This lesson introduces students to Sir Isaac Newton's Second Law of Motion. Topics include the two ways to study the dynamics of a system in which there is motion, a brief description of Newton's work in studying and measuring the motion of objects, and a summary of his three laws of motion. The lesson includes an activity in which students test Newton's Second Law by measuring the acceleration of a cart which is being pulled by increasingly larger masses. Educational levels: Undergraduate lower division, High school

John Pratte

English

Digital Library for Earth System Education

Newton’s Second Law 
 
Introduction 
There are two ways to study the dynamics of a system in which there is motion.  One of these is to study 
the kinematics of the system to see if there is any acceleration.  If there is an acceleration, then this 
implies that there is a net force on some part of the system.  If there is no acceleration, then this means 
that either there are no forces on the system, or that the forces within the system are all balanced.   
 
The other way to study the system is from an energy standpoint.  Energy can flow from potential to 
kinetic, or vice versa.  Furthermore, energy can be exchanged between different parts of the system.  If 
any object is giving energy to any other part of the system, then it must lose energy, either from its 
potential or kinetic energy.   
 
Each view has its advantages and disadvantages.  The energy view is nice mathematically, in that one is 
dealing with scalar quantities that are easy to handle.  However, studying the energy does not really allow 
one to predict the direction of motion, only the magnitude.  Studying the forces on the objects does allow 
this, but the mathematics involved in solving the motion is a little more complicated, in that one must use 
vectors.  In this particular activity, we are going study a system using the force approach, while the next 
activity in this module will use the energy approach. 
 
Fig. 1: Newton (NASA)
Newton 
Sir Isaac Newton was not the first person to study and measure the motion of 
objects.  As we stated in previous activities, there have been others like 
Aristotle and Galileo who did this before him.  Nor was he the first person 
who tried to model motion with equations that would allow for predictive 
behavior.  However, he was the first person to clearly state the basic laws of 
motion that allow us to analyze all systems.  These three laws, known as 
Newton’s Laws of Motion, are 
 
1. An object at rest or in a state of constant motion will remain in that state 
until acted upon by an unbalanced force. 
2. The net force on an object is proportional to its acceleration, with the 
proportionality constant being called the mass, i.e. m = Fnet/a. 
3. For every force on an object, there is an equal and opposite force 
produced by the object on the source of the first force. 
 
The first of these laws is a restatement of one of Galileo’s discoveries.  While seemingly obvious, 
especially to us now, it is a very powerful statement.  It says that an object that is noticed to be 
accelerating must have a net force acting upon it, even if the manner of the force is not readily noticeable.  
For instance, if one were to swing a rock on the end of a string in a circle, it would be obvious that the 
acceleration of the rock was due to the force exerted on it by the string, which is being accelerated by the 
hand holding it.  However, what of an electron that is above the Earth’s atmosphere that also moves in a 
circular orbit?  Since it is accelerating, we know that there must be some net force on it, even though 
there is nothing in contact with it (it turns out that the force operating on it is electromagnetism).  Or what 
of a distant star that is going around in an elliptical orbit, seemingly by itself?  We know that some other 
object must be applying a force on it, and by studying its motion, we might find its “neighbor” that is 
causing it to move thusly, even if its “neighbor” is a black hole that cannot be observed directly. 
 
The second law is often quoted as only F=ma, but this simplified version misses out on the fact that this 
law defines the term mass, which is one of the more misunderstood terms in science.  Many people 
confuse mass for weight, which is understandable since even scientists still persist in saying silly things 
like “one kilogram is equal to 2.25 pounds”.   But a fuller description of Newton’s Second Law shows that 
the mass is the proportionality constant between the applied force and the resulting acceleration.  Weight 
is merely the force that comes about because the gravitational attraction between two objects.  It is only 
important to know this when this one type of force is present.  Mass, on the other hand, is always an 
important factor, as it will determine the acceleration when any type of force is present. 
 
Possibly more misunderstood is the last of Newton’s laws.  It is used in many contexts where it should 
not, like economics, politics, and advertising, and is almost always used inappropriately.  Even when it is 
used in the context of physics and the dynamics of objects, it is often misused.  The reason is that the 
agents and receivers of the two forces involved in a situation are often left out, as when we state it merely 
as “For every force, there is an equal and opposite force.”  If one does not realize that these two forces 
act on different objects, then they might not think that acceleration is never possible, as there will never 
be a net force if every force creates an opposing force of equal strength.     
 
As an example, consider a person walking down the street.   If we concentrate on the foot in contact with 
the ground, we will notice that it applies a force on the ground in the reverse direction that the person is 
travelling.  According to Newton’s Third Law, the ground applies a force on the foot in the opposite 
direction.  It is this “reaction” force that propels the body forward.  However, we cannot forget about the 
force of the foot on the Earth.  This force is propelling the Earth backward at the same time that the body 
is being propelled forward.  Does the Earth really accelerate backward as the body goes forward?  Yes, 
but not by any noticeable amount.  This is because the Earth is so massive.  While the forces on the 
person and the Earth are the same, the differences in their mass means that the acceleration of the Earth 
is infinitesimal compared to that of the person. 
 
aEarth = (mperson/mEarth) aperson
 
Theory and Model 
In this week’s activity, we are going to 
test these laws.  In particular, we are 
going to test the validity of Newton’s 
Second Law using a quasi-frictionless 
cart attached to weights via a pulley 
(Figure 2).  Because gravity is allowed 
to operate on the weights, they will 
supply a force to the string that is equal 
to mwg.  This force will be applied to the 
cart via the tension T in the string, which should accelerate under its influence.  If there is no force lost in 
the pulley (frictionless pulley) or in the motion of the cart (frictionless surface), then the acceleration of the 
cart should be given by 
Fig. 2: Diagram of lab setup 
 
acart = T/mcart
 
The acceleration of the weights is a little more complicated, as gravity is acting down on the weights while 
the tension T in the string is pulling up on the weights.  This gives 
 
aw = (T – mwg)/mw
 
By noting that the acceleration of the cart to the right is the same as the acceleration of the cart 
downward (and accounting for the fact that we take the acceleration to the right or up is positive), we can 
relate these two equations, and solve for the tension T in the string, i.e. 
 
T = mwmcartg/(mw + mcart) 
  
 
Relating this back to the equation for the acceleration of the cart gives us 
 
acart = mwg/(mw + mcart) 
 
 
During the activity, we will use different masses of weights and measure the corresponding velocities of 
the cart.  If friction is negligible, the plot of the velocity of the cart versus the time should yield a straight 
line of slope given by the above equation. 
 
Procedure 
Figure 2 shows the necessary set up for this activity.  The actual equipment that you use for this will 
depend upon what is available at your institution.  Some locations will have “frictionless” air tracks that 
have carts that move down the track on a cushion of air.  Other locations will have carts that move along 
grooved tracks with low friction wheels.  The method of measuring the carts velocity will also vary with 
locations.  It is possible to measure the cart’s location using a pulley/photogate system that measures the 
how quickly the pulley turns and translates this into the object’s velocity.  It is also possible to use a 
motion detector that bounces ultrasonic sound waves off of the cart as it moves.  In any case, the 
instructions below should be followed, with any variations in methodology occurring in the process for 
measuring the cart’s velocity.   
 
As seen in Figure 2, a constant force is applied to a dynamics cart to pull it across the track.  The 
constant force will be provided by weights that are attached to a string that is suspended over a pulley.  
As gravity pulls these weights downward, it will pull the cart forward.  This motion will be measured by a 
photogate built into the pulley, which will relay the information to our computer.   
 
1. Make sure the track is level by adjusting any levelling screws.  Level can be determined 
either by a) using a bubble level or b) placing the cart on the track to see if it moves in either 
direction without being pushed. 
2. Measure the mass of the dynamics cart. 
3. Connect one end of the string to the dynamics cart and the other to the hanging mass (start 
with a total of 50 grams).  Place the cart on the track and run the string over the pulley.  Pull 
the cart as far back along the track so that the mass hanger almost touches the pulley.  Hold 
the hanger in place above the ground to prevent the system from moving.    
4. Initiate any software that you will be using to measure the velocity of the cart.  
5. Release the glider so it can be pulled by the falling mass hanger.  
6. Stop the data recording just before the mass hanger reaches the floor.  
7. Change the applied force (F = mg) by adding masses to the hanger. Repeat the data 
acquisition procedures above for total masses in 50 gram increments up to 250 grams.  After 
this, add the bar weights to the cart and repeat the above procedure starting at 50 grams on 
the hanger 
 
Data and Results 
From the computer, we will have tables of times versus velocity for each run. Since the acceleration 
should be constant, a plot of velocity versus time should yield a straight line whose slope is the 
acceleration. For each run, you need to have a table of the data like below, plus a plot of velocity versus 
time and a measure of the slope of the line.  When completed, answer the questions on the activity sheet.
ESA21: Environmental Science Activities Activity Sheet 
Newton’s Second Law
 
Name: 
Mass of the Cart = ________ gm 
Velocity Data 
Run 1 Run 2 Run 3 Run 4 Run 5 
Time Velocity Time Velocity Time Velocity Time Velocity Time Velocity 
          
          
          
          
          
          
          
          
          
          
          
 
Run 6 Run 7 Run 8 Run 9 Run 10 
Time Velocity Time Velocity Time Velocity Time Velocity Time Velocity 
          
          
          
          
          
          
          
          
          
          
          
 
After plotting each of these sets of data, calculate the acceleration from the best fit line of each graph 
 
 R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R 10 
ameasured           
atheoretical           
% Difference           
 
1. How close are your theoretical accelerations to the measured values? 
 
 
 
 
 
2. What are some possible characteristics of the experimental apparatus that are not accounted for in 
the model?  Could these characteristics account for the differences between measured and 
theoretical accelerations? 
 


Newton's Second Law

http://esa21.kennesaw.edu/activities/human_energy/P01_secondlaw.pdf

Newton's Second Law

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