Download Dynamics - Mr. Schroeder

Physics 20 - STA
Note Booklet
Unit 2 – Dynamics
Chapter 3 – Forces
USE THE
FORCE
Equation (F=ma) young
padawan.
Name: ______________________________________________________________________________
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Table of Contents
3.1 – The Nature of Force (Dynamics, Types of Forces, and Free-Body
Diagrams
3.2 – Newton’s First and Second Law
3.3 - Multi-Force Analysis
3.4 – The Force of Friction
3.5 – Newton’s Third Law
3.6 – Application Problems (Elevators, Pulleys, and Inclined Planes)
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3.1 – The Nature of Force
Previously, we have studied motion without regard to what caused the motion. We have already studied
kinematics. Dynamics, on the other hand, is the study of motion as we pay attention to what causes the
motion. Dynamics is the study of forces.
A force is defined as a push or a pull. Force is a vector, as it has a direction associated with it. The
typical SI unit used to describe force is the Newton (N). A force acting on an object can have one or
both of the following effects on the object:
1. accelerate the object (cause the object to speed up, slow down, or change direction)
2. deform the object
Before we continue, it is important to distinguish force from two other quantities: mass, and volume.
Mass is defined as the amount of matter possessed by something. Mass is a scalar quantity. The typical
SI unit used to describe mass is the kilogram (kg). Volume is the amount of space that something takes
up. Volume is a scalar quantity. The typical SI unit used to describe volume is the Litre (L).
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Types of Forces
There are different types of forces and scientists distinguish among them by giving these forces special
names. When an object is in contact with another, the objects will have a common surface of contact,
and the two objects will exert a normal force on each other. The normal force, FN, is a force that is
perpendicular to this common surface. Depending on the situation, another force called friction, Ff, may
be present, and this force acts parallel to the common surface.
The adjective “normal” simply means perpendicular. Figure 3.9 (a) shows a book at rest on a level table.
The normal force exerted by the table on the book is represented by the vector directed upward. If the
table top were slanted and smooth as in Figure 3.9 (b), the normal force acting on the book would not be
directed vertically upward. Instead, it would be slanted, but always perpendicular to the contact surface.
A free-body diagram is a powerful tool that can be
used to analyze situations involving forces. This
diagram is a sketch that shows the object by itself,
isolated from all others with which it may be
interacting. Only the force vectors exerted on the
object are included and, in this physics course, the
vectors are drawn with their tails meeting at the
centre.
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A stationary object may experience an applied
force, Fapp, if, say, a person pushes against the
object (Figure 3.10). In this case, the force of
friction acting on the object will oppose the
direction of impending motion.
Since force is a vector quanity, its direction is
important. This is especially important if two or
more forces are acting on an object.
The net force may be found using vector addition
Fnet = F1 + F2 + F3+…..
(Directions must be accounted for .. i.e. + and –
signs)
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Example
A car with a weight, Fg, of 10 000 N [down] is coasting on a level road. The car experiences a normal
force, FN, of 10 000 N [up], a force of air resistance, Fair, of 2500 N [backward], and a force of friction,
Ff, exerted by the road on the tires of 500 N [backward]. Draw a free-body diagram for this situation.
Example
Draw a free body diagram of a 20 kg object that is sliding along a horizontal surface, as it is pushed with
an applied force of 30 N and is resisted by a frictional force of 10 N.
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3.2 - Newton’s First and Second Law
First Law
Inertia is defined as the tendency of an object to stay in motion. Every object that is moving has a
tendency to stay in motion, in a straight line, at a constant speed. Every object that is at rest has a
tendency to remain at rest. In order to change an object’s tendency, an unbalanced force must be
applied.
The Law of Inertia (Newton’s First Law): An object at rest will remain at rest will remain at rest until
acted upon by an unbalanced force, and an object in motion will continue in motion in a straight line at a
constant speed until acted upon by an unbalanced force.
This of course implies that an object that remains at rest must have zero net, or total, force acting on it.
It is said to be in static equilibrium. If there were a net force, it would begin to move. Because force is
a vector, two forces of the same magnitude acting in opposite directions can balance each other to
produce zero net force. An object at rest therefore does not necessarily have zero force acting on it, just
no net force acting on it.
Similarly, an object is moving at a constant velocity must have zero net force acting on it. It is said to be
in translational equilibrium. If there were a net force, it would speed up, slow down, change direction,
or deform. Again, two forces of the same magnitude acting in opposite directions can balance each
other to produce zero net force. An object moving at a constant velocity therefore does not necessarily
have zero force acting on it, just no net force acting on it.
Fnet = 0 (If the object is stationary or moving at a constant rate)
When you are in a moving car, you can feel the effects of your own inertia. If the car accelerates
forward, you feel as if your body is being pushed back against the seat, because your body resists the
increase in speed. If the car turns a corner, you feel as if your body is being pushed against the door,
because your body resists the change in the direction of motion. If the car stops suddenly, you feel as if
your body is being pushed forward, because your body resists the decrease in speed.
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Second Law
Recall that the acceleration of an object is defined as the rate of change of an object’s velocity. This rate
of change depends on two things: the net force acting on the object, and the mass of the object. The
larger the net force acting on an object, the greater the acceleration. The larger the mass of the object,
the smaller the acceleration.
Newton’s Second Law: The acceleration of an object is directly proportional to, and in the same
direction as, the net force acting on the object. The acceleration of an object is inversely proportional to
the mass of the object.

