4.3: Newton's Third Law
Objectives
- Understand Newton's Third Law
- Understand the difference between countering force and action-reaction
- Use Newton's three laws to solve problems in one dimension
Vocabulary
- center of mass: The point at which all of the mass of an object is concentrated.
- dynamics: Considers the forces acting upon objects.
- free-body diagram (FBD): A diagram that shows those forces that act upon an object/body.
Equations
Newton’s Third Law: Forces or Pairs of Forces
It was Newton who realized singular forces could not exist: they must come in pairs. In order for there to be an “interaction” there must be at least two objects, each “feeling” the other’s effect.
Newton’s Third Law: Whenever two objects interact, they must necessarily place equal and opposite forces upon each other.
Mathematically, Newton’s Third Law is expressed as
Problem Solving
We use Newton’s laws to solve dynamics problems. Dynamics, unlike kinematics, considers the forces acting upon objects. Whether it is a system of stars gravitationally bound together or two colliding automobiles, we can use Newton’s laws to analyze and quantify their motion. Of Newton’s three laws, the major mathematical “workhorse” used to investigate these and endless other physical situations is Newton’s Second Law (N2L):
In using Newton’s laws, we assume that the acceleration is constant in all of the examples in the present chapter. Newton’s laws can certainly deal with situations where the acceleration is not constant, but for the most part, such situations are beyond the level of this book. A notable exception to this is when we investigate oscillatory motion. As a last simplification we assume that all forces act upon the center of mass of an object. The center of mass of an object can be thought of as that point where all of the mass of an object is concentrated. if your finger were placed at this point, the object would remain balanced. The 50 cm point is, for example, the center of mass of a meter stick.
Free-Body Diagrams
A diagram showing those forces that act upon a body is called a free-body diagram (FBD). The forces in a FBD show the direction in which each force acts, and, when possible, the relative magnitude of the each force by the length of the force vector. Each force in a FBD must be labeled appropriately so it is clear what each arrow represents.
Example 1: Sitting Bull
In the Figure below, a 1.0 kg bull statue is resting on a mantelpiece. Analyze the forces acting on the bull and their relationship to each other. There are two vertical forces that act upon the bull:
- The Earth pulling down on the center of mass of the bull with a force of
W=mg=(1.0)(9.8)=9.8 N - The floor pushing back against the weight of the bull, with a normal force
FN . The term normal force comes from mathematics, where normal means that the force is perpendicular to a surface. The normal force vector (often stated as “the normal”) is drawn perpendicular to the surface that the bull rests upon. Normal forces are usually associated with a push upon an object, not a pull.
Answer: Using N2L we write:
The statue is stationary so it has zero acceleration. This reduces the problem to
Example 2: Hanging Loose
In the Figure below, Mr. Joe Loose is hanging from a rope for dear life. Joe’s mass is 75 kg. Use
2a. Draw Joe’s FBD.
2b. What is the tension in the rope?
Answer: We assume the mass of the rope is negligible. Including the mass of the rope is not particularly difficult, but we’re just starting out!
The convention in physics is to use label
Once again, we apply
Example 3: Sliding Away
A 4900 N block of ice, initially at rest on a frictionless horizontal surface, has a horizontal force of 100 N applied to it.
Answer: Always begin by drawing an FBD of the problem.
Typically, applied forces are either written as
3a. Find the mass of the block of ice in Figure above, use
Answer:
3b. Find the acceleration of the block of ice in Figure above.
Answer:
3c. Find the velocity of the block at
Answer:
3d. Find the displacement of the block at
Answer:
Example 4: A Touching Story
In Figure below, Block
Note: When referring to more than one mass we often use the word “system.”
4a. Draw the FBD’s for Block
Answer: As Block
This force is labeled:
Block
This force is labeled:
4b. Find the acceleration of the system.
Answer: We use N2L applied to each block:
Block
Block
We have a system of two equations and two unknowns since the magnitudes of
Adding
Notice that on the left side of the resulting equation, the sum of
4c. What is the magnitude of the force between Block
Answer: This is answered by solving either the Block
The Atwood Machine
The Atwood Machine (invented by English mathematician Reverend George Atwood, 1746-1807) is used to demonstrate Newton’s Second Law, notably in determining the gravitational acceleration,
Example 5:
One end of the rope in Figure below is attached to a 3.2-kg mass,
5a. Draw FBDs for the
Answer:
5b. Determine the acceleration of the system. Use
Answer: Before we begin, we decide in which direction the system accelerates. Since the mass of the
The equations are set up so that the acceleration has a consistent sign.
Had we chosen the direction of the acceleration incorrectly, our answer would have been a negative number, informing us of our error.
Solving the system of equations we have:
5c. Find the tension in the rope.
Answer: Again, either equation will provide the answer. Using the second equation above, we have:
One possible check on the problem is to insure that:
Therefore:
It is always wise to check your results for consistency.
Image Attributions
- [1]^ Credit: Laura Guerin; Source: CK-12 Foundation; License: CC BY-NC 3.0
- [2]^ Credit: Image copyright artenot, 2014; modified by CK-12 Foundation - Raymond Chou; Source: http://www.shutterstock.com; License: CC BY-NC 3.0
- [3]^ Credit: Raymond Chou; Source: CK-12 Foundation; License: CC BY-NC 3.0
- [4]^ Credit: Christopher Auyeung, Raymond Chou; Source: CK-12 Foundation; License: CC BY-NC 3.0
- [5]^ Credit: Christopher Auyeung, Raymond Chou; Source: CK-12 Foundation; License: CC BY-NC 3.0
- [6]^ Credit: From Emmanuel Muller-Baden, Ed. (1905); Source: http://commons.wikimedia.org/wiki/File:Atwoods_machine.png; License: CC BY-NC 3.0
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Date Created:
Jan 31, 2013Last Modified:
Aug 18, 2015Vocabulary
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