Topic 2-01: Mass, weight, force and Newton's Second Law: The view of earthlings

Earthlings have a natural sense about the weight of an object. This is the sense whether one object is heavier or lighter than the other object. With the invention of the balance scale as shown in Fig.1, the comparison of the weights of different objects can be made precisely. It is natural to have some commonly agreed upon standards of weights to make communication easier. For example, a specifically manufactured object is defined as having the weight of 1 kilogram by an international agreement, one-thousandth of 1 kilogram is defined as 1 gram, 1000 kilograms is called 1 (metric) ton, and so on. Any object that balances with the standard weight of 1 kilogram on a balance scale is then defined to have the weight of 1 kilogram. The pound system is another such example of mutual agreement by defining the weight of some object as 1 pound of weight. Currently, among the major geographical regions around the world, only the United States of America is still sticking to the pound system, whereas other major regions, including the birth place of the pound system, England, have all shifted to the kilogram system.

Another important tool in the discussion of this topic is a spring. It was discovered that a vertically mounted spring stretches out by a certain amount when an object is suspended from the end of the spring as shown in Fig. 2. When different objects of the same weight are suspended from the end of the spring alternatively, the spring stretches by the same amount in each case. However, when objects of different weights are suspended from the end of the spring, the stretch of the spring varies according to the weight, that is, a heavier object makes the spring stretch more. Thus, a spring can also be used to compare the weights of different objects.

At this point it is tempting to consider weight as an intrinsic quantity of an object. Unfortunately, this cannot be done, since the weight of an object varies in different environments. For example, suppose a spring is hung from the ceiling of an elevator at rest, and an object, say with a weight of 1 kilogram, is suspended from the end of the spring. Of course, the spring stretches by a certain amount as the object is hung. Alongside the spring, a balance scale in the same elevator will show that another object of the weight of 1 kilogram balances with a standard weight of 1 kilogram on the other plate of the balance scale. When the elevator accelerates upward, we will observe that the amount of the stretch of the string increases, indicating that the weight of the suspended object has increased, whereas on the balance scale the object with the weight of 1 kilogram still balances with the standard weight of 1 kilogram. When the elevator accelerates downward, we will observe that the amount of the stretch of the spring decreases, indicating that the weight of the suspended object has decreased, whereas on the balance scale the object with the weight of 1 kilogram still balances with the standard weight of 1 kilogram. A similar phenomenon occurs when we carry the spring; two objects, each of which weighs 1 kilogram; a standard weight of 1 kilogram; and a balance scale up to a very high mountain. The amount of stretch of the vertical spring will decrease at the top of the high mountain compared to the stretch of the spring at the sea level with the same object hung from the end of the spring, indicating that the weight of the suspended object has decreased at the top of the mountain. But the object with weight of 1 kilogram still balances with the standard weight of 1 kilogram on the balance scale even at the top of the mountain. This series of observations clearly indicates that the "weight" of an object changes according to the environment and thus cannot be an intrinsic quantity of the object. However, the observation that two objects that balance on a balance scale will continue to balance in all the different environments suggests that we can indeed assign an intrinsic quantity, called "mass", to each object. We define the standard weight of 1 kilogram as having a mass of 1 kilogram, and any object that balances with the standard weight of 1 kilogram on a balance scale also has a mass of 1 kilogram. The intuitive "weight" of an object can be understood as some quantity that is proportional to its mass.

For example, an object of mass M will have a weight of cM. The proportional constant c will vary according to the environment, but c is universal in a fixed environment. The last statement means that, for example, at sea level every object of mass M will have a weight of cM, and c is universal no matter what kind of objects we are considering. If in a different environment, like in an accelerating elevator or at the top of a very high mountain, the constant c will change, but the change is universal for every object subjected to the same environment. Once "mass" and "weight" are defined like that, all the above mentioned observations, at the sea level on the ground, at the top of a very high mountain, or in an accelerating elevator can be explained consistently. We call the concept of "mass" intuitive because we can not explain what "mass" is beyond the definition given here; certainly the arrival of the concept of "mass" is far less straightforward compared to the intuitions of distance and time as discussed in Chapter 1. Mass is another basic concept of physics. Distance, time and mass are the three basic entities on which classical mechanics is constructed. The proportional constant c at sea level is usually written as g, and this g is the constant that has appeared repeatedly in the problems of Chapter 1.

To further understand what "weight" is, we first consider a setup as shown in Fig. 3. A spring is mounted on a horizontal surface. A string is attached to the spring, threads through a frictionless wheel, and its other end is tied to an object of mass M; the object of mass M is suspended vertically as shown in the figure. The spring will stretch by a certain amount. The amount of the stretch is equal to the case that the spring is vertically mounted as in Fig. 2. This experiment implies that the weight can be transmitted through a string. Now consider an experiment as shown in Fig. 4. It consists a very smooth (frictionless) horizontal surface and a frictionless wheel. A string threads through the wheel with its horizontal end tied to an object of mass m and its vertical end tied to an object of mass M. When released from the restrain, the object of mass m will slide toward the right. The positions of the object of mass m is carefully measured at various times and its speeds and accelerations are calculated as functions of time t. The result shows that the acceleration a is independent of time t. This experiment is repeated for various objects with masses m1, m2, m3, …. And their accelerations a1, a2, a3, …., are measured respectively. The measured accelerations are, of course, independent of time. We will also observe that

m1a1 = m2a2 = m3a3 = …..

In other words once the mass M is fixed, ma is a constant no matter what value m takes. We define this constant as the "weight" of the object of mass M, that is, we equate this constant with cM. As mentioned before the proportional constant c varies according to the environment. When the experiment of Fig. 4 is carried out at the sea level and the lab itself is at rest, the constant c is traditionally written as g. The value of g is 9.80 m/sec2. The constant g is sometimes called the gravitational acceleration. The result of the experiment of Fig. 4, performed at the sea level at rest, can be summarized as

ma = M g .                                                       (1)

"Force" is another intuitive feeling of earthlings. It is the strain that muscle feels. When a person is upholding an object of the weight Mg, the person's muscle feels a strain. It is said that the muscle exerts a force that cancels the weight, Mg, of the object. Thus the weight is a kind of force. The notion that "weight" is a force can be further confirmed by looking at Fig. 2. When a weight Mg is suspended on the spring, the spring stretches by a certain amount. If there is no weight attached to the spring in Fig. 2, a person can exert a force on the spring by pulling it to make it stretch exactly the same amount as the case when a weight of Mg is suspended. A force does not need to be confined to the vertical direction, and so is the weight as we have already seen in Fig. 3. We can tilt the horizontal surface of Fig. 3 to any angle to mimic a force exerted on the spring at any angle. Thus we can write Eq. (1) as

f = ma,                                                               (2)

and say that when an object of mass m is moving with an acceleration a, a force that equals to ma is being applied to the object. We already know that an acceleration can be a vector, a, in the two- or three-dimensional space. Thus generalizing Eq. (2), we can define a vector force f as

f = ma.                                                               (3)

So far we have only considered the case of a constant acceleration. However, Eq. (3) can be generalized to cover time varying accelerations, and thus a force f defined by Eq. (3) can also vary with time in both its magnitude and in its direction. Since acceleration a can be written in a differentiation form as

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where r(t) is the position vector of the object with a mass m, Eq. (3) can be written as
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Eq. (4) is known as Newton's Second Law and is used to define a force in general.