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Chapter
4
Forces in One Dimension
Section
4.1
Force and Motion
Force and Motion
A force is defined as a push or pull exerted on an object.
Forces can cause objects to speed up, slow down, or change
direction as they move.
Section
4.1
Force and Motion
Force and Motion
Consider a textbook resting on a table. How can you cause it to
move?
Two possibilities are that you can push on it or you can pull on it.
If you push harder on an object, you have a greater effect on its
motion.
The direction in which force is exerted also matters. If you push
the book to the right, the book moves towards right.
The symbol F is a vector and represents the size and direction
of a force.
Section
4.1
Force and Motion
Force and Motion
When considering how force
affects motion, it is important
to identify the object of
interest. This object is called
the system.
Everything around the object
that exerts forces on it is
called the external world.
A contact force exists when
an object from the external
world touches a system and
thereby exerts a force on it.
Section
4.1
Force and Motion
Contact Forces and Field Forces
If you drop a book, the gravitational force of Earth causes the
book to accelerate. This is an example of a field force.
Use the Force Obi-wan!
They are typically invisible fields which is why their discover is
relatively recent (last few hundred years)
What are some others?
Forces result from interactions; thus, each force has a specific
and identifiable cause called the agent.
Without both an agent and a system, a force does not exist.
A model which represents the ALL forces acting ON a system,
is called a free-body diagram or FBD for short.
Section
4.1
Force and Motion
Force and Acceleration – A Proof for NSL
A stretched rubber band
exerts a pulling force; the
farther you stretch it, the
greater the force with which it
pulls back.
Stretch the rubber band for a
constant distance of 1 cm to
exert a constant force on the
cart. Now let it go.
Section
4.1
Force and Motion
Force and Acceleration
You can construct a graph as
shown here.
The graph indicates that the
constant increase in the
velocity is a result of the
constant.
Section
4.1
Force and Motion
Force and Acceleration
Increase the force applied on
the cart gradually.
Plot a velocity-time graph for
each 2 cm, 3 cm and so on
and calculate the acceleration
(slope).
The relationship between the
force and acceleration is
linear, where the greater the
force, the greater the
resulting acceleration.
Section
4.1
Force and Motion
Force and Acceleration
To determine the meaning of
the slope on the forceacceleration graph, increase
the number of carts gradually.
The acceleration of two carts
is 1/2 the acceleration of one
cart, and the acceleration of
three carts is 1/3 the
acceleration of one cart.
Section
4.1
Force and Motion
Force and Acceleration
The slope, k, is defined as the
reciprocal of the mass
The equation
indicates that a force applied
to an object causes the object
to accelerate.
Section
4.1
Force and Motion
Force and Acceleration
The formula,
, tells you that if you double the force, you
will double the same object’s acceleration.
If you apply the same force to several different objects, the one
with the most mass will have the smallest acceleration and the
one with the least mass will have the greatest acceleration.
One unit of force causes a 1-kg mass to accelerate at 1 m/s2, so
one force unit has 1 kg·m/s2 or one newton and is represented
by N.
Vector sum of all the forces on an object is net force.
Section
4.1
Force and Motion
Newton’s Second Law
Newton’s second law
Section
4.1
Force and Motion
Newton’s Second Law
A useful strategy for finding how the motion of an object:
First, identify all the forces acting on the object.
Draw a free-body diagram showing the direction and relative
strength of each force acting on the system.
Then, add the forces to find the net force.
Next, use Newton’s second law to calculate the acceleration.
Finally, if necessary, use kinematics to find the velocity or
position of the object.
Section
4.1
Force and Motion
Newton’s First Law
In the absence of a net force, the motion (or lack of motion) of
both the moving object and the stationary object continues as it
was. Newton recognized this and generalized Galileo’s results in
a single statement.
Newton’s First Law: “an object that is at rest will remain at rest,
and an object that is moving will continue to move in a straight
line with constant speed, if and only if the net force acting on
that object is zero.”
If the net force on an object is zero, then the object is in
equilibrium.
Section
4.1
Force and Motion
Newton’s First Law
Newton’s first law is sometimes called the law of inertia – the
lazy force
Inertia is the tendency of an object to resist change.
If an object is at rest, it tends to remain at rest.
If it is moving at a constant velocity, it tends to continue moving
at that velocity.
Section
4.1
Force and Motion
Newton’s First Law
Section
Section Check
4.1
Question 1
Two horses are pulling a 100-kg cart in the same direction, applying
a force of 50 N each. What is the acceleration of the cart?
A. 2 m/s2
B. 1 m/s2
C. 0.5 m/s2
D. 0 m/s2
Section
Section Check
4.1
Answer 1
Answer: B
Reason: If we consider positive direction to be the direction of pull
then, according to Newton’s second law,
Section
Section Check
4.1
Question 2
Two friends Mary and Maria are trying to pull a 10-kg chair in
opposite directions. If Maria applied a force of 60 N and Mary
applied a force of 40 N, in which direction will the chair move and
with what acceleration?
