An interaction which has the potential to produce a change in the motion of an object.
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Motivation for Concept
Consider a bowling ball (or some other heavy object). If you want the ball to move, you have to interact with it. If you want the moving ball to turn, you have to interact with it. If you want the ball to stop moving, you have to interact with it. In physics, such interactions are called forces. If you want to move the ball, you will probably have to apply a contact force by using your hands or feet. There are other kinds of forces, however. The earth, for example, can alter the ball's motion through the invisible action-at-a-distance of gravity.
Newton's Laws
Newton's famous Three Laws form the basis of a scientific understanding of force.
First Law
Newton's First Law describes what happens in the absence of forces. If an object is moving with no force acting upon it, then it will move with constant velocity. Note that velocity is a vector, so this statement implies that the object will keep the same speed and the same direction of motion.
Second Law
Newton's Second Law defines force as the time rate of change of momentum:
\begin
[ \vec
\equiv \frac{d\vec{p}}
]\end
If many forces act upon an object, then the change in the object's momentum is equal to the combined effect of all the forces:
\begin
[ \sum_
^{N_{F}} \vec
_
= \frac{d\vec{p}}
] \end
It is important to note that the sum is only over forces that act on the object whose momentum change appears on the right hand side.
In most cases, the object under consideration will have a constant mass. If that is so, then the derivative of the momentum can be rewritten in the traditional formulation of Newton's Second Law:
\begin
[ \sum_
^{N_{F}} \vec
_
= ma ]\end
This form of the equation is the fundamental Law of Interaction for the Point Particle Dynamics model.
Third Law
Newton's Third Law is a rule for determining the effects of forces. This Law states that whenever one object applies a force on a second object, the second object must apply a force of the same size but opposite direction on the original object.
This law is often misunderstood. The wording makes it seem that forces are always a choice, but this is certainly incorrect. Objects apply forces without choosing to all the time. This law is simply describing well known consequences of action. Trying to change the motion of a bowling ball with a swift kick is a dangerous idea, because the bowling ball will automatically push back on your foot (the bowling ball has no "choice" in the matter).
Types of Forces
Newton's Laws describe the consequences of forces and give the rules they must obey, but the laws do not explain the types of forces that can be exerted. There are a vast array of ways for objects to interact with each other, but the ways that are commonly treated in introductory physics courses is a rather short list:
- contact forces occur when one massive body comes in contact with another.
- [gravity] is the attraction at a distance between massive objects. In introductory physics, we most often consider the force of gravity exerted by the earth on objects near its surface, in which special case the force is usually called weight.
- normal forces are a special case of contact force when an object is moving along a surface like a floor, ceiling or wall. The normal force is the portion of the contact force applied to the object by the surface that is directed perpendicular to the plane of the surface.
- tension is a force exerted by a string or rope.
- friction is a force exerted by a surface on an object moving along (or at rest on) that surface that is directed parallel to the plane of the surface.
Application of Newton's Laws
Force Diagram
Before setting up the equations of Newton's Second Law for an object, it is vital to quantitatively understand the forces acting on the object. The first step is to draw a force diagram for the situation, which is a sketch of the situation showing all relevant forces and their point of application. Here is an example:
This example shows a box that is acted upon by each of the five common categories of forces in introductory physics. The person pushing the box applies a contact force Fp. (The subscript "p" is chosen here to denote "push". You might also use "a" for applied, "c" for contact, or any other subscript that has meaning for you.) The person in front pulls on a rope, which transmits a tension force T to the box. The earth's gravity gives the box a weight force W. The floor provides both a normal force N and a friction force Ff. Note that the physical representation should always include a coordinate system.
Free Body Diagram
This physical representation is not very useful as a guide to using the equations of Newton's Second Law. If we wish to write Newton's 2nd Law algebraically, it is important to find the components of each vector. An alternate graphical representation that leads naturally to finding the vector components is the free body diagram. In a free body diagram, the center of mass of the box is represented as a point at the origin of a coordinate system. All the forces acting on the box are then drawn as [vectors] with their [tail] at the origin. For the example of the box, the free body diagram would be:
which leads naturally to the resolution into components of any vectors that do not point along an axis:
Beware of a temptation to say that in the free body diagram for this example that N is the y-component of T and Fp is the x-component of T. This is one drawback to the traditional way of drawing free body diagrams. The physical representation above and the vector addition diagrams below should make it clear that these forces are essentially independent (though they are related by Newton's 2nd Law).
