Numerical operations with scalars are simply a matter of adding or subtracting the numbers.
Example Example Example Example Example
We have 9 marbles and find 5 marbles. How many do we now have?
Solution Solution Solution Solution Solution
9 5 14
marbles marbles marbles ( )+
As you can see, simply add to perform the operation.
The same operation with vector quantities requires a bit more thought, as both the magnitude and direction must be included.
We’ll start with a straightforward addition problem.
Example
A student walks 4 blocks east and stops at an ice cream truck. After purchasing a snow cone, he walks another 10 blocks east. Where is the student in respect to his starting point?
Solution
4 blocks @ east (+)10 blocks @ east 14 blocks @ east
The student has walked a total of 14 blocks due east.
As in this example, when two vectors are added together, the result is conveniently called the resultant vector. The original vectors, which were added together, are the component vectors.
The problem above shows the nature of vectors. Both magnitude and direction must be included in operations with vectors.
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The next problem requires the use of the Pythagorean Theorem when we add a pair of vectors. We will also use the Cartesian Coordi-nate System (x and y axis).
Example
A bird is perched in the tree where it has its nest. The bird flies 500 m due east and lands on the ground in a field where it finds a worm.
When the bird takes off, it is chased by a hawk, so the bird flies 300 m due north before landing in a tree.
What direction must the bird fly to find its nest, and how far away is the nest?
Solution
When solving physics problems, always use a convenient method with which you are comfortable. This is how you should approach any problem you are solving. A method is suggested below.
1. Draw a diagram (above)
2. Isolate and label the parts of the problem 3. Identify both x and y components
4. Write the equation 5. Solve for the unknown
Let’s place the bird’s starting point at the x, y juncture, which is also its starting point at the nest. The bird is presently located at the head of the 300 m north vector.
Before solving the problem, convert the compass values to the x, y coordinate system. The due east vector is located directly along the x coordinate, which is 0º. The due north vector is located directly along the x coordinate, which is 90º.
SCALARS AND VECTORS
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As we inspect the problem it becomes clear that the Pythagorean Theorem is the best way to solve the problem.
c2 = a2 + b2 Change to the x, y coordinates and
This is the magnitude value of the resultant vector. The direction for the resultant can be found with a little right triangle trigonom-etry.
Thus the resultant vector (where the bird is located in respect to its starting point) is
583m @ 31°
Recall that the bird wanted to fly back to its nest. It can’t fly at 31° from its current position because that path takes the bird farther from its nest. The direction we have found must be reversed for the bird to return to its nest.
The resulting direction is exactly 180° opposite the direction the bird must fly. Take the resultant vector and add (or subtract) 180°to or from the vector’s direction.
180° + 31° = 211°
The bird must fly 583 m @ 211° from its present position to reach its nest. When this is done, the bird will effectively cancel out the resultant vector. That’s why its flight will be the equilibrant.
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An equilibrant vector is a vector that is exactly equal in magni-tude and opposite in direction from the resultant vector.
Example
Let us look at one more vector problem.
Suppose 4 ropes were used to pull on a stationary object.
• Rope a pulls in a direction of 15° with a force of 25N.
• Rope b pulls in a direction of 215° with a force of 16N.
• Rope c pulls in a direction of 75° with a force of 20N.
• Rope d pulls in a direction of 300° with a force of 30N.
If a single rope were to replace the four ropes, with what force and in what direction must the rope pull?
Solution
The diagram above allows us to see each vector in relation to all the other vectors.
We will isolate each vector in its turn and break each one into its x and y components. Then we can combine all the individual x and y components to find the resultant vector.
First, check and make sure that when you construct your tri-angles, the y side of the triangle is the opposite side of the triangle, and the x side is the adjacent side of the triangle.
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Rope a extends 15° into the first quadrant.
Side y = Side r (sin 15°) = (30N) (.26) = 7.8N Side x = Side r (cos 15°) = (30N) (.97) = 29.1N
Both components of rope a are located in the first quadrant.
y x
positive 7.8N positive 29.1N
= ∴+
= ∴+
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Rope b extends 35° into the third quadrant .
Side y = (side r) (sin 35°) = (16N) (.57) = 9.1N Side x = (side r) (cos 35°) = (16N) (.82) = 13.1N
Both components of rope b are located in the third quadrant.
y x
negative N negative N
= ∴−
= ∴−
9 1 13 1
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Rope c extends 75° into the first quadrant.
Side y = (side r) (sin 75° ) = (20N) (.97) = 19.4N Side x = (side r) (cos 75° ) = (20N) (.26) = 5.2N
Both components of rope c are located in the first quadrant.
y x
= ∴+
= ∴+
positive N positive N
19 4 5 2
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Rope d extends 30° into the third quadrant.
Side y = (side r) (sin 60°) = (30N) (.87) = 26.1 Side x = (side r) (cos 60°) = (30N) (.5) = 15.0N
Both components of rope d are located in the fourth quadrant.
y x
= ∴−
= ∴+
negative N positive N 26 1 15 0
. .
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All four ropes have been broken into their component vectors at this point. It is time to add them up. We can accomplish this by using a simple x, y chart. The chart is filled in with the individual components we have just found. Having done that, we then algebra-ically combine all the y components and do the same with the x components –8N in x direction
+36.2N in y direction
After combining the x components and the y components, the values are the two components of the new resultant vector. Apply the Pythagorean Theorem to find its magnitude.
r x y
Now that we know the magnitude of the resultant, we’ll use the tangent function to find the direction of the resultant vector.
Using the tangent function yields:
tan N
This number tells us that the direction of the resultant vector is –1°. That means that to find the value of the resultant vector, we simply subtract 1° from 360° to find the direction a single rope must pull to equal the pull from the original four ropes.
The resultant vector is 37N @ 359°
Simply stated: One rope pulling with a force of 37N in a direction of 359° accomplishes the exact same thing as the other four ropes.
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