 F
Newton’s second laws can be stated mathematically as a  , although it is typically rearranged to take
m


the form F  ma .
Example
A 100 kg car accelerates at a rate of 2.0 m/s2. What is the net force required to do this?
Example
A 100 kg car moving at 72 km/h slows to a stop in 10 s. How much force was required to stop the car?
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Multi-Force Problems
Recall that the net force acting on object is the total force acting on the object (the vector sum of all


forces acting on the object). When the net force results from only one force, F  ma can simply be used
to analyze the problem. When the net force results from two or more forces, however, a slightly more
in-depth analysis must be performed. The following are the steps of that analysis:
1. Draw a FBD. Cross off all forces that balance each other. If the only forces remaining are all in
one dimension, this method will work. If they are in two difference dimensions, another method
to be described later will have to be used.




2. Fnet is the sum of the forces, therefore Fnet  FA  FB  ... (all of the forces that are not crossed
off).

3. Fnet is also equal to the product of an object’s mass and acceleration (Newton’s second law),

 
therefore ma  FA  FB  ...
4. Plug in the values for the known variables, and solve. Remember that the accleration and forces
are vectors, therefore accelerations and forces that act to the left (or down, South, etc.) will have
a negative value and accelerations and forces that act to the right (or up, North, etc.) will have a
positive value.
Example
A 50 kg box is pushed to the right with a force of 150 N by one person. It is pushed to the left with a
force of 320 N by a second person. A 30 N force of friction acts on the box as well. Calculate the
acceleration of the box.
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3.3 - Multi-Force Analysis
Mass vs. Weight
To understand the differences we need to compare a few points:


Mass is a measurement of the amount of matter something contains, while Weight is the
measurement of the pull of gravity on an object.
The Mass of an object doesn't change when an object's location changes. Weight, on the other
hand does change with location.
To calculate the weight : Fg = ma or Fg = mg
Example
Calculate the weight of a 40.0 kg thumbnail.
Multi-Force Problems Involving Vector Components
If a force contributes to the motion of an object but the acceleration is not entirely in the direction of the
force, the component of the force in the direction of the acceleration must be found. The problem can
then be treated as any other multi-force problem.
To find the component of the force in the direction of the acceleration, follow the steps outlined in your
kinematics notes for finding vector components.
Example
A 500kg wagon is pulled by a force of 20N North and by 30N East.
a) What is the net force (magnitude and direction)?
b) If the force of friction is 8.0N, calculate the acceleration.
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Example
If a 10kg ball is being thrown up with an applied force of 5000N:
a) Draw a FBD of what is happening
b) Write an Fnet equation.
c) Calculate the acceleration of the ball being thrown.
Example
If a 20kg box is moving 2.0 m/s to the right at a constant rate, what is the applied force if the force of
friction is 13.4 N?
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3.4- The Force of Friction
The force of friction is a force that opposes the motion between two surfaces that are in contact
There are two types
 STATIC friction: opposes the start of motion between two surfaces that are not in
relative motion.
 KINETIC friction: the force between two surfaces that are in relative motion. It always
acts in the opposite direction to the motion of the object.
Static friction is ALWAYS bigger than kinetic friction
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Friction is directly proportional to the weight of an object (FN)
Ff   FN
“mew”
The coefficient of friction is a number (no units) that depends on the two surfaces in contact. The larger
the number is, the more “sticky” the situation. So, a very slippery surface (ice) would have a low
coefficient of friction.
0.06 – 0.20
0.8 – 1.2
1.0 – 1.6
Example
A 12 kg piece of wood is placed on a floor. There is 35 N of maximum static friction measured between
them. Determine the coefficient of friction between the piece of wood and the floor.
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Example
What frictional force must be overcome to start a 50 kg object sliding across a surface with a static
coefficient of friction equal to 0.35?
Example
A 10 kg box is dragged over a horizontal surface by a force of 40 N. If the box moves with a constant
speed of 0.5 m/s, what is the coefficient of kinetic friction for the surface?
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3.5 Newton’s Third Law
Consider any two objects, A and B, interacting with each other:
When object A exerts a force on object B, object B always exerts an equal but opposite force on object
A.
The above statement is known as Newton’s Third Law.
It is important to note that the net force is not necessarily zero here. The forces do not balance or cancel
each other out because they are each acting on different objects. A FBD of object A would look like the
following:
Object A
FBA
While a FBD of object B would look like the following:
FAB
Object B
It is entirely normal for one of the two objects to experience a greater acceleration than the other.
Remember that the forces acting on each one are equal, but the masses are not necessarily equal.

