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Forces
Forces are what make objects or systems in the world move and react. A force is any influence
that causes an object or system to accelerate. We will discuss certain important forces in this
document. Before that, we must learn about vector components in order to be able to evaluate
forces in the real world.
Vector Components
Forces will sometimes be pointing in various directions and the easiest way to solve a problem
will be to break the vector into ‘x’ and ‘y’ components. Breaking a vector into ‘x’ and ‘y’
components simply means to change it into two vectors, one being vertical in direction the other
being horizontal in direction. We can do this by using trigonometry and a little bit of problem
solving. Let us look at the vector ‘A’ below.
A=30 N
We can see that the vector is on some sort of angle in space. Let us put it
onto a Cartesian plane so we can investigate the angle.
Now we have a reference we can use in order to specify
the direction of the force. Let us assume that the angle of
the force is 30o off of the horizontal axis or x-axis. We can
now use this new information and draw two more forces
that add up to our original vector A.
A=30 N
30o
A=30 N
o
30
x
𝑎𝑑𝑗
ℎ𝑦𝑝
𝒙
cos(30) =
30
30 cos(30) = 𝒙
y
The new blue vectors labeled ‘x’ and ‘y’ are the vector
components of vector A. Now we can use trigonometry to
solve for the ‘x’ and ‘y’ sides of our new right angle
triangle. We can use the cosine function to solve for the
horizontal component because it has now become the
adjacent side. Likewise we can use sine function to solve
for the vertical side as it is now the opposite side.
𝑜𝑝𝑝
ℎ𝑦𝑝
𝒚
sin(30) =
30
30sin (30) = 𝒚
cos(𝜃) =
sin(𝜃) =
𝒙 = 25.98 𝑁
𝒚 = 15 𝑁
This means that of the 30 N of force in the original vector, 25.98 N of it is pushing directly right
and 15 N is pushing straight up. The vector has now been broken into components.
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Important Forces
Force of Gravity
The force of gravity is a special force. It has the property of always having the same acceleration
of approximately 9.8 m/s2. It also has the same direction no matter the situation. Gravity is
always pulling straight downwards on an object. Gravity is an ever present force as well so every
object on earth feels it. The force of gravity is usually denoted by:
�⃗𝒈 = 𝒎𝒈
��⃗
𝑭
Where ‘m’ is the mass of the object or system that is being discussed and ‘g’ is the acceleration
of gravity given usually as 9.8 m/s2. We can see the force of gravity acting on a few boxes below
in the section discussing the normal force.
Normal Force
The normal force is also a very unique force. This is the reaction force when dealing with
gravity. This force always points perpendicular to the surface the object is on. Let us look at a
few examples to demonstrate this.
Example 1: Box on a flat surface
The red vector is the force of gravity that we have learned and the
blue vector is the normal force. We can see that both have the exact
same magnitudes but exactly opposite directions. If this force was
not present, there would be a net force downwards, and the box
would be moving downward, into the earth. In the case of
completely flat surface like we have, the normal force can be shown
as:
�𝑭⃗𝑵 = −𝑭
�⃗𝒈
Example 2: Box on an inclined surface
The red vector is still the force of gravity and the blue vector is still
the normal force. It can be seen that gravity is still pushing straight
down but now the normal force is pushing up and to the left,
perpendicular to the surface which the box is resting on. The
relationship for the normal force and the force of gravity on an
inclined surface is described as:
�⃗𝑵 = −𝑭
�⃗𝒈𝒚
𝑭
In this case there is a subscript ‘y’ with the force of gravity which refers to the ‘y’ or vertical
component of the force of gravity. It may be confusing because it looks like all of the force of
gravity is in the ‘y’ direction. But we have to rotate the axes in this question in order to solve it
properly. On the next page we can see a new figure with a set of axes drawn on it.
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Usually the x and y axes of the Cartesian coordinate system are straight up and down and
straight left and right but we can manipulate that as needed. We have changed our axes in this
problem to fit our needs and allow us to solve the problem.
y
x
We have now put in our new axes that have been tilted so that the y axis
is parallel to the direction of the normal force. We can see now that
normal force and the ‘y’ component of gravity will cancel each other
out. You can clearly see the purple vector representing the component
of the force of gravity in the negative y direction. Now there is also the
orange vector that is pointing down the x axis. This is where the
acceleration down the slope comes from. This is the ‘x’ component of
the gravity force that is not cancelled out. We can use this force to
solve for the acceleration if that is what the question is asking.
