AN INTRODUCTION TO NEWTONIAN MECHANICS by Edward Kluk Dickinson State University, Dickinson ND

NEWTON SECOND LAW

        Definition of gravitational mass and how to compare forces
        This time we are facing a very formidable task to analyse and truly understand Newton laws of motion. In a process of doing it a few new ideas and related to them operational definitions must be established. These definitions will enable us to measure magnitudes which are characterizing quantitatively mentioned ideas.
         Long before Newton people were measuring, with help of simple two plate symmetric or spring balances, gravity forces that are pulling bodies down toward the center of Earth . Such measurements were mostly done for commercial purposes to compare amounts of matter, scientifically called masses,  with certain standards. In 1790 Paris Academy of Science established a standard of mass called 1 kilogram (1 kg) which is still used in International System of Units (SI). If a certain body placed on one plate of the balance needs to be equilibrated by two copies of 1 kg standard placed on the other plate, its mass is 2 kg. Moreover, the gravity force acting on this body is two times greater than the gravity force acting on 1 kg standard mass. In every day language these forces are called weights and the measuring process is called weighing. In this scheme masses and weights are evidently proportional. But exact relation between them (a coefficient of proportionality) is still missing. There is one more interesting thing. If an object is moved from the surface of Earth to the surface of another planet its weight will be different because a gravity force exerted by the planet on the object will be different. Its mass, however, will stay the same. For example, 2 kg object on the other planet must still be equilibrated on the two plate balance by two copies of 1 kg mass standard. This is why the idea of mass is more generic than the idea of weight. But comparing masses of other bodies with use of balances we are always employing gravitational forces. This is why such masses should be called gravitational masses. Later on you will learn why we do not use the added adjective too often. Now, try to figure out on your own why all these ideas are still working if a good spring balance is used. The good spring balance must have the spring which elongates proportionally to the applied force. Notice that using springs we can compare magnitudes of other forces to magnitudes of gravitational forces. For example, attach one end of an exercise spring to the ceiling and stretch it 0.2 m by pulling it down with your arm. Now, instead of stretching it yourself, hang on it such amount of mass which will stretch it 0.2 m. The gravity force acting on this mass must than be equal to the force you were exerting on the spring.

        Newtonian calculus
        Our investigation of rectilinear motion on a horizontal frictionless plane let us conclude that if a body is left alone (no a net force is acting upon the body) it moves with constant velocity. If the body stays at rest this constant velocity is equal to zero . So far we have learned how to compare forces and we can measure their relative magnitudes. We also know that forces are influencing body's motion. But we do not know any quantitative relations between the body's motion and applied forces. This situation will change as soon as Newton second law is introduced. A meaningfull introduction of it, however, is not possible without a little bit more sophisticated algebra. Limiting our discussion to 1D motion of a point like body along x axis we can mark a current position of the body at time t as x(t). Traditionally a small change of any variable is denoted with help of capital greek letter "delta" which looks like an equilateral triangle. Unfortunately HTML in its current form does not allow this kind of letter. Therefore it will be replaced by D. Consequently Dt will represent a small change of time and Dx(t) = x(t) - x(t - Dt) a small change of x that takes place in the time interval (t, t - Dt). The same kind of notation will be applied to other variables which are dependent on time. Right now it is not difficult to notice that  Dx(t) / Dt  represents a rate of change of body's position or a body's velocity. Assuming Dt  positive, if  Dx(t) is positive the velocity is positive too, and the body at this particular instant  t  moves in the positive direction of  x axis. If Dx(t) is negative then the velocity is negative too and body moves in the negative direction of  x axis. Understanding of this kind of math is very important for our further discussion.

        Formulation of the second law and problem of inertial mass 
        As we already have noticed a  change of body's velocity demands application of a force. More massive is the body more force is needed to induce the same change in its velocity. The resistance of the body against a change of its velocity is called the body's inertia. Notice that the body's inertia may have nothing to do with body's gravitational mass. The last demonstrates itself and can be measured only if the body is under influence of a gravitational force. The body's inertia demonstrates itself always in this senese that body's velocity cannot be changed without application of a force or inducting a change of its inertia. For sake of simplicity we will discuss here only bodies with constant inertia. Following Newton let introduce another idea which he called an amount of body's motion. Now it is known as a body's momentum p(t) and defined as a product of body's inertial mass m_I and its velocity v(t). Thus we may write

p(t) = m_I v(t) .

Newton proposed that a rate of change of momentum Dp(t)/Dt should be equal to the force F(t) acting upon the body which we can write as

Dp(t)/Dt = F(t) .

This formula represents Newton second law in its original formulation. In most of practical cases an inertial mass of the body is constant then it does not depend on time. Consequently

Dp(t) = p(t) - p(t - Dt) = m_I v(t) - m_I v(t - Dt) = m_I [v(t) - v(t - Dt)] = m_I Dv(t)

which leads to more popular form of Newton second law

m_I Dv(t)/Dt = F(t)    or   m_I a(t) = F(t)

where a(t) is a rate of change of velocity or a body's acceleration. It is very important to realize that the second law does not define force or inertial mass. But it makes possible to predict the motion if the force, inertial mass and some initial conditions for this motion are known. Another important and unsolved yet problem represents inertial mass. Formally at this stage we do not know how to measure it.