 F
Therefore, since a  , the heavier object will not accelerate at as great of a rate as the lighter object.
m
It is also possible that one of the two objects is attached to the ground or some other fixed object. In this
case, the attached object would not move unless the Newton III force acting on it was large enough to
overcome the attachment force (or force of static friction).
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Example
A 75 kg man and a 65 kg man push against each other as they stand on a horizontal, frictionless surface.
If the 75 kg man accelerates at a rate of 1.5 m/s2, at what rate does the 65 kg man accelerate?
Example
Two people, one of mass 50 kg and the other of mass 60 kg, are pushing against each other on a
horizontal, frictionless surface. If the first person accelerates from rest to 3.0 m/s in a time of 4.0
seconds, what is the acceleration of the second person?
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3.6 Applications of Newton’s Laws
Elevators
An elevator problem is just like any other multi-force problem. In an elevator problem, there will
always be the same two forces drawn on the FBD: the force of gravity (acting down) and the normal
force (acting up). The issue that sometimes makes elevator problems more difficult for students to solve
is knowing which variable to solve for. There are several things that can be asked for in an elevator
problem, but they all boil down to a few common variables:
Asked For:
actual weight
normal force
apparent weight
force of floor
scale reading
Solve For:
force of gravity (Fg)
normal force (FN)
normal force (FN)
normal force (FN)
normal force (FN)
Example
A 73 kg man stands in an elevator that accelerates downward at a rate of 2.0 m/s2.
a. With what force does the elevator floor push him up?
b. What is the man’s actual weight?
c. What is the normal force acting on the man?
d. What is the man’s apparent weight?
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Pulley Problems
Sometimes two or mare masses are joined to each other by a rope that goes around a pulley. The
analysis of a pulley problem is very much like the method previously learned for analyzing multi-force
problems, with one addition: the system should be “straightened-out” before a FBD is drawn. This
means that all forces should be treated as if they are acting left or right.
For example, the following:
m1
m2
Fg1
Fg2
would become:
m1
Fg1
m2
Fg2
Once the system is straightened out and the FBD is drawn, the problem is like any other multi-force
problem. Note that in the above example, Fg1 is equal to m1g while Fg2 is equal to m2g. The mass that is
used for the net force is the entire mass of the system (m1 + m2 in this case) because the entire mass is
being accelerated. Also note that in the above example, Fg1 is negative because in the FBD it is drawn to
the left, while Fg2 is positive because in the FBD it is drawn to the right.
Example
m= 2.0 kg
FF= 20 N
What is the acceleration of the system?
m= 10 kg
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Inclined Planes
An object on a hill experiences at least two forces (the normal force and a force of gravity) that do not
cancel each other out. A FBD must therefore be drawn. The problem can then be “straightened-out”
and analyzed as a multi-force problem.
To “straighten-out” the problem, the force acting parallel to the hill must be found. Call this force FH.
This is the force that acts to pull the object down the hill, and is really just a combination of the normal
force and the force of gravity acting on the object.
FH
 FN
Fg

FH
The angle of incline of the hill () will always be the same as the angle between the force of gravity and
the normal force acting on the object. This can be justified by using trigonometry. The resultant vector
can be found using the angle, , and a trigonometric function (typically the sine function). The resultant
vector is the force that acts on the object parallel to the hill, FH.
Now redraw the object horizontally with FH instead of Fg and FN. The diagram above “straightened-out”
horizontally looks like the following:
FH
Don’t forget to include any other forces that act on the object, such as an applied force or force of
friction. Now solve the problem as any other multi-force problem.
Example
If a 5.0 N force of friction acts to oppose the motion of the block in the diagram below, what is the
acceleration of the block?
F= 20 N
m= 2.0 kg
= 300
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