Chapter
5
Forces in Two Dimensions
In this chapter you will:
Add vectors graphically
and mathematically.
Use Newton’s laws to
analyze motion when
friction is involved.
Analyze force vectors in
equilibrium.
Section
5.1
Warmup
Consider a person who walks 100 m due north
and then loses all sense of direction. Without
knowing the direction, the person walks another
100 m. What could the magnitude of displacement
be relative to the original starting point? Hint:
Draw a picture using two vectors.
Chapter
Table of Contents
5
Chapter 5: Forces in Two Dimensions
Section 5.1: Vectors
Section 5.2: Friction
Section 5.3: Force and Motion in Two Dimensions (what does twodimensions mean?)
Homework for Chapter 5
Read Chapter 5
Study Guide 5, due before the Chapter Test
HW 5.A, HW 5.B: Handouts. You will need a ruler / protractor.
Section
5.1
Vectors
In this section you will:
Measure and report the magnitude of vectors.
Evaluate the sum of two or more vectors in two dimensions.
Determine the components of vectors.
Section
Vectors
5.1
Vectors and Scalars
vector – A quantity with magnitude (size) and direction.
ex: displacement, velocity, acceleration, and force
Vectors are represented by arrows.
tail
tip or head
scalar - A quantity with magnitude only.
ex: distance, speed, mass, time, and volume
Section
Vectors
5.1
Vectors in Two Dimensions
The direction of a vector is where it points towards. You can specify
the angle in degrees, as measured counter-clockwise from the x-axis.
N
NW
More generally,
give the cardinal
direction. Here,
the vector’s
direction is
southwest, or
SW.
W
180°
SW
90°
θ = 225°
270°
S
NE
0°
360°
E
SE
Section
Vectors
5.1
Adding Vectors

Vectors can be added by placing arrows tip to tail.

resultant - The arrow that extends fro the tail of the first vector to
the tip of the last vector. It indicates both the magnitude and the
direction of the vector sum.

Vectors may be added in any order (commutative operation).
A+B=B+A
 Vectors may be shifted around, but as long as the keep the same
size and direction, they are the same vector.
Section
Vectors
5.1
Adding Vectors Graphically
The figure below shows two force vectors on a free-body diagram.
vector a
Shift vector b so a
and b are tip to tail.
→
vector a
vector b
Instead, shift vector b so it
is tip-to-tail with vector a.
vector b
vector b
Vectors a and b are originally
drawn tail-to-tail. They can not
be added this way.
Section
Vectors
5.1
Adding Vectors Graphically
You can draw the resultant vector pointing from the tail of the first vector
to the tip of the last vector and measure it to obtain its magnitude.
vector b
Practice: Adding
Vectors Graphically
Worksheet.
vector a
Section
5.1
Vectors
Adding Vectors Mathematically
You may also use mathematical methods to determine the
length or direction of resultant vectors.
If you are adding together two vectors at right angles, vector A
pointing north and vector B pointing west, you could use the
Pythagorean theorem to find the magnitude of the resultant, R.
R
A
B
Section
5.1
Vectors
Finding the Magnitude of the Sum of Two Vectors
Find the magnitude of the sum of a 15-km displacement and a 25-km
displacement when the angle between them is 90° and when the
angle between them is 135°.
Section
Vectors
5.1
Finding the Magnitude of the Sum of Two Vectors
Sketch the two displacement vectors, A and B, and the angle
between them.
B
A
θ1
Section
Vectors
5.1
Finding the Magnitude of the Sum of Two Vectors
Identify the known and unknown variables.
A
θ1
R
B
Known:
Unknown:
A = 25 km
R=?
B = 15 km
θ1 = 90°
θ2 = 135 °
Section
5.1
Vectors
Finding the Magnitude of the Sum of Two Vectors
When the angle is 90°, use the Pythagorean theorem to find the
magnitude of the resultant vector.
Substitute A = 25 km, B = 15 km
Section
Vectors
5.1
Finding the Magnitude of the Sum of Two Vectors
When the angle does not equal 90°, use the graphical method to find
the magnitude of the resultant vector:
1) Draw the vectors to scale. Pick a scale that will fit on your paper.
example: 15 km = 1.5 cm
2) Measure the resultant.
