Download Vectors Part II

Vectors Part II
Ch. 6
Introduction
• Although we already talked a little about vectors,
there’s still quite a bit to learn about them (I’m
sorry ).
– By studying the vector nature of forces, the concept of
equilibrium will be developed.
• This study of the forces acting on an object that is in
equilibrium is called statics.
6:1 Review
• In adding 2 vectors, place the tail of one vector at
the head of the other vector.
– Next, add a third vector going from the tail of the first
vector to the head of the 2nd vector. This is the resultant.
– The direction is expressed as an angle measured
clockwise from north (0°).
6:2 Independence of Vector Quantities
• Vectors act independently.
– One vector does not change another.
• Example: A motorboat heads due east at 8.0 m/s across a river flowing due south
at 5.0 m/s.
– The boat will travel 8.0 meters due east in 1 second. It will also travel 5.0
meters due south in that same second.
• These velocities cannot change one another.
• What is the resultant vector?
– When the vectors are added together, the resultant velocity is:
– This means the same thing as saying the boat traveled east 8.0 m/s and south
5.0 m/s at the same time.
6:3 Vector Addition on Forces
• Force vectors are added in the same way as velocity vectors.
– Forces that act on the same point at the same time are called
concurrent forces.
• Example: A force of 45 N and a force of 65 N act concurrently on
point P. The smaller force acts in the direction of 30° and the larger
force acts at 90 °.
• Draw the original, then move the vectors to make a triangle.
• You may have to do the measuring of this with a protractor… 
Guided Practice/HW:
• Let’s do #1-4, 6 on pg. 97 and pg. 98, # 9.
6:4 Vector Addition- Mathematical Method
• The Pythagorean Theorem can be used to find the
length of the 3rd side of a right triangle.
– Review: a2 + b2 = c2.
– Example: Consider an airplane that is flying due east at
90.0 km/h and is being blown north at 50.0 km/h. What
is the resultant velocity of the plane?
• Non-right triangles:
– The addition of vectors doesn’t always result in
the creation of right triangles.
• If this is the case, you must use the Law of Cosines and
the Law of Sines.
• Law of Cosines:
– When we know two sides of a triangle and their included
angle, then the Law of Cosines enables us to find the
third side.
– Thus, if we know sides a and b and their included angle,
then the Law of Cosines states:
c2 = a2 + b2 – 2ab cos C
• Law of Sines:
– Allows us to solve triangles that are not right-angled,
and are called oblique triangles.
– The sides of a triangle are to one another in the same
ratio as the sines of their opposite angles.
Practice!
• #14-17 on pg. 100
6:5 Addition of Several Vectors
• Quite often, 3 or more vectors act
concurrently on the same point.
– You follow the same procedure as with adding 2
vectors.
6:6 Equilibrium
• Sometimes, the sum of these forces equals 0.
– If this occurs, the object is in equilibrium and the
object has no acceleration.
6:7 The Equilibrant
• An equilibrant force balances the forces acting on a
point to produce equilibrium.
• The equilibrant is equal in magnitude to the
resultant, but opposite in direction.
• The equilibrant is the force that is equal and
opposite to the resultant.
– It produces equilibrium.
• Equilibrium does not mean
that there is no motion! There
is no CHANGE in motion.
Practice!
• Pg. 102, #20-22
6:8 Perpendicular Components of Vectors
• As we’ve seen in the past, it is possible to regard a
single vector quantity as the resultant of 2 vectors,
each acting in directions other than the original
vector.
– These vectors are called components.
– The process of finding the effective value of a
component in a given direction is called vector
resolution.
Example:
• A 58-N force is being exerted on a rope held at an
angle of 30 degrees with the horizontal. What are
the vertical and pulling forces being exerted on the
rope?
• One thing that we find is that as the direction of the
force changes, the magnitude of the components
also change.
• In adding non-perpendicular vectors, first find the
components of those vectors. Then add the vertical
and horizontal components separately. Finally
combine the sums vectorally to form the resultant
vectors.
Example: Resolving a Velocity Vector
into its Components
• A wind with a velocity of 40.0 km/h blows at 60.0
degrees.
A. What’s the North component of the wind’s velocity?
A. What is the East component of the wind’s velocity?
Practice!
• Pg. 105: #25-27
6:9 Gravitational Force and Inclined Planes
• The gravitational force acting on an incline is
directed toward the center of the Earth.
– This means that the object’s weight, W, must act
perpendicular to the surface of the Earth.
• The weight of the trunk FW can be resolved into 2 components
that are perpendicular to each other.
• The parallel force and force of friction are parallel to
the surface. The weight goes straight down to the
earth’s surface. The perpendicular force is
perpendicular to the surface of the incline. The
normal force is numerically equal to the
perpendicular force.
Example:
• A trunk weighing 562 N is resting on a plane
inclined at 30.0 degrees from the horizontal.
Find the magnitude of the parallel and
perpendicular components of the weight.
Practice!
• Pg. 107: #30, 32
6:10 Nonperpendicular Components of
Vectors
• A vector doesn’t always have to be resolved into 2
components at right angles.
– It can be resolved into components that lie in any
direction as long as their vector sum is equal to the
original vector.
• The direction of the component depends on the
direction of the object exerting the force.
Example
• A sign weighing 40 N is supported by ropes A
and B.
– There are 3 forces acting on this sign.
• Force in rope A, force in rope B, and the weight of the
sign.
– Because the sign is in equilibrium, the forces in
the 2 ropes must produce a resultant of 40 N
straight up to balance the weight of the sign.
• This makes an equilateral triangle, so the force on ropes
A and B must also be 40 N.