Editor's note: The June 20, issue of "DC" featured #49 in this Nature of Science Series: Pinning Down Mass. The phrase at the end of paragraph three is actually meaningless and correctly should read, "A mole then is the amount of a substance that contains as many elementary entities as there are atoms in exactly 0.012 kg of Carbon-12." We apologize for any pursuant confusion this may have caused. -- Ed.
Physics: Force and Mass
Mass as a measure of inertia of a body makes no reference to the position in space of the body, though, of course, mass must occupy some volume of space somewhere in order to be mass at all. Mass is a fundamental and unchanging property of material bodies, though a specific mass does change (1) with time due to the effects of the motions of its internal components (K.E.), (2) with increased speed through space, or (3) with increased temperature, if not in amount then at least in volume. Pressure and density also may play a part in decreasing the amount of space a given mass can occupy without actually increasing or decreasing the amount of mass.
A body with mass resists motion if it is at rest; this tendency to resist motion is called inertia. Or, if the body-mass is already in motion in a given direction (linearly or curvilinearly) at a given speed, it will resist either an imposed change in the direction of its travel or in its speed (speeded up or slowed down). We recognize these resistances as Newton's First Law of Motion. Incidentally, the first two Newtonian Laws are in reality the same law of motion, the Second one being a more precise statement of the First Law, while the Third Law of Motion is not entirely a law of motion, as much as it is a statement of fact about the behavior of things.
Inertia is not measured directly; it is not said that a body has so many units of inertia. We do say that a body has mass, and mass is the quantitative measure of the amount of inertia the body has; thus, mass measures inertia. Inertia is that property of mass that resists (or opposes) change in a motion state. Inertia accounts for the fact that an object tossed from a moving car, for example, will continue to move forward in the same direction the car is traveling after leaving the car. Inertia also accounts for the fact that a person in an accelerating car feels as though he is being pushed in the direction opposite to the acceleration.
Newton discovered that the acceleration of a given body is proportional to ( ) the unbalanced force acting upon it. Thus, the harder you push or pull against a body, the faster it will accelerate: the acceleration produced by a given amount of force is inversely proportional to the weight of that body: F Wa, or, a F/W. The acceleration is always in the direction of the unbalanced, or applied, force. (The dyne is a unit of force and is equal to 1/980 grams; the dyne is the gram-centimeter/second unit. One dyne = one gm-cm per second, per second, or 1 gm-cm/sec2.)
Mass is a scalar quantity. The multiplication of a vector by a scalar results in a new vector of different magnitude but of the same direction as before the operation was performed.
Density is the ratio of mass to volume for a given material objects, i.e., the amount of matter in a unit volume of a substance. It is usually expressed in grams per cubic centimeter, grams per liter, or pounds per cubic foot, and is calculated by dividing mass by the volume occupied (at standard temperature and pressure). From the above it is easily seen that density applies to liquids, gases, as well as to solids. The weight-density of a body = weight divided by the volume, while the mass-density of a body = mass/volume.
In dealing with bodies in angular motion (curvilinear), the shape of the rotating bodies (or mass distribution in space) has differing effects of resistance to rotational motion. If two rotating bodies have the same mass (i.,e., amount of matter) but the mass in one is in the shape of a disk and in the other, the shape of an elongated but compact cylinder, their "rotational inertias" will be significantly different. Which object will have "more" inertia?
Forces: The four basic forces (now referred to as "interactions") of gravitational, electromagnetic, strong and weak act through "empty" space, i.e., action-at-a-distance, or AAAD, by some kind of energy-particle contact, except, so far, the g-force. Other forces (F) are exerted on objects by direct mechanical contact; thus, the body doing the pushing or pulling (force) must make contact with the body being pushed or pulled. (Attraction = pulling and repulsion = pushing). Force is proportional to mass times acceleration (F = kma); if k = unity (i.e., one), then the proportionality factor k can be eliminated from the equation and it then becomes simply F = ma.
Often when we think of force, we conceptionalize it as some vague invisible, generalized, "just there" kind of something, like a gravitational force. But mechanical forces, such as surface-to-surface contact pushings and pullings, have specific components. Some vector quantities, like force, are not completely specified by their magnitude and direction alone. The effects of an applied, unbalanced force also depend on its line of action and its point of application. The point of application of a given force acting on a rigid body may be transferred to any other point on the line of action without altering the effect of the force. A force, therefore, applied to a rigid body may be regarded as acting anywhere along its line of action. How can one direct and apply a mechanical force to a plastic substance such as a liquid or a gas without first providing such substances with a rigid container? Some measurable effect has to be produced by a force or, obviously, we could never know of its existence. One such effect is to change the shape or dimensions of an object on which the force is acting. Another effect is to alter the state of motion (speed or direction or both) of an object.
Experience tells us that an object-mass at rest will never start to move of itself; a push or pull originating externally to the object-mass must be exerted on it by some other object-mass. Actually, the force that moves the resting body (thus overcoming its inertia) has to be an unbalanced force (a resultant force or net force) acting on the object-mass. Such a force is required to slow down or stop a body already in motion, also. A sidewise force must be exerted on a moving body to deviate it from a straight line.
Rest (or changing velocity) results from more than one force acting on a given body. Thus, a state of "balance" or equilibrium is obtained because of the cancellation of all acting forces by other, opposite forces (technically: by oppositely-acting forces). The vector sum of all these acting forces is zero. If the vector sum of all the acting forces is not zero, a net sum in one direction results and there is, therefore, an unbalanced force (i.e. unopposed or at least not equally opposed) operating, and the object moves. This may be called a "motive force" and is expressed mathematically thus: Unbalanced force = Fu = mVf -mVo.
In short, a force is that which imposes a change in the speed of a body (its magnitude) or in its direction of movement in space, or both (change in the velocity of a body). Since a change in velocity is an acceleration, a force, then, is that which imposes an acceleration on a body-mass, the acceleration and force being in the same direction. Do you recognize this as Newton's Second law of Motion - F=ma?
In general, a force usually causes motion, except in certain special cases. The so-called "force of friction" is always considered "operating" in the opposite direction to the applied force because no matter from what direction the force is applied, the "force of friction" tends to oppose it. Friction as a force is a bothersome concept to me; it certainly belongs in the "special case" exception. First, frictional resistance to the direction of an applied and unbalanced force (the motive force) can occur only in a gravitational force field whereby the downward pull of the g-force keeps the unmoving frictional surface of a stationary surface in contact with the surface of the object-mass moved by an unbalanced force. Thus, secondly, the frictional force is not active but passive and dependent, and so only comes into operation as a result of the force-action on the object-mass receiving the unbalanced, motive force. To me, the term "frictional resistance" instead of "force" seems more appropriate and distinguishes between the drag of object-in-motion and in contact with each other due to the real force of gravity and the motion induced by active forces. I have a similar type "problem" with so-called centripetal "force," which will be discussed later on in this series.