Writing Newton's 2nd Law
Once the free body diagram is complete, Newton's 2nd Law can be written. It is important to note that due to its vector nature, Newton's 2nd Law is really three laws: one for each coordinate direction. Often, as is the case in our example, there are no forces in the z direction and so that direction is ignored and we write:
\begin
[ F_
+ T_
+ W_
+N_
+ F_
= ma_
]
[ F_
+ T_
+ W_
+N_
+ F_
= ma_
]\end
At this stage, we have made no use of the free body diagram. We have just blindly listed all five forces and added them up. The free body diagram is used to simplify these equations. For example, we can see from the diagram that many of the components are zero. Using this information, the equations simplify to:
\begin
[ F_
+ T_
+ F_
= ma_
]
[ T_
+ W_
+ N_
= ma_
]\end
There are two important things to note here. First, only the tension T appears in both equations, because it is the only vector in our diagram that is not directed along a coordinate axis. Second, all the force components are listed with a plus sign, even though our diagram makes it clear that Ff,x and Wy will be negative. This is not a mistake. Solving the equations would give negative values for these components. Explicitly including the negatives in the equation would mean that the algebra would actually give positive values for these two components.
Since some students prefer to explicitly assign the negatives in the equation, there is another way to write these equations. They can also be written in terms of the magnitude of each force. An equivalent set for our example is:
\begin
[ F_
+ T\cos\theta - F_
= ma_
]
[ T\sin\theta - W + N = ma_
]\end
Because the negatives have been explicitly included, all quantities will emerge as positive numbers, which is appropriate for magnitudes (but not appropriate for vector components!).
Applying Constraints
Successful application of Newton's 2nd Law often requires an understanding of constraints on the acceleration of an object. For instance, in the example above, it is likely that you would be told to assume the box slides along the floor. In that case, it is clear that the box will not move in the y direction at all. If the box does not move in the y direction, then it certainly has no y-acceleration. Thus, the y-equation should be modified to:
\begin
[ T_
+ W_
+ N_
= 0 ]\end
or, equivalently:
\begin
$ T\sin\theta - W + N = 0 $\end
It is important to note that acceleration can be zero even if an object is moving. If you were told in the example above that the box was sliding with constant velocity along the floor, then both ax and ay will be zero.
Graphical Representation of Net Force
The free body diagram is useful for finding vector components, but it can sometimes confuse the concept of net force. The net (or total) force acting on a body is the vector sum of all the individual forces, and this quantity is equal to the mass times the acceleration. Graphically, a vector sum is accomplished by arranging the vectors [tail-to-tip], but a free body diagram arranges all the vectors with their [tail] at the origin. Thus, the free body diagram does not clearly show the net force. It is possible to draw a graphical representation of the net force by arranging the vectors tail-to-tip, beginning at the origin.
Note that the order of the vectors is irrelevant (the answer will be the same no matter how they are arranged).
Newton's Laws Applied to Systems
Internal Forces and Newton's 3rd Law
Consider a situation like that shown here, where there is negligible friction from the ground, but the boxes produce friction on each other:
We can choose to view the two blocks independently or as a single system. Suppose that we first choose to consider the blocks as independent objects. In that case, a separate free body diagram must be made for each block. Shown here are [physical representations] and free body diagrams for each object:
Physical Picture, Box A
Free Body Diagram, Box A
Physical Picture, Box B
Free Body Diagram, Box B
Note in box A's physical picture we have removed objects or people that do not interact with box A. The ground pushes up on box B, which in turn pushes up on box A, but the ground does not directly push on box A. The person pushes on box B, which in turn attempts to move box A, but the person does not directly push on box A.
We have labeled two of the forces in a new way. The friction force and the normal force that act between the objects have been labeled with two subscripts each. Each of the subscripts include the labels of the blocks. The order of the subscripts conveys meaning:
FAB = "the (friction) force on A from B"
NBA = "the normal force on B from A"
Note that when double subscripts are used in this wiki, the best way to remember their meaning is to insert the word "from" between them.
The convention of which subscript should be listed first differs from book to book. Our convention may not match that of texts or other reference materials you may be using.
Our inclusion of these subscripts has a point. When you are constructing free body diagrams for more than one object, Newton's 3rd Law will apply to any forces that are exerted on one of those objects by another of the objects. Thus, in our example, any forces from block A on block B or from block B on block A will require the application of Newton's 3rd Law. When the double subscript notation is used, Newton's 3rd Law is easy to write:
\begin
[ F_
= -F_
][F_
= - F_
][ N_
= - N_
][N_
= -N_
]\end
In other words, for the vector components, when the order of the subscripts is reversed, a negative sign applies.
Note that we wrote the y direction equation for the friction force and the x direction equation for the normal force even though these components are clearly zero. The reason is to reinforce that Newton's 3rd Law constrains both the relative magnitude and the relative direction of the forces. In terms of magnitudes only, the relationship is: FAB = FBA (the magnitudes are the same, even though the directions are different)!
