Chapter 5
Forces in Two
Dimensions
Vectors:
• Vectors have both magnitude and
direction.
• Vectors must be added using vector
addition.
– You will have to treat vertical and horizontal
vectors separately.
• You can add vectors in any order as long
as you do not change there length or
direction.
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Measuring Angles
• GEOGRAPHICAL :
• 40 degrees North of
West
• 50 degrees West of
North
• MATHEMTATICAL:
140 degrees
counterclockwise from
+x axis
Vector Direction Examples
•Vector 35
m/s, due
South
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5
Resultant Vector
• The vector that results from the addition of
2 or more vectors.
• Always drawn from the “tail” of the first
vector to the “tip” of the last vector.
• Direction should always be measured
between the first vector and the resultant.
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Adding Vectors and Finding the
Resultant: Method 1
1. Scaled Vector Diagram/Graphically
•
Decide on a scale (EX: 1 km = 1 cm)
•
Use a ruler to measure the vectors and a
protractor to measure angle direction. Draw the
vectors tip to tail.
•
Draw the resultant vector from the tail of the first
to the tip of the last vector.
•
Use a ruler to measure the magnitude of the
resultant vector
•
Use a protractor to measure the angle of
direction (angle between the 1st vector and the
resultant vector).
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Adding Vectors and Finding the
Resultant: Method 2
1. Mathematical Method
•
If the two vectors being added are
at right angles, the magnitude can
be found using the Pythagorean
Theorem and the direction can be
found using trig ratios (SOH CAH
TOA).
•
If the two vectors being added are
at some angle other than 90, the
magnitude and direction can be
found by using the Law of Cosines
and the Law of Sines.
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Adding Vectors
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Law of Cosines
a2 = b2 + c2 - 2bc (cos A)
2
2
2
b = a + c - 2ac (cos B)
2
2
2
c = a + b - 2ab (cos C)
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Law of Sines
a
b
c__
=
=
sin A
sin B
sin C
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Adding Vectors - perpendicular
Add these vectors -determine the resultant.
• 2.0 m/s, 90 deg
• 7.0 m/s, 0 deg
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Adding Vectors
Add the following vectors - determine the resultant.
• 3.0 m/s, 45 deg
• 5.0 m/s, 135 deg
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Examples:
1. A person walks 100m N and loses all sense if direction. Without
knowing the direction, she walks 100m again. Draw a vector
representation and determine the range of her displacement.
2. You are traveling from SMCC to Jackson for the cross country
meet. You travel 30 km west, 20 km north, and 10 km west. Find
your displacement (magnitude and direction) both graphically and
mathematically.
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Examples:
1. In your basketball game, you start at half court
and run straight down the sideline 20 m. You
then make a sharp 90 degree cut towards the
lane. You run 15 m before the ball is thrown to
you and you catch it with a jump stop. What is
the magnitude and direction of your
displacement?
2. A person jogs 15 km and then turns to the right
at a 45 degree angle and continues to run 25
more kilometers. Find the resultant vector
(magnitude and direction) for the jogger.
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Components of a Vector – the
horizontal and vertical vectors
that make up the resultant
Components
• You can use trig
to find the
components.
*Be careful if the
angle is bigger
than 90 degrees.
You may have to
use a reference
angle.
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Method 3: Vector
Resolution/Components
Two or more vectors can be added by:
• Resolving each vector into its x and y
components.
• Add all the x-components to form the xcomponent of the resultant:
Rx = Ax + Bx + Cx…
• Add all the y-components to form the ycomponent of the resultant:
Ry = Ay + By + Cy…
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• Use the Pythagorean
Theorem to find the
magnitude of the
resultant R.
R2 = Rx2 + Ry2
• Use tangent to find the
direction of R.
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Examples:
1. A bus travels 23 km on a straight
road that is 30º N of E. What are
east and north components of its
displacement?
2. A hammer slides down a roof that
makes a 40 angle with the
horizontal. What are the
magnitudes of the components of
the hammer’s velocity at the edge
of the roof if it is moving at a speed
of 4.25 m/s?
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EX:
Add the following three vectors using the
component method: A is 4 m south, B is
7.3 m northwest, C is 6 m 30⁰ south of
west.
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Example:
2. A GPS receiver told you that
your home was 15 km at a
direction of 40º north of west, but
the only path led directly north. If
you took that path and walked 10
km, how far and in what direction
would you then have to walk to
reach your home?
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FRICTION
• Kinetic friction force: the force exerted on
one surface by another surface when the
objects are in motion
EX: Sliding your book across your desk
Ff = µkFN
• µ = “mu” = coefficient of friction
• Ff is proportional to the force pushing
one surface against the other (FN)
FRICTION
• Static friction force:
the force exerted on
one surface by
another surface when
there is no motion
between the two
surfaces.
EX: Pushing on a
car or an extremely
heavy crate…
FRICTION
• Eventually there is a limit to this static
friction force – once the applied force is
greater than the maximum static friction
force, the object will begin to move.
• If the applied force increases, the static
friction force will increase up to a
maximum value. Ff ≤ μsFN
• At the instant before motion: Ff = μsFN
FRICTION
• Besides the normal force, friction
also depends on the types of
surfaces that are in contact.
• Different surfaces have different
coefficients of friction (for static
and kinetic)
• Table 5-1 p.129
EX: You push a 25 kg wooden
box across a wooden floor at a
constant speed of 1 m/s. How
much force do you exert on the
box?
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EX:
A small child is dragging a heavy, rubbersoled shoe by its laces across a sidewalk
at a constant speed of 0.35 m/s. If the
shoe has a mass of 1.56 kg, what is the
horizontal component of the force exerted
by the child?
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EX:
• If the child pulls with an extra 2 N in the
horizontal direction, what will be the
acceleration of the shoe?
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Inclined Planes
• A tilted surface is an inclined
plane.
• Objects accelerate down
inclined planes because of an
unbalanced force.
• The force of gravity acts in
the downward direction.
• The normal force acts in a
direction perpendicular to the
surface.
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Components of vectors
Analyzing forces on inclined planes will
involve resolving the weight vector (Fgrav)
into two perpendicular components.
- one parallel to the surface
- one perpendicular to the inclined surface.
The parallel force causes acceleration
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Components of vectors


Wt
F
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The angle of incline always equals the angle
between the weight vector and its
perpendicular component.
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Use sine and cosine
to find the components.
Practice
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Example
A trunk weighing 562 N
is resting on a plane
inclined 30º above the
horizontal. Find the
components of the
weight force parallel
and perpendicular to
the plane.
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Example
A 62 kg person on skis is
going down a hill sloped
at 37º. The coefficient
of kinetic friction
between the skis and
the snow is 0.15. How
fast is the skier going 5
s after starting from
rest?
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Equilibrant
Equilibrant – a force that puts an object in
equilibrium.
To find the equilibrant:
 Find the resultant of all the forces on the
object.
 The equilibrant is the same in magnitude
but opposite in direction.
 Equilibrant
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EX: What is the equilibrant for an 8 N force
applied at 0º,a 6 N force applied at 90º,
and a 7 N force applied at 60º?
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EX: What is the tension in each
cable?
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