Download Ch 4 Vectors b

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)
b)
c)
d)
25 deg
14 deg
35 deg
45 deg
Angle θ is =
•a)
b)
c)
d)
25 deg
14 deg
35 deg
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)
b)
c)
d)
15 N
12 N
20 N
11N
R is = ?
a)
b)
c)
d)
15 N
12 N
20 N
11N
Θ is = ?
a)
b)
c)
d)
53 deg
35 deg
25 deg
45 deg
Θ is = ?
a)
b)
c)
d)
53 deg
35 deg
25 deg
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)
b)
c)
d)
produce a net force.
produce a triangle for a vector diagram.
are called an equilibrant.
produce an acceleration.
3 Forces in Equilibrium:
a)
b)
c)
d)
produce a net force.
produce a triangle for a vector diagram.
are called an equilibrant.
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