For our example, we can write the equations of Newton's 2nd Law for each object as (removing components that are zero for clarity):
\begin
[ F_
= m_
a_
]
[ F_
+ F_
= m_
a_
]
[ W_
+ N_
= m_
a_
= 0]
[ W_
+ N_
+N_
= m_
a_
= 0]\end
For the double subscript forces, only forces with the first subscript the same as the object whose mass is on the right hand side should be included!
Substituting from Newton's 3rd Law lets us alter the equations for object A:
\begin
[ -F_
= m_
a_
]
[W_
- N_
= 0 ]\end
Two notes. First, we could just as simply have substituted into B's equations, but A's are shorter. (Substituting into both A's and B's equations is foolish – can you see why?) Second, after the substitution, all the two-subscript forces in the equations for A are "backwards" (B is the first subscript).
After the substitution, adding the equations for the x direction and those for the y direction gives:
\begin
[ F_
= m_
a_
+m_
a_
]
[ W_
+ W_
+ N_
= 0]\end
We have constructed two equations from which all two-subscript forces have dropped out!
External Forces and the System Viewpoint
It is no accident that the two-subscript forces are so easy to eliminate from the equations of Newton's 2nd Law. It is a simple consequence of Newton's 3rd Law. The fact that the presence of FBA,x in object B's equations guarantees the presence of FAB,x in object A's, and that FBA,x = – FAB,x makes it automatic that these forces will cancel. For this reason, it is possible to skip some of the automatic algebra by taking the system viewpoint.
In the system viewpoint, if we are told that box A does not slip along box B, we might as well treat them as one object. In that case, the forces from A on B and vice versa are applied internal to that object. In other words, any forces that would have to be "two-subscript" forces can be described as internal forces. These forces will cancel from the equations of motion of the complete system as a consequence of Newton's 3rd Law. Thus, when writing Newton's 2nd Law for the composite A+B object, we can completely ignore the internal forces. The free body diagram for the composite object, then, is constructed as shown here:
The forces that remain in the free body diagram after internal forces are removed are called external forces. As the name implies, each of these forces must involve an object outside the system. In our example, the force from the person on block B clearly involves the person, who is not part of the two block system. The normal force from the ground on block B involves the ground. The weight of each block involves the earth.
We can now write the equations of Newton's 2nd Law:
\begin
[ F_
= m_
a_
]
[ W_
+ N_
= m_
a_
= 0]\end
If we use the fact that the mass and the weight of a composite object is simply equal to the sum of the mass and weight (respectively) of the parts, we have recovered exactly the same equations as we did by considering box A and box B as separate objects, if we remember that we have assumed A and B each have the same acceleration (aA,x = aB,x).
Center of Mass
The system viewpoint can actually be applied even to a collection of objects that are not fixed relative to each other. Suppose we make a minor change to the example above by assuming that block A begins to slide back along the top of block B when the force F is applied. Now, the x-equation from the individual free body diagrams:
\begin
[ F_
= m_
a_
+ m_
a_
]\end
is in conflict with the system version:
\begin
[ F_
= m_
a_
]\end
Clearly, the system version is in error, since the objects should not have the same acceleration! It can be fixed, however, if we state that the system acceleration should be defined as the acceleration of the system's center of mass. The position of the center of mass is defined as:
\begin
[ x_
\equiv \frac{m_
x_
+ m_
x_{B}}{m_
+m_{B}} ] \end
Taking two time derivatives, then (assuming the masses are constant) immediately gives:
\begin
[ a_
= \frac{m_
a_
+ m_
a_{B,x}}{m_
+m_{B}} ] \end
Thus, if we replace the acceleration in the system equation by acm, we find:
\begin
[ F_
= m_
a_
= m_
\frac{m_
a_
+ m_
a_{B,x}}{m_
+m_{B}} =
m_
a_
+m_
a_
]\end
in complete agreement with the result of using individual free body diagrams. This realization leads us to a general statement of Newton's 2nd Law applied to a system with constant mass.
Statement of the Law for a System
For a system with constant mass, we have:
\begin
[ \sum_
^{N_{F}} \vec
^
_
= m_
\vec
_
]\end
It is important to note that the sum is only over external forces. Internal forces will always cancel as a result of Newton's 3rd Law. You can show using similar arguments that the most general form of Newton's 2nd Law is:
\begin
[ \sum_
^{N_{F}} \vec
^
_
= \frac{d\vec
_{cm}}
]\end
where
\begin
[ \vec
_
\equiv m_
\vec
_
]\end