chapter 4 vectors the cardinal directions. vectors an arrow-tipped line segment used to represent...
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Chapter 4 Vectors
The Cardinal Directions
Vectors
• An arrow-tipped line segment used to represent different quantities.
• Length represents magnitude.
• Arrow head represents direction.
Vector Addition in 1 - Dimension
• When vectors point in the same direction we add them just as we would add any two numbers.
Vector Addition in 1 - Dimension
• When vectors point in opposite directions we subtract them just as we would with any two numbers.
Vector Addition in 2-Dimensions
• Vectors in 2-dim are added by placing the tail of one to the head of another.
Remember This?
Addition of Several Vectors
• The order of addition is not important.
• R is called the resultant.
Independence of Vector Quantities
• Perpendicular vectors can be treated independently of each other.
Analytical Method of Vector Addition
• The sum of any two vectors can be determined using trigonometry.
Adding Perpendicular Vectors
Angle θ is =
a) 25 deg
b) 14 deg
c) 35 deg
d) 45 deg
•
Angle θ is =
a) 25 deg
b) 14 deg
c) 35 deg
d) 45 deg
Vector Components
• We can take two vectors and replace them with a single vector that has the same effect. This is vector addition.
• We can start with a single vector and think of it as a resultant of two perpendicular vectors called components.
• This process is called vector resolution.
Example
Example 2
Problem Solving Strategy
• In resolving vectors choose the most convenient axis according to the specifics of the problem.
• Choose the axis that simplifies the solution.• Axis may be up-down, left-right, east-west
or north-south.• Be sure to specify the positive direction for
each.
Adding Vectors at Any Angle
• Vector resolution is the method used.
• Resolve all vectors into x and y components.
• Add all x’s and all y’s together.
• Use xtot and ytot to create a right triangle.
• Use Pythagorean formula to calculate resultant and trig to find angle.
R is = ?
a) 15 N
b) 12 N
c) 20 N
d) 11N
R is = ?
a) 15 N
b) 12 N
c) 20 N
d) 11N
Θ is = ?
a) 53 deg
b) 35 deg
c) 25 deg
d) 45 deg
Θ is = ?
a) 53 deg
b) 35 deg
c) 25 deg
d) 45 deg
Applications of Vectors
Vectors can be used to represent:
- displacement
- velocity
- acceleration
- force
Equilibrium
• When the net force is zero, the object is in equilibrium.
• When the vector sum of the forces is not zero, a force can be applied that will produce equilibrium. This force is called the equilibrant.
• It is equal in magnitude but opposite in direction to the resultant.
3 Forces in Equilibrium:
a) produce a net force.
b) produce a triangle for a vector diagram.
c) are called an equilibrant.
d) produce an acceleration.
3 Forces in Equilibrium:
a) produce a net force.
b) produce a triangle for a vector diagram.
c) are called an equilibrant.
d) produce an acceleration.
Gravitational Force and Inclined Planes
• Gravitational force always points towards center of Earth.
• This is weight.
• Choose one axis parallel to the plane and the other perpendicular to it.
Formulas
• R2 = A2 + B2 – 2AB cos Θ
• Ax = A cos Θ
• AY = A sin Θ
• A = Ax + AY