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Scalars & Vectors Tug of War Treasure Hunt Scalars Completely described by its magnitude Direction...
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Transcript of Scalars & Vectors Tug of War Treasure Hunt Scalars Completely described by its magnitude Direction...
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Tug of War
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Treasure Hunt
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Scalars
• Completely described by its magnitude• Direction does not apply at all• e.g. Mass, Time, Distance, etc.
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Vectors
• Characterised by its magnitude & direction• Knowledge of direction is necessary• e.g. Displacement, Velocity, Acceleration,
Force, etc.
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Vector Quantity
How to specify a velocity vector?
1. By scaled drawing:Draw an arrow of definite length
and direction to represent the vector.
2. By a statement:A car is travelling eastward at a velocity of 5 m/s.
5 m/s
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Vector QuantityFor example:
A boy travels 10 m along a direction of 200 east of north.
200
10 m
north
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Adding & Subtracting Scalars
• Same as in algebra• You only have to add algebraically the
variables togetheri.e. x units + y units = (x + y) units
e.g. Adding Time: 10s + 15s = 25se.g. Subtracting volumes: 15cm3 - 10cm3 = 5cm3
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Adding Vectors
• If the vectors are acting along the same line:
10 N
8 N12 N
the resultant force = (10 + 8 - 12) N
= 6N to the East
Just add them up algebraically!
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Adding Vectors
• If the vectors are acting at an angle to each other: Eric leaves the base camp and hikes 11.0 km, north and
then hikes 11.0 km east. Determine Eric's resulting displacement.
?
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Method 1: Graphical Method
Graphical Method / Scaled Vector Diagram
1. Decide on a scale (e.g. 1cm : 1 km)
2. Draw the vectors in the desired directions
11 km
11 km
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Graphical Method
1. Complete a parallelogram using the 2 sides given.
2. Draw the diagonal that represents the resultant.
3. Measure the length that represents the magnitude.
4. Use a protractor to measure the angle the resultant makes with a specified reference direction.
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11.0 km
11.0 km
15.6 km
In this example, Eric’s final displacement is 15.6 km (because the red line is 15.6 cm long) and is at 450 East of North.
450
Graphical Method
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Mathematical MethodWe use the Pythagoras’ Theoremc = (a2 + b2)where c is the resultant
Method 2- Mathematical Method
112 + 112 = R2
R = 15.6 m
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To find the direction of the resultant, we use the definition of tangent.Tan = opposite side / adjacent side = tan-1 (opposite side / adjacent side)
Mathematical Method
Mathematical Method
= tan-1 (11.0 / 11.0) = 45o
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Class Practice Question 1
• A barge is pulled at a steady speed through still water by two cables as shown in the plan view below. By means of a vector diagram, determine the magnitude and direction of the resultant force exerted on the barge by the cables. [3]
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• [1] -- for an appropriate scale (take up more than ½ of the space provided)
• [1] – R = 1.1 x 105 N (tolerance of 0.1 x 105 N )
• [1] – R is 37o clockwise from F2
Class Practice Question 1
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Question?
• Can we still use Pythagoras's method for mathematical method if the vectors are not perpendicular to each other?
?
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• Solve this problem by Mathematical method.
Class Practice Question 1
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Mathematical Method –when the vectors are not perpendicular
N
120o
Hint:
Apply cosine rule to this triangle to find magnitude of R
Apply sine rule to find direction of R
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To find magnitude:
c2 = a2 + b2 - 2ab cosc = 75 0002 + 50 0002 – 2(75 000) (50 000)cos120o
c = 1.09 x 105
The magnitude of resultant is 1.09 x 105 N.
To find direction:75 000 / sinA = 109 000/ sin120A = 37o
Cosine Rule
sine Rule
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Question?
• But can we still use the graphical method is there are more than 2 vectors to be added?
20 m
25 m
15 m
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Graphical Method – Head-to-tail Method
• The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position.
• Where the head of this first vector ends, the tail of the second vector begins (thus, head-to-tail method).
• The process is repeated for all vectors which are being added.
• Once all the vectors have been added head-to-tail, the resultant is then drawn from the tail of the first vector to the head of the last vector; i.e., from start to finish.
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Graphical Method – for more than two vectors
• Head-to-tail method
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Example
weight
drag
Lift
Thrust
What is the resultant force on the plane?
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Using Graphical method• Head-to-Tail Method
(for addition of more than 2 vectors)
weight
drag
Lift
Thrust
Resultant
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Question?
• But can we still use the mathematical method is there are more than 2 vectors to be added?
20 m
25 m
15 m
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Mathematical Method – for more than two vectors
• When there are more than two vectorsSimply use any of the above methods and solve this two
vectors at a time.A
B
C
D
E
First find the resultant of A and B, and name it D.
Then find the resultant of D and C, which is E and which is also the resultant of the three vectors.
It doesn’t matter which two vectors you resolve first, be A & C or B & C, the answer will still be the same.
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Addition & Subtraction of Vector Quantities
• A VERY IMPORTANT NOTEIf the vector sum is 0 the object that the vectors are
acting on is in equilibrium; it doesn’t move at all.
8N 8N
The vector sum is 0.
10N
6N
8NThe vector sum is 0.
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Equilibrium
• For example, if a box stays in equilibrium,the resultant of F1 and F2 must be equal and opposite to F3.
F1 = 4 N
F2 = 3 N
F3 = 7N
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Equilibrium
• For example, if a box stays in equilibrium,the resultant of F1 and F2 must be equal and opposite to F3. F1
F2
F3
R
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Equilibrium
• Equilibrium means • the forces acting on that
object are balanced• the resultant force is zero• the object does not move
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Example• This system is in equilibrium.
Find the weight of the car by graphical method.
736 g
425 g
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Ans
• Draw a free-body diagram to show all the forces.
T1 = 4.25 N
T2 = 7.36 N
30o
W
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Ans• From the free-body diagram, it is clear that Resultant of T1
and T2 must be equal and opposite to W so that the system remains in equilibrium. Hence, to find W, just find resultant of T1 and T2 by graphical method.
T1 = 4.25 N
T2 = 7.36 N
Ans: W = 8.5 N