Force of Friction
Friction is a force that acts against the motion of an object. Its direction is always opposite to
that of the motion of the object in question. The force of friction comes from the contact
between the object or system and the ground that it is resting on. The formula for the force of
friction looks like:
�𝑭⃗𝒇 = 𝝁𝑭
�⃗𝑵
The symbol mu ‘µ’ is the coefficient of friction. The greater the value of the coefficient of
friction, the higher the friction will be. The coefficient of friction changes depending on
whether the object is at rest or if it is moving. The coefficient will also change due to the surface
that the object or system is on. For example the coefficient of friction for ice will be much less
than the coefficient of friction for carpet. A lot of friction means that you need a more powerful
force in order to cause the object or system to move or begin its motion.
There are two main kinds of friction, static friction and kinetic friction.
Static Friction
Static friction is the force that must be overcome in order to start moving in any way. The
formula for static friction looks like:
𝐹⃗𝑠𝑓 = 𝜇𝑆 𝐹⃗𝑁
The only difference between this formula and the one above is the subscript ‘s’, which refers to
static. The coefficient of static friction is typically larger than that of kinetic friction.
Kinetic Friction
Kinetic friction is used when a body or a system is already in motion. This force tries to stop the
motion that is happening to the system. The formula for kinetic friction looks like:
𝐹⃗𝑘𝑓 = 𝜇𝑘 𝐹⃗𝑁
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The only difference between this formula and that of static friction is the subscript. This type of
friction is also referred to as dynamic friction.
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Free Body Diagram
A free body diagram is a method of labeling all of the forces that are acting on a body. This type
of diagram allows you to visualize all of the forces and work out what your next step should be.
Below are a few examples of free body diagrams. Remember to draw the forces coming from
the center of the object in question
Example 1: Box at rest on a floor
FN
With a box at rest on a floor there are only two forces; the force of
gravity going straight down and the normal force pointing straight
up. These two forces cancel each other out and the box will remain
at rest.
Fg
Example 2: Box freefalling
Ff
With a box freefalling there is a very large force of gravity as the box
is accelerating downwards throughout its fall. There is a small force
of friction due to air resistance. If the box is in freefall for a short
time the air resistance is usually negated.
Fg
Example 3: Box being pushed on a carpet with high coefficient of friction
FN
Fa
Fsf
This example shows four different forces acting upon the box. The
applied force of ‘Fa’ is not great enough to overcome the force of
static friction hence the box remains at rest. The force of gravity as
well as the normal force are both there as in the first example.
Fg
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Net Force
Let us look at the diagrams again and see if we can predict what is going to happen to each box.
To do this we can look at the net force that is present in the system. A formula for the net force
can be shown in order to look at the examples mathematically.
The net force is also known as the resultant force. It is the acceleration that is left over after all
the forces have been accounted for. A simple formula for the net force can be written as:
�⃗𝑵𝒆𝒕 = 𝑭
�⃗𝒙 + 𝑭
�⃗𝒚
𝑭
This simply means that all of the forces in the ‘x’, or horizontal direction, and all of the forces in
the ‘y’, or vertical direction, are added together to see where the system will accelerate to or if it
will accelerate at all.
Example 1 where the box is at rest on an even floor there are only two vectors acting on the
object. These two forces as previously noted are the force of gravity and the normal force. As
discussed these forces are the same in magnitude and opposite in direction such as the formula
𝐹𝑁 = −𝐹𝑔 states. Let us write in our forces for the x direction and for the y direction.
𝐹⃗𝑥 = 0
𝐹⃗𝑦 = 𝐹⃗𝑔 − 𝐹⃗𝑁
𝐹⃗𝑦 = 0
As discussed there are no forces in the horizontal direction and we know that the force of
gravity cancels out the normal force. Now to find the net force:
𝐹⃗𝑁𝑒𝑡 = 𝐹⃗𝑥 + 𝐹⃗𝑦
𝐹⃗𝑁𝑒𝑡 = 0
Therefore there is no net force and the box remains at rest in this situation.
Example 2 where the box is in freefall we again have no x direction forces and only the y
direction has any forces. Let us negate the force of air resistance in this case to simplify our
question. Now let us find the net force:
𝐹⃗𝑁𝑒𝑡 = 𝐹⃗𝑥 + 𝐹⃗𝑦
𝐹⃗𝑁𝑒𝑡 = 0 + 𝐹⃗𝑔
𝐹⃗𝑁𝑒𝑡 = 𝐹⃗𝑔
Therefore we can see that the box is accelerating due to gravity and thus its acceleration is 9.8
m/s2 straight down.
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Example 3 has four forces to worry about but we can already see that the vertical direction forces
will cancel out because it is simply the same situation as example 1. So we can worry about the
forces in the x direction only.
𝐹𝑁𝑒𝑡 = 𝐹𝑥 = 𝐹𝑎 − 𝐹𝑠𝑓
Since we do not know the values of the applied force or the force of friction we cannot go any
further than this. We are assuming that the applied force is lower than the force of friction and
thus the object remains at rest.
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This document last updated: 7/27/2011