An interaction which produces 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.
Statement of 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. Trying to tackle a 200 pound athlete running at full speed is going to hurt, even if they do not purposely push against you.
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 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 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
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:
PHYSICAL REP
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:
FBD
which leads naturally to the resolution into components:
COMPONENTS
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!).