         Solution of the inertial mass problem
        Historically verification of Newton second law took many years because of lack of proper technologies and enough advanced mathematics. Remember that to formulate the second law Newton was forced to invent a calculus similar to what we are using in our lectures. Latter developed advanced mathematics helped to confront Newtons second law with astronomical data and confirm its correctness. In our verification we will rely mostly on unusual properties of our fictitious planet like its very low gravity acceleration and lack of friction. The second law describes correctly motion of a body if a net force acting upon this body is equal to zero. A zero force implies a zero acceleration which in turn implies a constant velocity. In a case of free fall  motions experiments show a constant and equal for everybody acceleration g. On Earth g is about 9.8 m / s² and on our strange planet about 0.01 m / s². In both cases the second law implies a constant force F = m_I g. On the other hand we know that this case F is a gravitational force and then it must be proportional to the gravitational mass of the body. Consequently for everybody inertial mass is proportional to its gravitational mass. Because there are not other constrains on inertial mass it is very convenient to choose for a proportionality coefficient the plain number 1. Thus both masses will have the same units and the same values, but not necessary the same nature. Additionally, from now on we will skip subscript I used with inertial mass.

        The verifying "experiment"
        Now we know enough about the second law to verify it "experimentally". The "experimental" set contains (see the applet above):

• the block of mass m_B which can slide on the frictionless horizontal surface

• the hanger of mass m_H connected with the block with help of massless string running through the pulley

• the pulley which will not rotate because there is not friction between the pulley and the string.

        We will investigate motion of the system containing the block, hanger and string. A total mass of the system  m = m_B + m_H  will be kept constant. If we let it go, the system will move with a constant acceleration caused by a constant gravitational force F acting on the hanger. Knowing the hanger mass we can find this force as F = m_H g . Remember you are on the surface of the strange planet, not Earth. Timing the block or hanger every two meters and making the graph of the travelled distance versus t² we can find the system acceleration exactly the same way as we did it for the free fall motion. Remember to include into your graph the point t = 0 with the travelled distance also equal to zero.
        If you are already convinced about constant acceleration of the system the "experiment" could be simplified. Just measuring only a flight time of the block from 0 m to 8 m and using an appropriate formula we could calculate an acceleration. Please avoid this simplification and time the block or hanger every two meters because the collected data will be used again for the study of system mechanical energy.
        For each consecutive run a portion of mass should be moved from the block to the hanger. It will increase the force F running the system , without changing the total mass of the system. When enough data pairs force - acceleration are collected than a graph force vs. acceleration can be made. As the total mass of the system was kept constant, then according to the second law this graph should be a straight line through (0,0) point and its slope should be equal to the total mass of the system. If you understand the idea of this "experiment " go ahead find a partner and make it. It is easier for two people to make the "experiment". One person shall observe the block or hanger and signal when to read the time, whereas the other person shall read and record the time. Make all graphing and calculations needed to verify the second law. Compare the inertial mass of the system recovered from the slope on the graph force vs. acceleration with the real mass of the system. They should not differ more than 5%. Any discrepancy between them is related to both experimental errors in time and distance measurements and imperfectness of graphing.

        Please save all the data related to this "experiment" because they will be used again in conjunction with mechanical energy problems.

         Short epilogue
        Laws formulated by scientists have their limitations. That means they apply only under certain conditions and if these conditions are not met they do not work. To make them work with less restrictive conditions they have to be generalized. If Newton second law were exact for bodies on and around Earth we would not have stationary satellites nor hurricanes. Shortly you will learn about the second law limitations.

          Evaluation
          If at this point you can solve the following problems:

• Using general numbers, m_B for mass of the block , m_H  for mass of the hanger and g for gravity acceleration, express with their help the acceleration of the system from our "experiment". Using this result and Newton second law find general expressions for total force acting on the block and total force acting on the hanger. If you add them together you should obtain the total force acting on the system block - hanger. If the resultant is equal to the gravitational force acting on the hanger and you understand why, you have made it.
• Analysing our "experiment" you can conclude that the hanger exerts the force on the block through the string . The block is also exerting some force on the hanger through the string. Otherwise only force acting on the hanger would be the gravity force and the hanger should be falling down with gravity acceleration. In the former problem you have found the total force acting on the hanger. Now subtract from it the gravity force acting on the hanger and the result represents the force with which the block act on the hanger. What is the relation between the force with which the hanger acts on the block and the force with which the block act on the hanger. Newton third law states that if a body A acts on a body B with a certain force then the body B acts on the body A with the force of the same magnitude but opposite direction. If you recognize that forces acting along of piece of string (even if this string for some reason is not straight) can have only two directions, does the third law works for the block and hanger?
• Using Newtonian calculus finish the following: Dy²(t) = y²(t) - y²(t - Dt) = ... and prove that it is equal to [y(t) + y(t + Dt)] Dy(t). The last expression for very small Dt is approximately equal to 2y(t) Dy(t).
• Newton first law is often stated as follows: If a net force acting on the body is zero then this body moves with a constant velocity or stays at rest. If you take this formulation exactly as it sounds, does it follows from Newton second law?

the objectives of this lesson are fully achieved. If you have doubts try to read it once more concentrating on them, but do not try to memorize this text. Physics is not about memorizing, it is about understanding.