For our problem, the resultant measures 37 cm.
3) Convert back to the original units.
3.7 cm = 37 km
Section
5.1
Vectors
Finding the Magnitude of the Sum of Two Vectors
Are the units correct?
Each answer is a length measured in kilometers.
Do the signs make sense?
The sums are positive.
Are the magnitudes realistic?
The magnitudes are in the same range as the two combined
vectors, but longer. The answers make sense.
Section
Vectors
5.1
Finding the Sum of Two Vectors
p. 121 Practice Problems:
1 & 2: Use the mathematical method (Pythagorean theorem and
calculator) to find the magnitude of the displacement (length of the
resultant).
Also, report the direction of displacement for #1. Possible answers
are N, NE, E, SE, S, SW, W, NW.
3 & 4: Use the graphical method (ruler) to find the magnitude of the
displacement.
Also, report the direction of displacement for #4.
Section
Vectors
5.1
Vector Components
Because a vector has both magnitude and direction, you can separate
it into horizontal (or x-) and vertical (or y-) components.
To do this, draw a rectangle with horizontal and vertical sides and a
diagonal equal to the vector.
Draw arrow heads on one horizontal and one vertical side to make the
original vector the resultant of the horizontal and vertical components.
Label the vector and its x- and ycomponents.
F
Fy
vector resolution - Breaking a vector
into its x- and y- components.
Fx
Section
Vectors
5.1
Practice with Vector Resolution
Use a ruler to draw and label the x- and y- components of these vectors.
2.
1.
d
v
3.
F
4.
a
HW 5.A Handout
Section
Section Check
5.1
Question 1
Jeff moved 3 m due north, and then 4 m due west to his friends
house. What is the magnitude of Jeff’s displacement?
A. 3 + 4 m
B. 4 – 3 m
C. 32 + 42 m
D. 5 m
Section
Section Check
5.1
Answer 1
Answer: D
Reason: When two vectors are at right angles to each other as in
this case, we can use the Pythagorean theorem of vector
addition to find the magnitude of resultant, R.
Section
Section Check
5.1
Answer 1
Answer: D
Reason: Pythagorean theorem of vector addition states
If vector A is at right angle to vector B then the sum of
squares of magnitudes is equal to square of magnitude of
resultant vector.
That is, R2 = A2 + B2
 R2 = (3 m)2 + (4 m)2 = 5 m
Section
5.1
Section Check
Question 2
Is the distance that you walk equal to the magnitude of your
displacement? Give an example that supports your conclusion.
Answer 2
Not necessarily. For example, you could walk around the block (one km
per side). Your displacement would be zero, but the distance that you
walk would be 4 kilometers.
Section
Section Check
5.1
Question 3
Could a vector ever be shorter than one of its components? Equal in
length to one of its components? Explain.
Answer 3
It could never be shorter than one of its components, but if it lies
along either the x- or y- axis, then one of its components equals its
length.
Section
Section Check
5.1
Question 4
The order in which vectors are added does not matter.
Mathematicians say that vector addition is commutative. Which
ordinary arithmetic operations are commutative?
Answer 4
Addition and multiplication are commutative. Subtraction and
division are not.
Section
Section Check
5.1
Question 5
Which of the following actions is permissible when you graphically
add one vector to another: moving the vector, rotating the vector, or
changing the vector’s length?
Answer 5
Allowed: moving the vector without changing length or direction.
Section
Section Check
5.1
Question 6
In your own words, write a clear definition of the resultant of two or
more vectors. Do not explain how to find it; explain what it
represents.
Answer 6
The resultant is the vector sum of two or more vectors. It represents
the quantity that results from adding the vectors.
Section
5.1
Vectors
Please Do Now
Write the three most important things you’ve learned and
want to remember about adding vectors.
Section
5.2
Friction
In this section you will:
Define the friction force.
Distinguish between static and kinetic friction.
Section
5.2
Friction
Static and Kinetic Friction
Push your hand across your desktop and feel the force called
friction opposing the motion.
A frictional force acts when two surfaces touch.
When you push a book across the desk, it experiences a type of
friction that acts on moving objects.
This force is known as kinetic friction, and it is exerted on one
surface by another when the two surfaces rub against each
other because one or both of them are moving.
