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Combining vectors in 2D
Components,
Overall Velocity or force
Equilibrium, Equilibriants
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BREAKING A VECTOR DOWN INTO ITS COMPONENTS
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Components of a vector
• Portion of a vector value that affects 1 dimension
• Any vector moving at an angle not parallel or perpendicular to a coordinate axis can be broken down into 2 perpendicular components
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How to find components
• Draw 2-d vector
• Create a triangle using the 2-D vector as hypotenuse
• Drawn legs should intersect at a right angle
• Use trig to calculate unknown sides or angles
• Indicate direction through drawing or name
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To solve for components
• Need to be able to use right triangle trigonometry
• Trigonometry focuses on the ratio values of sides of a right triangle
• As angles change vary in different right triangles, so do the ratio of sides to each other
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If you walk 32 m towards the NE, how far do you travel east?
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Components
Horizontal component
Vertical
Component
Resultant side
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Adding vectors not in cardinal directions
• Break all vectors down into components, that’s is smaller vectors in cardinal directions that add up to the original vector.
Example:
4 m/s [E]
Horizontal component
3 m/s [N]
Vertical component
5 m/s
[E 36.9 N]
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Find the components of :
• 57 N [N 57W]
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Find the components of:
• 87.6 m ( 300°)
• Remember, here the cardinal directions are 0/360, 90, 180, 270
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Overall velocity problem
• If the projectile landed with a vertical velocity of 37 m/s downward and a horizontal velocity of 29 m/s to the right, what would be the overall final velocity (magnitude and direction)
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Add the following vectors
• 30 m/s [ NE] + 45 m/s [S 67 W]
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Calculate sides or angles
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PUTTING ALL VECTOR IDEAS TOGETHER TO SOLVE A PROBLEM
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Friday’s in class question
173 N [W 57.352 S] + 142 [E 51.22 N]
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Answer
The vectors Drawn with Components
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Calculations for the answer
The 4 component vectors:
93.3N [W], 146N [S]
88.9N [E], 111N [N]
These combine to : 4.4 N [W], 35 N [S]
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Final Recombination and answer
• Final triangle is drawn as:
Final answer is: 35.3 N [W 82.8 S] or
35.5 N [S 7.2 W]
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FINDING THE EQUILIBRIANT
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Equilibrium
• When the sum of all forces acting on an object cancel each other out.
• Net force = 0
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Equilibrant
• A force added to all other forces applied to an object to achieve equilibrium
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equilibrium problems
• These are problems that require an object to be at equilibrium.
• Equilibrium = all forces applied to object balance each other out. Net force on object = 0
• Equilibriant: the force applied to an object that produces equilibrium
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ALL VECTOR VALUES CAN BE COMBINED USING THE ABOVE METHODS
Displacement, Velocity, Acceleration, Force
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SCALAR VALUES COMBINED WITH VECTORS
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Effect of scalar values on2-D force problems
• Scalar values: Those that do not indicate a direction (time, mass)
• Scalars affect magnitude, not direction
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APPLICATION OF VECTOR MATH TO SOLVING 2D PROBLEMS