Physics Lecture 4 - Newton's Laws

Newton's Laws

Isaac Newton developed the prescription for how to think about dynamics, the study of why things move as they do. The prescription is based on the three Laws of Motion as stated below.

Mass is the measure of inertia, i.e. objects only change their motion in response to actions from external objects. What we call mass is the amount of opposition to changes in motion that an object possesses.
The definition of force is that it equals the product of mass and acceleration. Since acceleration is the measure of the change in motion (remember that Galileo's definition of motion is velocity) of an object, the action required to exert that change in motion should be proportional to mass as is consistent with the first law.
Forces only occur as pairs. That is, in order for object 1 to act on object 2 and change its motion, object 1 must exert a force on object 2. However, this action necessitates that object 2 also act to change the motion of object 1.

Again, just as with the discussion of Galileo's ideas, a lot is being said! The words can be replaced by mathematical equations that condense the length of the statement of these laws, but still, the deep ideas are in there. One "deep" piece of information is that the laws as stated above are not very specific about what we mean by the term "force". Forces are operationally defined. You know a force is acting if something which has a mass is changing its motion. If a force is acting, it is due to an external object acting on the thing whose motion is changing. This says nothing about how the force is exerted!

Many forces are contact forces, i.e. they act only while two objects are physically touching. Other forces are action-at-a-distance forces for which no physical contact is necessary. Examples are gravitional, electrical, and magnetic forces. Newton's Laws of Motion apply to both kinds of forces.

The simplified expression of Newton's Second Law in math-speak is

F_{net, ext} = ma

where F_{net, ext} is the net (i.e. the vector sum of all forces), external force acting on an object. The bold face type indicates that force has the characteristics of a vector, namely it must be specified by direction and magnitude. Force is not exactly a vector since you must also specify what the force acts on before it is truly defined. The equation F_{net, ext} = ma means we can relate actions to changes in motion, hence the equation a = F/m is referred to as the equation of motion.

Units

We will want to make direct measurements of forces in physics experiments (if we were in fact doing such experiments), so it is appropriate to introduce an SI unit for such measurements. For obvious reasons, the unit is called the newton and is equal to (1 kg) · (1 m/s²).

Types of Forces

There are various types of forces to deal with in freshman physics. Ultimately, we know that all of these ultimately reduce to just two fundamental types: electromagnetism and gravity. That, however, is only the result of a long historical succession of reductionism. For the most part, the properties of the various manifestations of the electromagnetic force in nature are so different as to make it acceptable to simply deal with them as essentially different kinds of force. In either the extreme reductionist view or the more complicated real-world view, however, all forces obey Newton's Three Laws of Motion. Note that even though the term ``Law'' is used, these are really principles in that Newton defined terms like mass and force through these laws: hence they cannot really be proven wrong although they might not be particularly useful for all occasions.

The various force types are reviewed briefly here as they are in the Online Textbook.

Gravitational force - the force exerted by any massive body on another massive body. Always straight down towards the center of the earth with a magnitude equal to the mass of the object times the gravitational acceleration (usually g = 9.8 m/s²).
Spring forces - for compression or stretching of a spring. The magnitude and direction are given by F = -kx. The minus sign indicates that the direction of the force is opposite to the direction of pull or push on the spring.
Normal forces - these are the reaction forces that result from contact between two objects. Normal forces are always directed perpendicularly away from the surface which exerts the normal force. Their magnitude depends on some external agent which maintains the contact between objects. To determine their direction for any situation, note that a normal force can only push the object it acts on
Tension - this reaction force is the force of a string or rope on an object to which the string or rope is attached. The direction of the tension is always along the rope or string and away from the surface of the object to which the rope or string is attached. Tension forces can only pull the objects they act on. NOTE: It is important to remember that we always assume a non-stretching string or rope (unless explicitly told otherwise) so that the magnitude of the tension is constant along the string or rope. This enormously simplifies the mathematics of using the tension by adding important constraints to the solution of the problem. One of the most important constraints is that the length of the string or rope is constant.
Friction - these forces always resist the motion occuring or the motion that would occur if friction were not present.

We should examine each of these more closely to be sure the words used in their description actually form useful ideas when you apply them.

A pictorial example of normal forces for two cases are below. Note that there have to be other forces acting on the objects in order to have the normal forces be non-zero in magnitude, but the directions must always be opposite each other and equal magnitude in accordance with Newton's Third Law, since, by definition, normal forces come from the interaction of objects with each other.

Why Is There a Third Law of Motion?