A. The chair will move towards Mary with an acceleration of 2 m/s2.
B. The chair will move towards Mary with an acceleration of 10 m/s2.
C. The chair will move towards Maria with an acceleration of 2 m/s2.
D. The chair will move towards Maria with an acceleration of 10 m/s2.
Section
Section Check
4.1
Answer 2
Answer: C
Reason: Since the force is applied in opposite direction, if we
consider Maria’s direction of pull to be positive direction
then, net force = 60 N – 40 N = 20 N . Thus, the chair will
move towards Maria with an acceleration.
Section
Section Check
4.1
Question 3
State Newton’s first law.
What is it in ONE word?
Section
Section Check
4.1
Answer 3
Newton’s first law states that “an object that is at rest will remain at
rest, and an object that is moving will continue to move in a straight
line with constant speed, if and only if the net force acting on that
object is zero”.
Inertia
Section
4.2
Using Newton's Laws
Using Newton’s Second Law
Newton’s second law tells
you that the weight force,
Fg, exerted on an object of
mass m is
Why is Fg, or weight, location specific? Hint:
you weight less on the moon, right?
Consider a free-falling ball in midair. It is
touching nothing and air resistance can be
neglected, the only force acting on it is Fg.
Both the force and the acceleration are
downward – speeds up! Force causes
acceleration which is Δvelocity! BAM!!
Section
4.2
Using Newton's Laws
Using Newton’s Second Law
How does a bathroom scale work?
When you stand on the scale, the
spring in the scale exerts an
upward force on you because you
are in contact with it.
Because you are not accelerating,
the net force acting on you must
be zero (Fnet=ma=0).
The spring force, Fsp, upwards
must be the same magnitude as
your weight, Fg, downwards.
Section
4.2
Using Newton's Laws
Fighting Over a Toy
Anudja is holding a stuffed dog, with a mass of 0.30 kg, when Sarah
decides that she wants it and tries to pull it away from Anudja. If
Sarah pulls horizontally on the dog with a force of 10.0 N and Anudja
pulls with a horizontal force of 11.0 N, what is the horizontal
acceleration of the dog?
Section
4.2
Using Newton's Laws
Fighting Over a Toy
Sketch the situation (FBD) and identify the dog as the system and
the direction in which Anudja pulls as positive.
Section
4.2
Using Newton's Laws
Fighting Over a Toy
Identify known and unknown variables.
Known:
Unknown:
m = 0.30 kg
a=?
FAnudja on dog = 11.0 N
FSarah on dog = 10.0 N
Section
4.2
Using Newton's Laws
Fighting Over a Toy
Substitute FAnudja on dog = 11.0 N, FSarah on dog = 10.0 N, m = 0.30 kg
Section
4.2
Using Newton's Laws
Drag Force and Terminal Velocity
When an object moves through any fluid, such as air or water,
the fluid exerts a drag force on the moving object in the
direction opposite to its motion.
As the ball’s velocity increases, so does the drag force. The
constant velocity that is reached when the drag force equals
the force of gravity is called the terminal velocity.
Section
Section Check
4.2
Question 1
If mass of a person on Earth is 20 kg, what will be his mass on
moon? (Gravity on Moon is six times less than the gravity on Earth.)
A.
B.
C.
D.
Section
Section Check
4.2
Answer 1
Answer: C
Reason: Mass of an object does not change with the change in
gravity, only the weight changes.
Section
Section Check
4.2
Question 2
Your mass is 100 kg, and you are standing on a bathroom scale in
an elevator. What is the scale reading when the elevator is falling
freely?
A.
B.
C.
D.
Section
Section Check
4.2
Answer 2
Answer: B
Reason: Since the elevator is falling freely with acceleration g, the
contact force between elevator and you is zero. As scale
reading displays the contact force, it would be zero.
Section
Section Check
4.2
Question 3
In which of the following cases will your apparent weight be greater
than your real weight?
A. The elevator is at rest.
B. The elevator is accelerating in upward direction.
C. The elevator is accelerating in downward direction.
D. Apparent weight is never greater than real weight.
Section
Section Check
4.2
Answer 3
Answer: B
Reason: When the elevator is moving upwards, your apparent
weight
(where m is your mass and a is
the acceleration of the elevator). So your apparent
becomes more than your real weight.
Section
4.3
Interaction Forces
Identifying Interaction Forces
When you exert a force on your
friend to push him forward, he
exerts an equal and opposite
force on you, which causes you
to move backwards.
An interaction pair of forces is
two forces that are in opposite
directions and have equal
magnitude.
