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Newton's Third
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by David Castro
Charles A. Dana Center
University of Texas
Austin, Texas
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Introduction
Newton's three laws serve as both the philosophical and contextual underpinnings of all mechanics. The first two laws are readily understood by students since they can be expressed in simple mathematical form and are used extensively in problem solving. In brief:
Newton's first law can be expressed as:
If 
then 
while Newton's second law is:
.
Unfortunately, most students' understanding of Newton's third law is limited. For example, when asked to explain the third law, many students reply, "Every action causes an equal and opposite reaction."
This definition is often memorized by students in order to hide a partial or incomplete understanding of this important law. For example, students will often state, incorrectly, that the force of gravity exerted by the Earth on an apple is greater than the force of gravity exerted on the Earth by that same apple. Similarly, they often believe that the force exerted by a bug on the windshield of a moving car is far less than the force exerted by the windshield on the bug!
A more complicated, yet accurate, statement of Newton's third law may prove useful:
If object A exerts a force on object B, then object B exerts a force on object A that has the same magnitude as the first force, but whose direction vector is rotated 180°. These two forces are called an action-reaction pair and are never applied on the same object.
Unfortunately, substituting a more precise definition often does little to help students understand the subtleties inherent in the third law. In practice, most students still simply fall back on the shorter, less precise (and misleading) definition that they first memorized sometime in middle school. More importantly, no matter how carefully a definition is written, unless students have spent time applying Newton's third law in a variety of contexts, they are unlikely to master its subtleties.
The following exercise provides physics students (B and C) with an opportunity to apply Newton's third law to a variety of physical situations, while also providing an excellent opportunity to review vectors. Double subscript notation is used throughout and serves to help students think through force problems.
Problem Statement
In this exercise, we will use force diagrams to better understand the interplay between the forces acting on more than one object. In each case, we will draw force diagrams for each object in our system, express our results algebraically, and then combine the equations for each individual object to better understand the behavior of the system as a whole. Since this problem is designed to help you think through Newton's third law, we will not seek numerical answers.
To assure uniformity within our results, we will adopt the following conventions for all of our diagrams. These conventions will also help guide you through each situation.
- Forces are represented by arrows drawn from the point at which that force is being applied. If the force is gravity, it is assumed to act on the object's center of mass.
- The length of the arrow is proportional to the amount of force being applied. For example, if a scale of 1 cm = 10 N is used, a 35 N force would be represented by an arrow 3.5 cm in length.
- In order to emphasize that velocity and acceleration are consequences of forces -- and are not in themselves causes -- these vectors are never directly attached to the object being analyzed.
- Every force will be assigned two subscripts. The first represents the object to which the force is applied, while the second represents the object applying the force. If you cannot identify both the target and the source of a particular force, then it is probably not a "real" force. Careful use of subscript notation will help you avoid including "pseudo-forces" in your force diagram, as well as accidentally counting a single force more than once.
- We will use the following symbols to differentiate between different types of forces:
woe (weight): the force on the object due to the gravitational attraction of the Earth. This is always a noncontact force.
Nos (normal force): the contact force that keeps two objects from moving through each other. It is called a normal force (not a "natural" force), since it is always applied at right angles to the surfaces in contact. In this case, Nsub>OS represents the normal force exerted on the object by the surface. Never use "e" as a subscript for a normal force in order to avoid confusion between a force applied by the surface and the gravitational (Earth) force.
fos (frictional force): the force that prevents objects from sliding past each other. fos represents the frictional force on the object by a surface.
Tor (tension force): the force applied by a rope, chord, or chain to an object. A tension always "pulls," so don't use tension for a spring, since it can both push and pull. In this case, Tor represents the tension on the object by a rope.
This notation can be extended to cover other types of forces. Examples include:
FoA (applied force): an external force applied to an object. This is normally given in problems as a given value (e.g., 20 N) -- usually without indicating its source. In this case, an applied force is exerted on object o. In the solution to this exercise, a capital A will always be used to indicate an applied force.
FaE (electrical force): the electrical force exerted on object a by an electric field. Fields are indicated using uppercase letters.
FaB (magnetic force): the magnetic force exerted on object a by a magnetic field.