For high school physics courses, the importance of Newton's Third Law of Motion often gets shrouded in mystery. In fact, it seems less ``fundamental'' than the first two Laws as those two ``define'' two important quantities, mass and force. The Third Law, however, is essential to making the Second Law consistent in an inertial frame however. In some sense, the Third Law is necessary to complete the definition of force because it specifies *how* objects must interact in order to exert forces on each other. Since all forces are ultimately the result of an interaction between objects, you cannot specify a force until you know what it acts on, and you cannot specify what it acts on until you know the consistency rules that govern that interaction. The example below is a good case in point.

Suppose you have two blocks, labeled A and B. They have masses m_A and m_B, respectively. These blocks slide without friction.

If the blocks are welded together, then they can be considered as just one object of mass m_A + m_B. An external force, F applied to this object in the horizontal direction as shown in the figure above must push the object with an acceleration of a_AB = F/( m_A + m_B).

Now suppose the weld is broken so that the blocks are truly separate. When force F is applied to block A as before, the blocks will be kept in contact since A must push on B as long as F is acting as shown in the figure above. As far as the unseeen external object providing force F is concerned, nothing is changed. This object cannot distinguish between the blocks welded or separate but maintaining contact. Therefore, we expect the acceleration of the pair of blocks to remain at a_AB = F/( m_A + m_B). Thus, the forces acting in the horizontal direction on each block must yield this acceleration. The force diagrams for the two are shown below (horizontal direction only).

Note that the force diagram must show a force F_BA (read this as the force on B due to A), otherwise we cannot explain block B's acceleration in Newton's scheme that all accelerations must be due to external forces acting on the object. By the same token, we must have a force F_AB acting, in the direction shown, on block A, otherwise, the acceleration of block A would have to be higher than a_AB (since only block A is being directly affected by force F now). So, we have established that the forces F_AB and F_BA must be in opposite directions as per Newton's Third Law or we cannot explain why the accelerations of the blocks can both be a_AB. What about their relative magnitudes? Here, we resort to quantitative determination of the forces acting in order to explain the accelerations of the blocks. Let the direction to the right be positive. Then, for each block to have acceleration a_AB, we need

Block B:	m_B a_AB	= F_BA

Block A:	m_A a_AB	= F - F_AB

Add these two equations together and use our previous result for a_AB in terms of F and the masses of the blocks...

(m_A + m_B) a_AB	= F + F_BA - F_AB
(m_A + m_B) [F/( m_A + m_B)]	= F + F_BA - F_AB
F	= F + F_BA - F_AB ==>
F_BA	= F_AB

So, the magnitudes of the contact forces are equal and their directions are opposite. The same analysis must hold true for any objects in contact. We always find that the forces they exert on each other must be equal magnitude and opposite directions in order for Newton's Second Law, that accelerations are due only to the net external forces acting, to be consistent.

Some Example Problems

Let's look at applications of these ideas to some standard problems.

First, we do the half-Atwood machine.

The free-body diagrams are as shown below

Newton's prescription says that the net force determines the change in motion, hence if we wanted to find the accelerations of the two blocks after they are released, we need to write down for the horizontal motion of m_A and the vertical motion of m_B.

m_A:	T	= m_Aa

m_B:	m_Bg - T	= m_Ba

Note that we listed the accelerations of both blocks as having the same magnitude and sign. We have to be careful in doing this even though it is easy to simply pass this point by. In choosing the directions for our coordinate systems (note that we chose different coordinate systems for each mass), we have determined the relationship between the accelerations of the blocks. m_B moves downward. The acceleration must be the same magnitude for both blocks since the string does not change length (hence the amount of rope separating the two blocks stays constant as the blocks accelerate). The signs are the same only because of the coordinate systems we chose. Had the positive direction for m_B been upward, then the acceleration of m_B and the acceleration of m_A would have had opposite signs.

We add our two equations together (after all, they are just algebraic equations now) to get

m_Bg = ( m_A + m_B)a ==> a = (m_Bg)/ (m_A + m_B)

We could plot this solution for the acceleration, but the equation is simple enough that we can just state some things about it and see that they are true:

the acceleration is never greater than the free-fall acceleration g since, unless there were such a thing as negative mass, the ratio of (m_Bg)/ (m_A + m_B) must always be less than 1
the acceleration is constant with time. We didn't explicitly assume this (and note that Newton's Laws don't care unless we have to deal with a non-inertial reference frame)
the tension is easily derived once the acceleration is known since we can go back to either of our equations of motion and find that

T = (m_Am_B)g/ (m_A + m_B)

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