Use sine and cosine functions to find the
horizontal and vertical components of the given vector.
25 m
Rx
Ry36º
Cos(36) = Rx/25
Rx = 25cos(36)
Rx = 20.2 m
Sin(36) = Ry/25
Ry = 25sin(36)
Ry = 14.7 m
Example:
5 N6 N
x y5 cos 30° = +4.33 5 sin 30° = +2.5
6 cos 45 ° = - 4.24
6 sin 45 ° = + 4.24
+ 0.09 + 6.74
R = (0.09)2 + (6.74)2 = 6.74 N
q = arctan 6.74/0.09 = 89.2°
135°45° 30°
R
Rx
Ry
Problem 1
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Problem 2
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Problem 3
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Problem 4
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