Combining vectors in 2D

Components,

Overall Velocity or force

Equilibrium, Equilibriants

BREAKING A VECTOR DOWN INTO ITS COMPONENTS

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

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

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

If you walk 32 m towards the NE, how far do you travel east?

Components

Horizontal component

Vertical

Component

Resultant side

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]

Find the components of :

• 57 N [N 57W]

Find the components of:

• 87.6 m ( 300°)

• Remember, here the cardinal directions are 0/360, 90, 180, 270

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)

Add the following vectors

• 30 m/s [ NE] + 45 m/s [S 67 W]

Calculate sides or angles

PUTTING ALL VECTOR IDEAS TOGETHER TO SOLVE A PROBLEM

Friday’s in class question

173 N [W 57.352 S] + 142 [E 51.22 N]

Answer

The vectors Drawn with Components

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]

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]

FINDING THE EQUILIBRIANT

Equilibrium

• When the sum of all forces acting on an object cancel each other out.

• Net force = 0

Equilibrant

• A force added to all other forces applied to an object to achieve equilibrium

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

ALL VECTOR VALUES CAN BE COMBINED USING THE ABOVE METHODS

Displacement, Velocity, Acceleration, Force

SCALAR VALUES COMBINED WITH VECTORS

Effect of scalar values on2-D force problems

• Scalar values: Those that do not indicate a direction (time, mass)

• Scalars affect magnitude, not direction

APPLICATION OF VECTOR MATH TO SOLVING 2D PROBLEMS