In review, we've looked at forces and put them into two categories: contact and non contact. In addition, we have examined what happens to objects when combinations of forces are applied. We have seen that even though a number of forces may act on an object, if the forces cancel each other, that object will keep moving at the same speed in a straight line (keep its state of motion). Along the same line, if the forces don't cancel, a net force will result and that object will change its state of motion (i.e. accelerate).
In addition, we have come to realize that net forces not only cause changes in translation, but they can cause changes in rotation as well. When a force is applied at some distance away from an axis of rotation, it can create a torque, which alters the rotational state of motion of the object (angular acceleration). We've developed techniques that make it easier for us to setup and solve problems. Free body diagrams and setting up Newton's 2nd law equations will remain with you for the remainder of the year.
What we haven't done with force is look inside what creates the forces themselves. Non contact forces are created by field interactions which can be evaluated in terms of the physical characteristics of the objects. Contact forces occur as a result of the direct interaction of two objects. Consequently, our analysis of force can get a little more in depth.
Net Force and Units
Although I have referred to these concepts already , it never hurts to re-examine them in more detail to gain mastery of the subject. Net forces result from combinations of forces that don't cancel each other out. When this occurs, an object's state of motion is changing over time, what we have come to call acceleration. The net force is characterized by the product of the mass of the object experiencing the net force and it's acceleration.
Or, more simply put,
To analyze the units of force, we need a little background. This year we will use the SI (international standard--metric system) of measurement. Within this system there are two common sets of measurements: cgs (centimeter-gram-second) for small puny things--like Rattet, and mks (meter-kilogram-second) for large scale things--like Quiocho. For our purposes, it will be most convenient to use the mks system of nomenclature (names).
In the mks system, the unit of mass is the kilogram (kg) , length is the meter (m), and time is the second (s) . In mechanics, all of the other units result from combinations of these. For example, as we study kinematics, you find that velocity has units of meters per second (m/s) and acceleration has units of meters per second squared (m/s2) . With this knowledge, we can figure out what the units of force ought to be.
Since F = ma, and m has units of kg, while a has units of m/s2, then force must have units of kg•m/s2. Well , any real physicist is far too lazy to write down kg•m/s2 every time they need to express the units of force, so we invented a name for the unit of force--the newton (N). Thus, whenever you compute any force, net or otherwise, the units that follow a numerical answer ought to be newtons (N), so long as you are working with the mks system.
For argument's sake: Earlier we showed that the net force acting on an object traveling in a circular path has a special version of "ma", which is
Show that this expression yields the same units as "ma".
Non Contact Forces
All non contact forces occur as a result of the interaction of fields. As previously discussed, three non contact forces exist: gravity, electric, and magnetic. We will examine each of these "fields" of study in depth.
Gravitation:
Any thing that has mass has a gravitational field surrounding it. This field spreads out radially, forever, in all directions. As far as we know the gravitational field surrounding any object extends into the farthest reaches of the universe. Anytime two fields interact, a gravitational force is exerted on the objects generating those fields. Consequently, nothing is ever truly "weightless," because its field interacts with all of the other fields in the universe, no matter its location.
Since mass is what generates a field, it mak es sense that the more mass an object has, the larger the field is that surrounds that object. As far as we know, all gravitational fields are attractive. The result of which implies that gravitational fields do not have a net effect of canceling each other out at great distances. That is, the combined field from two masses, when observed from a very large distance from the masses, appears like the field from one, larger mass.
However, if we want to evaluate gravitational fields and how they interact with each other, then we need to know how big they are at some point. So, the other factor that needs consideration is distance. How does distance affect the size of the gravitational field? If you were to take an educated guess, you'd probably say that the field would decrease in strength as you move farther away from an object. Whether it would decrease as 1/r (where r represents the distance away from the object), 1/r2, o r 1/r3, is a little harder to guess. In order to understand what the relationship actually is we need to look at the geometry surrounding an object in 3 dimensions. To assist, I'm going to attempt to come up with a fitting analogy .
Imagine, if you will , a ball of chalk dust with a tiny explosive device placed in the center of the ball. You then take the ball into space such that it is far from any other object that might influence how it scatters once it explodes. Around the ball you setup 5 observation shells that will record all of the dust particles that pass through them. Each observation shell forms a spherical shell around the ball at one meter intervals. When the ball is exploded, it spreads out evenly in all directions and eventually drifts beyond all of the observation points. All of the shells count the same number of dust particles that passed by. However, how close the dust particles were to each other at each shell varied tremendously. In fact, since the area of each observation shell can be computed as A = 4πr2, the density of the particles at the 1 meter shell were 4 times greater than the 2 meter shell , 9 times greater than the 3 meter shell, 16 times greater than the 4 meter shell , and 25 times greater than the 5 meter shell . Therefore, the particle density decreased with the square of the distance.
The number of dust particles passing an observation point are analogous to something called gravitational flux. The density of the dust at a particular shell equates to gravitational field strength. Consequently , as you get farther away from a mass, the strength of its field decreases with the square of the distance. Equation wise, the field strength appears as:
where m is the mass and r is the distance from the center of the mass.
Notice that the field strength is only proportional to and not equal to m/r2. The reason for this has to do with the units we chose to measure gravitational field. It would be possible to manipulate the values of the units such that an equality resulted, but it is simpler to keep the same units of measurement and introduce something called a proportionality constant. Essentially, what a proportionality constant does, is change a proportional equation to an equality . So, if you graphed field strength versus m/r2, you'd get a direct variation graph with a small slope. The small slope is your proportionality constant. For gravitation, the value of the proportionality constant is 6.67 x 10-11 N•m2/kg2. The number has a special name--the universal gravitation constant. The stuff behind the number are merely the units of the constant.
The significance of this number is that it applies to any gravitational field or force interaction in the universe. In reality , the constant was not determined by a field strength vs. m/r2 graph, but a gravitational force versus m1m2/r2 graph. Even so, the concepts behind how the constant was found are the same.
In 1798 a guy named Henry Cavendish used a torsional balance (this is a scale that measures force by an axle being twisted) to determine the value of the universal gravitation constant. By bringing two large known masses close to two small known masses suspended from the torsional axle, the gravitational force between them would cause the axle to twist until it resisted being twisted further. Thus, he could record the force and then carefully measure the distance that separated the centers of the masses. After taking gazillions of measurements he was able to graph the gravitational force versus m1m2/r2 data, establish a line of best fit, and arrive at a value for the universal gravitational constant. Not much different than what you were supposed to do in your spring lab (except for the gazillions of measurements part).
Cavendish was working from the model of gravitation proposed by Isaac Newton in his theory of gravitation. Wherein, Newton proposed that all objects are gravitationally attracted to each other according to the product of the masses divided by the square of the distance between their centers of mass. Or, in equation form:
Cavendish's contribution allows us to write a more exact version of this equation.
Once again physicists are pretty lazy , so we called the co nstant "G", thus making the equation:
The only thing missing here is a direction. Since gravity always pulls one object toward the center of the other (and vice versa), regardless of where that object is located, then the force must be radially directed.
EXERCISES:
