
AN INTRODUCTION TO NEWTONIAN
MECHANICS by Edward Kluk Dickinson State University, Dickinson ND 
Introduction
In our discussion of rectilinear frictionless
motion on a horizontal plane we already noticed what friction does to any
motion. Namely it tries to stop the motion, creating a force which acts in
the direction opposite to the direction of motion. Finding how this force
depends on such magnitudes as velocity, mass, area of contact with the horizontal
plane, gravity acceleration and possibly other parameters was not a simple
task. Purely on an experimental basis, scientists have found out that
a magnitude of the force of friction
F_{f}_{ } is approximately proportional
to the weight of the moving mg body and it does not depend
practically on the magnitude of body's velocity or area of the contact.
Therefore the magnitude F_{f}  of the friction force
has a simple form
F_{f } = u_{k} mg
where u_{k} is a kinematic friction coefficient. This coefficient depends on contacting materials and smoothness of contacting surfaces. Its value for ice on ice is about 0.035 whereas for rubber on concrete is about 1.00.
Generally every force is a vector. This means that beside of its magnitude the force also has a direction. Magnitudes of vectors are usually marked the same way as absolute values of numbers with help of two embracing vertical lines . Notice that for every nonzero number c the value of c / c is 1 for positive c and 1 for negative c. These two values as we look on them from the origin of the number line are showing respectively positive and negative directions along this line. Similarly if we take a velocity vector v in a case of rectilinear motion, then v / v is a vector of a unit length showing the direction of body's motion. As a matter of fact, a set of all vectors along a straight line is equivalent to the set of real numbers. In particular, an absolute value of a real number is equivalent to the magnitude of the vector associated with this number. All positive numbers are associated with the vectors directed in the positive direction of the number line and all negative numbers are associated with the vectors directed in the negative direction along this line. On a top of it, all operations (like addition, subtraction etc.) on real numbers have their equivalences for this set of vectors.
Using the vector properties which are described above we may write the friction force (vector) in the following form
F_{f } =  (v / v) u_{k} mg .
This formula tells us that the friction force acts always in an opposite direction to the direction of velocity. Moreover, it tells us that the friction force depends on the velocity direction but does not depend on the velocity magnitude.
A body moving through liquid or gas also experiences a resistance of the medium. The resistance tries to stop the body's motion. This time, however, the force of resistance depends on a magnitude of the body's velocity. The greater velocity causes the greater resistance force.
Newton second law and friction
Analysing Newton second law for a vertical motion of the body under influence of a constant gravitational force we have found out that the kinetic energy of the body was decreasing when the body was moving up and increasing when the body was moving down. We also discovered the existence of potential energy and the conservation of mechanical energy.
For a rectilinear horizontal motion of the body under influence of the friction force, Newton second law has the following form
m Dv/Dt =  (v / v) u_{k} mg .
Multiplying this formula by v, and using the relation v^{2} = v^{2 } we obtain
D(1/2 mv^{2}) / Dt =  v u_{k} mg =  u_{k} mg v .
This result shows that the rate of change of kinetic energy is always negative. Consequently the kinetic energy is always disappearing (dissipating) and never can be regained for this kind of motion. Then we do not deal here with any kind of potential energy. As you remember, potential energy always can be changed into kinetic energy. This time kinetic energy is converted into heat energy which is not a macroscopic mechanical energy. We may say that heat energy is a microscopic mechanical energy, or a sort of mechanical energy related to unordered (random) atomic and molecular motions. Heat energy can be converted into kinetic energy only with help of heat engines which are relatively complicated devices and must obey the laws of thermodynamics.
Our discussion of energy conversions in motions with friction let us conclude that for such motions principle of mechanical energy conservation does not hold. It holds only if for each acting force a respective potential energy exists or this force does not convert kinetic energy into other kinds of energy. Such forces are called conservative forces and so far we have discussed only two kinds of them. Namely, gravitational and elastic forces.
One more look at the Newtons second law for the rectilinear horizontal motion of the body under influence of the friction force shows us that its acceleration Dv / Dt is always in the opposite direction to the velocity of the body. It simply means that the body is slowing down until its kinetic energy is exhausted and the body stops. Moreover this acceleration (or deceleration if you prefer) is constant, consequently the velocity v(t) and position x(t) of the body as functions of time have the following forms
v(t) = v_{o}  u_{k} g t , x(t) = x_{o} + v_{o} t  (u_{k} g / 2) t^{2 } ,
where v_{o}_{ }and x_{o} are initial velocity and initial position respectively.
Our mathematical model of the motion with friction, however, has some clearly unphysical properties related to the oversimplified choice of the form of friction force. As long as the body's velocity differs from zero this force stays constant. But as soon as the velocity reaches zero the force must instantly drop to zero too. Otherwise the body would start to move in opposite direction which in reality for the investigated motion does not happen. Such instant drop of the friction force or any other force to zero is physically impossible. It may take a very short but always nonzero time interval. Therefore, for low enough velocities a real friction force must be dependent on velocity. In practice, however, the velocities for which the real friction force starts to exhibit a velocity dependence are very low and we do not need to worry about them.
"Experimental" determination of friction coefficient
If indeed for rectilinear motion with friction there is a constant rate of deceleration then we can apply to it the results which were obtained above. According to these results an average velocity for the time interval (t_{2}, t_{1}) can be calculated as follows
[x(t_{2})  x(t_{1})] / (t_{2}  t_{1}) = [v_{0} (t_{2}  t_{1})  (u_{k} g / 2) (t_{2}^{2}  t_{1}^{2})] / (t_{2} t_{1}) = v_{0}  u_{k }g (t_{2} + t_{1}) / 2 .
Thus this average velocity is equal to a true velocity at time (t1 + t2) / 2. Consequently a collection of data of covered distance versus time will make possible to obtain a velocity versus time data collection. This in turn allows to graph velocity versus time and obtain an experimental friction coefficient from the slope and initial velocity from the vertical intercept of this graph.
For data collection start the applet and set the initial velocity to 0.25 m/s and friction coefficient to 0.30. From now on please pretend that you do not know any of these values and your task is to find them experimentally. Prepare your stopwatch and simultaneously click on the applet's start button and start the stopwatch. Now try to stop the moving object at the distance of 5m and simultaneously stop the stopwatch. Read both the covered distance as accurately as possible and time elapsed, and record them in a table which is arranged as the table below. In next five trials try to stop the object as
trial # 
1 
2 
3 
4 
5 
6 
distance x [m] 






elapsed time [s] 






close as possible to 6, 7, 8, 9 and 10 m. When all data are collected use them to calculate average velocities and average times for each consecutive distance interval starting with the trial #1. To do it for the first interval take n = 2. This means that you have to take for distances and elapsed times data from the first two columns of the table above. Do the same for the next four distance intervals.
pair  12  2 3  34  45  56 
(t_{n+1} + t_{n}) / 2 [s]  
(x_{n+1}  x_{n}) / (t_{n+1}  t_{n}) [m/s] 
Remembering that the average velocity for each interval is a real velocity for the average time in this interval graph these velocities versus time using a standard size graph paper. Draw the best straight line through the graphed data, and find its slope and vertical intercept. The vertical intercept represents an initial velocity which should be close to 0.25 m / s. The slope represents an experimental value of  u_{k} g. Because we are still on the strange planet g = 0.01 m / s^{2}. Now calculate u_{k } which should be very close to 0.30.
Planing the whole experiment first five 1 m intervals were deliberately avoided. There the object moves relatively fast and time differences between edges of these five intervals are relatively small. Thus we shall expect bigger errors in measurements of these time differences and less reliable values of velocities.
Evaluation
If at this point you do understand:
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.
Last update: Jan 10, 1997  E  mail to Edward Kluk 
Copyright (c) 1996 Edward Kluk 