Section
4.3
Interaction Forces
Identifying Interaction Forces
An interaction pair is also called an action-reaction pair of
forces.
This might suggest that one causes the other; however, this is
not true.
The two forces either exist together or not at all.
Section
4.3
Interaction Forces
Newton’s Third Law
Newton’s third law, which states that all forces come in pairs.
Newton’s Third Law states that the force of A on B is equal in
magnitude and opposite in direction of the force of B on A.
The two forces in a pair act on different objects and are equal in
magnitude and opposite in direction. They are vectors!
Section
4.3
Interaction Forces
Earth’s Acceleration
When a softball with a mass of 0.18 kg is dropped, its acceleration
toward Earth is equal to g, the acceleration due to gravity. What is
the force on Earth due to the ball, and what is Earth’s resulting
acceleration? Earth’s mass is 6.0×1024 kg.
Section
4.3
Interaction Forces
Earth’s Acceleration
Draw free-body diagrams for the two systems: the ball and Earth and
connect the interaction pair by a dashed line.
Section
4.3
Interaction Forces
Earth’s Acceleration
Identify known and unknown variables.
Known:
Unknown:
mball = 0.18 kg
FEarth on ball = ?
mEarth = 6.0×1024 kg aEarth = ?
g = 9.80 m/s2
Section
4.3
Interaction Forces
Earth’s Acceleration
Use Newton’s second and third laws to find aEarth.
Section
Interaction Forces
4.3
Earth’s Acceleration
Substitute a = –g
Section
4.3
Interaction Forces
Earth’s Acceleration
Substitute mball = 0.18 kg, g = 9.80 m/s2
Section
4.3
Interaction Forces
Earth’s Acceleration
Use Newton’s second and third laws to solve for FEarth on ball and
aEarth.
Section
4.3
Interaction Forces
Earth’s Acceleration
Substitute FEarth on ball = –1.8 N
Section
4.3
Interaction Forces
Earth’s Acceleration
Use Newton’s second and third laws to find aEarth.
Section
4.3
Interaction Forces
Earth’s Acceleration
Substitute Fnet = 1.8 N, mEarth= 6.0×1024 kg
Section
4.3
Interaction Forces
Forces of Ropes and Strings
The force exerted by a string or rope is called tension.
At any point in a rope, the tension forces are pulling equally
in both directions.
You cannot PUSH a rope.
Section
4.3
Interaction Forces
The Normal Force
The normal force is a
perpendicular contact force
exerted by a surface on
another object.
It exists because the object has
mass and accelerates towards
the Earth. This is the force that
keeps you in your chair or a
book from falling to the ground.
What happens when the
normal forces is smaller than
the objects weight?
The normal force is critical
when calculating friction forces.
Section
Section Check
4.3
Question 1
Explain Newton’s third law – statement, equation, and what happens
when the applied force is larger then the reaction force that can
be generated?
Section
Section Check
4.3
Answer 1
Suppose you push your friend, the force of you on your friend is
equal in magnitude and opposite in direction to the force of your
friend on you. This is summarized in Newton’s third law, which
states that forces come in pair. The two forces in a pair act on
different objects and are equal in strength and opposite in direction.
Newton’s third law
The force of A on B is equal in magnitude and opposite in direction
of the force of B on A.
Something fails. It will break.
Section
Section Check
4.3
Question 2
If a stone is hung from a rope with no mass, at which place on the
rope will there be more tension?
A. The top of the rope, near the hook.
B. The bottom of the rope, near the stone.
C. The middle of the rope.
D. The tension will be same throughout the rope.
Section
Section Check
4.3
Answer 2
Answer: D
Reason: Because the rope is assumed to be without mass, the
tension everywhere in the rope is equal to the stone’s
weight .
Section
Section Check
4.3
Question 3
In a tug-of-war event, both teams A and B exert an equal tension of
200 N on the rope. What is the tension in the rope? In which
direction will the rope move? Explain with the help of Newton’s third
law.
Section
Section Check
4.3
Answer 3
Team A exerts a tension of 200 N on the rope. Thus, FA on rope = 200 N.
Similarly, FB on rope = 200 N. But the two tensions are an interaction pair,
so they are equal and opposite. Thus, the tension in the rope equals
the force with which each team pulls (i.e. 200 N). According to
Newton’s third law, FA on rope = FB on rope. The net force is zero, so the
rope will stay at rest as long as the net force is zero.
Section
5.2
Friction
Static and Kinetic Friction
There are two types of translation friction, and both always
oppose motion. There is also rolling friction which is for later.
Kinetic friction is exerted when the two surfaces rub against
each other.
To understand the other friction, imagine trying to push a heavy
couch across the floor but it does not move.
Newton’s 3rd law 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.
Section
5.2
Friction
Static and Kinetic Friction
You might push harder and harder 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
When you push hard enough the couch will begin to move.