Section
5.2
Friction
Static and Kinetic Friction
To understand the other kind of friction, imagine trying to push a
heavy couch across the floor. You give it a push, but it does not
move.
Because it does not move, Newton’s laws tell you that there
must be a second horizontal force acting on the couch, one that
opposes your force and is equal in size.
This force is static friction, which is the force exerted on one
surface by another when there is no motion between the two
surfaces.
Section
5.2
Friction
Static and Kinetic Friction
You might push harder and harder, as shown in the figure below,
but if the couch still does not move, the force of friction must be
getting larger.
This is because the static friction force acts in response to other
forces.
Section
5.2
Friction
Static and Kinetic Friction
Finally, when you push hard enough, as shown in the figure
below, the couch will begin to move.
Evidently, there is a limit to how large the static friction force can
be. Once your force is greater than this maximum static friction,
the couch begins moving and kinetic friction begins to act on it
instead of static friction.
Section
5.2
Friction
Static and Kinetic Friction
Friction depends on the materials in contact. For example, there is
more friction between skis and concrete than there is between skis and
snow.
The normal force between the two objects also matters. The harder one
object is pushed against the other, the greater the force of friction that
results. Since the normal force depends on the mass of the object,
greater mass means greater friction.
Surface area does NOT contribute to friction.
Give an example of low friction:
Give and example of high friction:
Section
5.2
Friction
Static and Kinetic Friction
If you pull a block along a surface at a constant velocity,
according to Newton’s laws, the frictional force must be equal
and opposite to the force with which you pull.
You can pull a block of known
mass along a table at a constant
velocity and use a spring scale,
as shown in the figure, to
measure the force that you exert.
You can then stack additional
blocks on the block to increase
the normal force and repeat the
measurement.
Section
5.2
Friction
Static and Kinetic Friction
Plotting the data will yield a graph like the one shown here.
There is a direct proportion between the kinetic friction force and
the normal force.
The different lines correspond
to dragging the block along
different surfaces.
Note that the line corresponding
to the sandpaper surface has a
steeper slope than the line for
the highly polished table.
Section
5.2
Friction
Static and Kinetic Friction
You would expect it to be much harder to pull the block along
sandpaper than along a polished table, so the slope must be related
to the magnitude of the resulting frictional force.
The slope of this line, designated μk, is called the coefficient of
kinetic friction between the two surfaces and relates the frictional
force to the normal force, as shown below.
Kinetic friction force
The kinetic friction force is equal to the product of the coefficient
of the kinetic friction and the normal force.
Section
5.2
Friction
Static and Kinetic Friction
The maximum static friction force is related to the normal force
in a similar way as the kinetic friction force.
The static friction force acts in response to a force trying to
cause a stationary object to start moving. If there is no such
force acting on an object, the static friction force is zero.
If there is a force trying to cause motion, the static friction force
will increase up to a maximum value before it is overcome and
motion starts.
Section
Friction
5.2
Static and Kinetic Friction
Static Friction Force
The static friction force is less than or equal to the product of the
coefficient of the static friction and the normal force.
μ
In the equation for the maximum static friction force, s is the
coefficient of static friction between the two surfaces, and
μsFN is the maximum static friction force that must be overcome
before motion can begin.
Section
5.2
Static and Kinetic Friction
Note that the equations for the kinetic and maximum static
friction forces involve only the magnitudes of the forces.
The forces themselves, Ff and FN, are at right angles to each other.
The table here shows coefficients of friction between various
surfaces.
Although all the listed coefficients are less than 1.0, this does not
mean that they must always be less than 1.0.
p.129
Section
5.2
Friction
Balanced Friction Forces
You push a 25.0 kg wooden box across a wooden floor at a constant
speed of 1.0 m/s. How much force do you exert on the box?
Identify the forces and establish a coordinate system.
Section
5.2
Balanced Friction Forces
Draw a motion diagram indicating constant v and a = 0.
Draw the free-body diagram.
Section
5.2
Balanced Friction Forces
Identify the known and unknown variables.
Known:
Unknown:
m = 25.0 kg
Fp = ?
v = 1.0 m/s
a = 0.0 m/s2
μk = 0.20
Section
5.2
Balanced Friction Forces
Solve for the Unknown
The normal force is in the y-direction, and there is no acceleration.
