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{excerpt}An interaction which produces a change in the motion of an object.{excerpt}

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h2. 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.  

h2. Statement of Newton's Laws

Newton's famous Three Laws form the basis of a scientific understanding of force.

h4.  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.  

h4.  Second Law

[Newton's Second Law] defines force as the time rate of change of [momentum]:

{latex}\begin{large}\[ \vec{F} \equiv \frac{d\vec{p}}{dt}\]\end{large}{latex}

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:

{latex}\begin{large}\[ \sum_{k=1}^{N_{F}} \vec{F}_{k} = \frac{d\vec{p}}{dt} \] \end{large}{latex}

{note}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.{note}

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:

{latex}\begin{large}\[ \sum_{k=1}^{N_{F}} \vec{F}_{k} = ma \]\end{large}{latex}

This form of the equation is the fundamental Law of Interaction for the [Point Particle Dynamics] model.


h4. 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.  

{warning}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).{warning}

h2. 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|contact force] occur when one rigid 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|normal force] 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.

h2. Application of Newton's Laws

h4. Physical Representation

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 [physical representation] of the situation.  Here is an example:

!forcephysrep1.png!

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 _F_~p~.  (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 _F_~f~.  Note that the physical representation should always include a [coordinate system].

h4. 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|Introduction to Vectors] with their [tail] at the origin.  For the example of the box, the free body diagram would be:

!forcefbd1.png!

which leads naturally to the resolution into components of any vectors that do not point along an axis:

!forcecomp1.png!

h4. 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:

{latex}\begin{large} \[ F_{p,x} + T_{x} + W_{x}+N_{x} + F_{f,x} = ma_{x} \]
\[ F_{p,y} + T_{y} + W_{y}+N_{y} + F_{f,y} = ma_{y}\]\end{large}{latex}

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:

{latex}\begin{large}\[ F_{p,x} + T_{x} + F_{f,x} = ma_{x}\]
\[ T_{y} + W_{y} + N_{y} = ma_{y} \]\end{large}{latex}

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 _F_~f,x~ and _W_~y~ 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:

{latex}\begin{large}\[ F_{p} + T\cos\theta - F_{f} = ma_{x} \]
\[ T\sin\theta - W + N = ma_{y}\]\end{large}{latex}

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!).

h4. 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:

{latex}\begin{large}\[ T_{y} + W_{y} + N_{y} = 0 \]\end{large}{latex}

or, equivalently:

{latex}\begin{large}\[ T\sin\theta - W + N = 0\] \end{large}{latex}

{note}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* _a_~x~ *and* _a_~y~ will be zero.{note}

h4. 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.

NETFORCEPIC!forcenetpic1a.png!

Note that the order of the vectors is irrelevant (the answer will be the same no matter how they are arranged).

ALTERNATE!forcenetpic1b.png!

h2. Newton's Laws Applied to Systems

h4. Internal Forces and Newton's 3rd Law

Consider a situation like that shown here:


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|physical representation] and [free body diagrams|free body diagram] for each object:




Note that we have labeled two of the forces in a strange 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:

_F_~AB~ = "the (friction) force *on* A *from* B"

_N_~BA~ = "the normal force *on* B *from* A"

{note}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.{note}
{warning}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.{warning}

Our inclusion of these subscripts has a point.  When you are constructing free body diagrams for more than one object, [Newton's 3rd Law|Newton's Third 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:

{latex}\begin{large}\[ F_{AB,x} = -F_{BA,x} \]\[F_{AB,y} = - F_{BA,y}\]\[ N_{AB,x} = - N_{BA,x}\]\[N_{AB,y} = -N_{BA,y}\]\end{large}{latex}

In other words, for the *vector components*, when the order of the subscripts is reversed, a negative sign applies.  
{note}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: _F_~AB~ = _F_~BA~ (the magnitudes are the same, even though the directions are different)!{note}

For our example, we can write the equations of Newton's 2nd Law for each object as (removing components that are zero for clarity):

{latex}\begin{large}\[ F_{AB,x} = m_{A}a_{A,x}\]
\[ F_{x} + F_{BA,x} = m_{B}a_{B,x}\]
\[ W_{A,y} + N_{AB,y} = m_{A}a_{A,y} = 0\]
\[ W_{B,y} + N_{y} +N_{BA,y} = m_{B}a_{B,y} = 0\]\end{large}{latex}

{note}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!{note}

Substituting from Newton's 3rd Law lets us alter the equations for object A:

{latex}\begin{large}\[ -F_{BA,x} = m_{A}a_{A,x}\]
\[W_{A,y} - N_{BA,y} = 0 \]\end{large}{latex}

{note}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).{note}

After the substitution, adding the equations for the _x_ direction and those for the _y_ direction gives:

{latex}\begin{large}\[ F_{x} = m_{A}a_{A,x}+m_{B}a_{B,x} \]
\[ W_{A,y} + W_{B,y} + N_{y} = 0\]\end{large}{latex}

We have constructed two equations from which all two-subscript forces have dropped out! 

h4. 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 _F_~BA,x~ in object B's equations guarantees the presence of _F_~AB,x~ in object A's, and that _F_~BA,x~ = -- _F_~AB,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|internal force].  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:

PICTURE

The forces that remain in the free body diagram after internal forces are removed are called [external forces|external force].  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:

{latex}\begin{large}\[ F_{x} = m_{A+B}a_{x}\]
\[ W_{A+B,y} + N_{y} = m_{A+B}a_{y} = 0\]\end{large}{latex}

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 (_a_~A,x~ = _a_~B,x~).

h4. 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:

{latex}\begin{large}\[ F_{x} = m_{A}a_{A,x} + m_{B}a_{B,x} \]\end{large}{latex}

is in conflict with the system version:

{latex}\begin{large}\[ F_{x} = m_{A+B} a_{x}\]\end{large}{latex}

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:

{latex}\begin{large}\[ x_{cm} \equiv  \frac{m_{A}x_{A} + m_{B}x_{B}}{m_{A}+m_{B}} \] \end{large}{latex}

Taking two time derivatives, then (assuming the masses are constant) immediately gives:

{latex}\begin{large}\[ a_{cm} = \frac{m_{A}a_{A,x} + m_{B}a_{B,x}}{m_{A}+m_{B}} \] \end{large}{latex}

Thus, if we replace the acceleration in the system equation by _a_~cm~, we find:

{latex}\begin{large}\[ F_{x} = m_{A+B} a_{cm} = m_{A+B}\frac{m_{A}a_{A,x} + m_{B}a_{B,x}}{m_{A}+m_{B}} =
m_{A}a_{A,x}+m_{B}a_{B,x} \]\end{large}{latex}

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.

h4. Statement of Newton's 2nd Law for a System

For a system with constant mass, we have:

{latex}\begin{large}\[ \sum_{k=1}^{N_{F}} \vec{F}^{ext}_{k} = m_{\rm system}\vec{a}_{cm}\]\end{large}{latex}

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:

{latex}\begin{large}\[ \sum_{k=1}^{N_{F}} \vec{F}^{ext}_{k} = \frac{d\vec{p}_{cm}}{dt} \]\end{large}{latex}

where

{latex}\begin{large}\[ \vec{p}_{cm} \equiv m_{\rm system}\vec{v}_{cm} \]\end{large}{latex}