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.
Section
5.2
Friction
Static and Kinetic Friction
Frictional force depends on the materials that the surfaces are
made of.
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 pulled (weight) against the other, the
greater the force of 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 at a
constant velocity and use a spring scale
to measure the force that you pull.
You can then stack additional blocks on
the block to increase the normal force
and repeat.
Section
5.2
Friction
Static and Kinetic Friction
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 sandpaper
surface has a steeper slope
than the polished table.
Section
Friction
5.2
Static and Kinetic Friction
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.
Kinetic friction force
Ff, kinetic = μkFN
Section
Friction
5.2
Static and Kinetic Friction
Static Friction Force
Ff, static  μsFN
In the equation for the maximum static friction force, μs 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
Friction
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.
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?
Section
5.2
Friction
Balanced Friction Forces
Identify the forces and establish a coordinate system.
Section
5.2
Friction
Balanced Friction Forces
Draw a motion diagram indicating constant v and a = 0.
Section
Friction
5.2
Balanced Friction Forces
Draw the free-body diagram.
Section
Friction
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
Friction
Balanced Friction Forces
The normal force is in the y-direction, and there is no acceleration.
FN = Fg
= mg
Section
5.2
Friction
Balanced Friction Forces
Substitute m = 25.0 kg, g = 9.80 m/s2
FN = 25.0 kg(9.80 m/s2)
= 245 N
Section
5.2
Friction
Balanced Friction Forces
The pushing force is in the x-direction; v is constant, thus there is no
acceleration.
Fp = μkmg
Section
5.2
Friction
Balanced Friction Forces
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
Section Check
5.2
Question 1
Define friction force.
Did you think to ask which one first?
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 a friction force depends?
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
5.2
Section Check
Question 4 – 4!!! How could I do that to you!
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.
E. All of the above…
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 and Motion in Two Dimensions
Equilibrium Revisited
Recall that when the net force on an object is zero, the object is
in equilibrium.
An object in equilibrium is motionless or moves with constant
velocity.
Equilibrium can occur no matter how many forces act on an
object. As long as the resultant is zero, the net force is zero.
Section
5.3
Force and Motion in Two Dimensions
Equilibrium Revisited
What is the net force acting on
the object?
Vectors may be moved if you
do not change their direction
(angle) or length.
Section
5.3
Force and Motion in Two Dimensions
Equilibrium Revisited
The addition of the three forces, A, B, and C.
There is no net force; thus, the
sum is zero and the object is
in equilibrium.
Section
5.3
Force and Motion in Two Dimensions
Equilibrium Revisited
Suppose that two forces are added are not zero.
How could you find a third force that would result in zero, and
therefore cause the object to be in equilibrium?
Find the sum of the two forces already on the object.
This single force that produces the same effect as the two
individual forces added together, is called the resultant force.
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 and Motion in Two Dimensions
Equilibrium Revisited
Finding the equilibrant for two vectors.
This works for any number of vectors.
Section
5.3
Force and Motion in Two Dimensions
Motion Along an Inclined Plane
Section
5.3
Force and Motion in Two Dimensions
Motion Along an Inclined Plane
Because an object’s acceleration is usually parallel to the slope,
one axis, usually the x-axis, should be in that direction – the ramp.
With this coordinate system, there are two forces—normal and
frictional forces. These forces are in the direction of the coordinate
axes. However, the weight is not –> vector components.
SOH, CAH, TOA.
When an object is placed on an inclined plane, the normal force will
not be equal to the object’s weight.
You will need to apply Newton’s laws once in the x-direction and
once in the y-direction.
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
How do you decide the coordinate system when the motion is along
a slope? Is the normal force between the object and the plane the
object’s weight?
Section
Section Check
5.3
Answer 2
An object’s acceleration is usually parallel to the slope. One axis,
usually the x-axis, should be in that direction. The y-axis is
perpendicular to the x-axis and perpendicular to the surface of the
slope. With these coordinate systems, you have two forces—the
normal force and the frictional force. Both are in the direction of the
coordinate axes.
However, the weight is not. This means that when an object is
placed on an inclined plane, the magnitude of the normal force
between the object and the plane will usually not be equal to the
object’s weight.
Section
Section Check
5.3
Question 3
A skier is coming down the hill. What are the forces acting parallel to
the slope of the hill?
A. Normal force and weight of the skier.
B. Frictional force and component of weight of the skier along the
slope.
C. Normal force and frictional force.
D. Frictional force and weight of the skier.
Section
Section Check
5.3
Answer 3
Answer: B
Reason: The component of the weight of the skier will be along the
slope, which is also the direction of the skier’s motion. The
frictional force will be in the opposite direction from the
direction of motion of the skier.
Chapter
5
Forces in Two Dimensions
End of Chapter 4 & 5