Additional forces should be noted in a similar fashion. Your goal is to be consistent and to clearly indicate which object is receiving the force and which object is applying the force.
Using this notation, our cumbersome third law definition reduces to:
.
Problem 1: Example
A block of mass ma is sitting on the ground.
(A) Draw a force diagram for each object shown below. Make sure to include appropriate subscripts.
Note that although Nas and wae have the same magnitude, they are not action-reaction pairs. The action-reaction parts for this situation are (Nas:Nsa) and (was:wsa). Action-reaction pairs always have the same symbol and reversed subscripts.
(B) Write a 
expression for each object shown above.
(C) Combine the expressions for each object to find an equation that represents the behavior of the system as a whole.
This simple example confirms that the Earth-block system is in equilibrium. Note that if the block was falling through the air, then both the Earth and the block accelerate toward each other.
Since both the block and the Earth move independently, the two force equations cannot be combined. However, replacing w with mag, we find that:
The expression for ae underscores why the acceleration of the Earth is not readily apparent in most everyday situations: me >>> ma.
Problem 2
Block A of mass ma is sitting on table t of mass mt, which is in turn resting on the Earth.
(A) Draw a force diagram for each object shown below. Make sure to include appropriate subscripts.
(B) Write a 
expression for each object shown above.
(C) Combine the 
expressions for each object to find an equation that represents the behavior of the system as a whole.
Problem 3
A man of mass mm is riding in an elevator cage of mass mc. The elevator is accelerating downward at ½ g, and mm = 2mc.
(A) Draw a force diagram for each object shown below. Make sure to include appropriate subscripts.
(B) Write a 
expression for each object shown above.
(C) Combine the
expressions for each object to find an equation that represents the behavior of the system as a whole.
Problem 4
A man of mass mm is lifting himself in an elevator cage of mass mc. The elevator is accelerating upward at ½ g and mm = 2mc. Assume that the pulley and rope are both massless.
(A) Draw a force diagram for each object shown below (do not include the pulley). Make sure to include appropriate subscripts.
(B) Write a
expression for each object shown above.
(C) Combine the
expressions for each object to find an equation that represents the behavior of the system as a whole.
Problem 5
Two granite blocks are being accelerated across a rough table by an applied force F as shown below. The masses of the two blocks are mb and ma, where ma > mb.
(A) Draw a force diagram for each object shown below. Make sure to include appropriate subscripts.
(B) Write a 
expression for each object shown above.
(C) Combine the
expressions for each object to find an equation that represents the behavior of the system as a whole.
Problem 6
Two wooden blocks are accelerated across a rough table by an applied force F, as shown below. The masses of the two blocks are ma and mb, and the connecting rope has a mass of mr, where ma > mb > mr.
(A) Draw a force diagram for each object shown below. Make sure to include appropriate subscripts.
(B) Write a
expression for each object shown above.
(C) Combine the
expressions for each object to find an equation that represents the behavior of the system as a whole.
Problem 7
Two cardboard boxes are accelerated across a rough floor by an applied force F as shown below. The masses of the two blocks are ma and mb, where ma > mb. Both blocks A and B are accelerating to the right. Block A and block B do not move relative to each other.
(A) Draw a force diagram for each object shown below. Make sure to include appropriate subscripts.
(B) Write a
expression for each object shown above.
(C) Since blocks A and B do not slip relative to each other, then the force equations for the two blocks can be meaningfully combined. Combine the
expressions for each object to find an equation that represents the behavior of the system as a whole.
Click here to view the answers and commentary!
David Castro taught AP Physics (B and C), AP Calculus (AB and BC), and AP U.S. and European History in a teaching career spanning 14 years, including 5 years as a master AP Physics teacher. In 1997, he received a Special Recognition Teaching Award, and in 2002 his combined AP Physics and AP Calculus syllabus was published in the AP Physics Teacher's Guide. Active as an AP Physics consultant in the Southwest Region since 1995, his areas of expertise include Pre-AP middle school science, AP Vertical Teams, as well as interdisciplinary physics/calculus. He also serves as a Reader for AP Physics. Mr. Castro recently joined the Charles A. Dana Center at the University of Texas, where he continues to focus on providing support for science educators.
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