FN = Fg
= mg
Substitute m = 25.0 kg, g = 9.80 m/s2
FN = 25.0 kg(9.80 m/s2)
= 245
N
Section
5.2
Balanced Friction Forces
The pushing force is in the x-direction; v is constant, thus there is no
acceleration.
Fp = μkmg
Substitute μk = 0.20, m = 25.0 kg, g = 9.80 m/s2
Fp = (0.20)(25.0 kg)(9.80 m/s2)
= 49
N
Section
5.2
Balanced Friction Forces
Are the units correct?
Performing dimensional analysis on the units verifies that
force is measured in kg·m/s2 or N.
Does the sign make sense?
The positive sign agrees with the sketch.
Is the magnitude realistic?
The force is reasonable for moving a 25.0 kg box.
Section
5.2
Friction
Practice Problems p.128: 17-19.
Practice Problems p.130: 22,23.
HW 5.B
Section
Section Check
5.2
Question 1
Define friction force.
Section
Section Check
5.2
Answer 1
A force that opposes motion is called friction force. There are two
types of friction force:
1)
Kinetic friction—exerted on one surface by another when the
surfaces rub against each other because one or both of them
are moving.
2)
Static friction—exerted on one surface by another when there is
no motion between the two surfaces.
Section
Section Check
5.2
Question 2
Juan tried to push a huge refrigerator from one corner of his home to
another, but was unable to move it at all. When Jason accompanied
him, they where able to move it a few centimeter before the
refrigerator came to rest. Which force was opposing the motion of
the refrigerator?
A. Static friction
B. Kinetic friction
C. Before the refrigerator moved, static friction opposed the
motion. After the motion, kinetic friction opposed the motion.
D. Before the refrigerator moved, kinetic friction opposed the
motion. After the motion, static friction opposed the motion.
Section
Section Check
5.2
Answer 2
Answer: C
Reason: Before the refrigerator started moving, the static friction,
which acts when there is no motion between the two
surfaces, was opposing the motion. But static friction has a
limit. Once the force is greater than this maximum static
friction, the refrigerator begins moving. Then, kinetic
friction, the force acting between the surfaces in relative
motion, begins to act instead of static friction.
Section
Section Check
5.2
Question 3
On what does the magnitude of a friction force depend?
A. The material that the surface are made of
B. The surface area
C. Speed of the motion
D. The direction of the motion
Section
Section Check
5.2
Answer 3
Answer: A
Reason: The materials that the surfaces are made of play a role.
For example, there is more friction between skis and
concrete than there is between skis and snow.
Section
Section Check
5.2
Question 4
A player drags three blocks in a drag race, a 50-kg block, a 100-kg
block, and a 120-kg block with the same velocity. Which of the
following statement is true about the kinetic friction force acting in
each case?
A. Kinetic friction force is greater while dragging 50-kg block.
B. Kinetic friction force is greater while dragging 100-kg block.
C. Kinetic friction force is greater while dragging 120-kg block.
D. Kinetic friction force is same in all the three cases.
Section
Section Check
5.2
Answer 4
Answer: C
Reason: Kinetic friction force is directly proportional to the normal
force, and as the mass increases the normal force also
increases. Hence, the kinetic friction force will hit its limit
while dragging the maximum weight.
Section
5.3
Force in Two Dimensions
Equilibrants and the Parallelogram Method
In this section you will:
Determine the force that produces equilibrium when
multiple forces act on an object.
Learn the parallelogram method of vector addition.
Section
5.3
Force in Two Dimensions
Equilibrium Revisited
Now you will use your skills in adding vectors to analyze
situations in which the forces acting on an object are at angles
other than 90°.
Recall that when the net force on an object is zero, the object is
in equilibrium.
According to Newton’s laws, the object will not accelerate
because there is no net force acting on it; an object in
equilibrium is motionless or moves with constant velocity.
Section
5.3
Force in Two Dimensions
Equilibrium Revisited
It is important to realize that equilibrium can occur no matter
how many forces act on an object. As long as the resultant is
zero, the net force is zero and the object is in equilibrium.
The figure here shows three
forces exerted on a point
object. What is the net force
acting on the object?
Remember that vectors may
be moved if you do not
change their direction (angle)
or length.
Section
5.3
Force in Two Dimensions
Equilibrium Revisited
The figure here shows the addition of the three forces, A, B, and
C.
Note that the three vectors
form a closed triangle.
There is no net force; thus, the
sum is zero and the object is
in equilibrium.
Section
5.3
Force in Two Dimensions
Equilibrium Revisited
Suppose that two forces are exerted on an object and the sum is
not zero.
How could you find a third force that, when added to the other
two, would add up to zero, and therefore cause the object to be
in equilibrium?
To find this force, first find the sum of the two forces already
being exerted on the object.
This single force that produces the same effect as the two
individual forces added together, is called the resultant force.
Section
5.3
Force in Two Dimensions
Equilibrium Revisited
The force that you need to find is one with the same magnitude
as the resultant force, but in the opposite direction.
A force that puts an object in equilibrium is called an
equilibrant.
Section
5.3
Force in Two Dimensions
Equilibrium Revisited
The figure below illustrates the procedure for finding the
equilibrant for two vectors.
This general procedure works for any number of vectors.
Section
5.3
Force in Two Dimensions
Example: Forces of 6.0 N east and 8.0 N south are
applied to the top of a pole. What is the equilibrant?
R =  (6.0 N)2 + ( 8.0 N)2
Equilibrant = 10 N, NW
= 10 N
Section
5.3
Force in Two Dimensions
The Parallelogram Method for Vector Addition
This is a graphical method that can only be used for
adding two vectors.
1) Place the vectors tail-to-tail
2) The vectors are two sides of the parallelogram.
Draw in the other two sides.
3) Draw the resultant from the tails to the tips.
Worksheet: The Parallelogram Method
Section
Section Check
5.3
Question 1
If three forces A, B, and C are exerted on an object as shown in the
following figure, what is the net force acting on the object? Is the
object in equilibrium?
Section
Section Check
5.3
Answer 1
We know that vectors can be moved
if we do not change their direction
and length.
The three vectors A, B, and C can
be moved (rearranged) to form a
closed triangle.
Since the three vectors form a
closed triangle, there is no net force.
Thus, the sum is zero and the object
is in equilibrium. An object is in
equilibrium when all the forces add
up to zero.
Section
Section Check
5.3
Question 2
What is the difference between a resultant and an equilibrant?
A resultant is the sum of two or more vectors. An equilibrant is the
same magnitude as the resultant, but opposite in direction.
Section
Section Check
5.3
Question 3
Compare and contrast the tip-to-tail method for vector addition with
the parallelogram method.
The tip-to-tail method can be used for adding any number of vectors.
The parallelogram method can only be used to add two vectors.
In the tip-to-tail method, vectors are lined up head-to-tail. In the
parallelogram method, vectors are lined up tail-to-tail.
They are both graphical methods.
Chapter
Chapter 5 Review
5
Graphical Method of Vector Addition
• add vectors tip-to-tail; the sum of the vectors is the resultant
• the resultant is from the tail of the first to the tip of the last
• vector addition is commutative
• vectors have magnitude and direction (ex: force, velocity,
displacement)
• scalars only have magnitude (ex: mass, speed, total distance)
• displacement ≠ total distance
Chapter
Chapter 5 Review
5
Mathematical Method for Adding Vectors (use when you have a right
triangle)
magnitude:
R = √ A2 + B2
direction:
Give the general cardinal direction ( north, west,
southwest, etc.)
For vectors, you must specify both magnitude and direction.
Finding the Component of Vectors
• The x- component is the part of the vector parallel to the x-axis.
• The y- component is the part of the vector parallel to the y-axis.
•  is labeled so the x- component is adjacent and the ycomponent is opposite.
Chapter
Chapter 5 Review
5
Friction
Ff = μ FN = μ mg
• Friction only depends on μ and m (not surface area).
• Ff static > Ff kinetic
Chapter
5
Physics Chapter 5 Test Information
The test is worth 50 points total.
True/False: 6 questions, 1 point each.
Multiple Choice: 7 questions, 1 point each.
Matching: 7 vocabulary, 1 point each.
Problems: 5 questions for a total of 30 points.
Know:
- graphical and mathematical methods for adding vectors
- the Pythagorean Theorem
- x- and y- components of vectors
- how to work friction problems