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### 401-F: The Reaction Force of Weight

• Topic Cluster: Dynamics
• Topic: Identifying Forces (Newton's Third Law)
• Objective: Properly use the terms gravity and weight; properly identify the direction of the force of weight; identify the reaction force to weight
• Content: Physicists use the term weight to describe the force of gravity on a planet; the directions humans call downward is actually inward, towards the center of the earth; the reaction force of our weight is a force we exert on the earth.
• Level: 3

#### BACK to Ladder Identifying Forces (Newton's Third Law)

This pod is very closely related to another pod which focuses on the words "up" and "down" in gravity.

The weight is the force of gravity that pulls a person inward (which people call downwards) while they are on a planet.

Whenever A exerts a force on B, B exerts a force on A with equal magnitude in opposite direction. These are called an action-reaction pair, or, more accurately, an interaction pair.

Whenever two forces are an interaction pair, they have these three qualities:

• The same type of force.
• The same magnitude.
• Opposite directions.
• Agent and object flipped.
A balanced force pair is a pair of forces that have equal magnitude and opposite direction, but that act on the same object. A balanced force pair are not an interaction pair, because they do not have their agent and object reversed.

### Guided Problem:

A person with a weight of 600 N is standing on a flat surface. Two forces act on this person: gravity and the normal force. They are represented in the table below.

`
Force Magnitude Direction Agent Object
A. Gravity 600 N Down (inwards) Earth Person
B. Normal Force 600 N Up (outwards) Floor Person

These two forces have equal magnitude and opposite direction, but they are NOT an interaction pair. How can you tell? They fail the third requirement: these forces do not have their agent and object reversed. In fact, they have the same object, they both act on the person. Therefore, these forces are a balanced force pair, but not an interaction pair.

However, Newton's Third Law state that all forces must be a part of an interaction pair. So, if these two forces aren't an interaction pair, there need to be two other forces that complete two pairs. We can figure out what they are by following the rules above: create forces with equal magnitude, opposite direction, and agent and object reversed:

Force Magnitude Direction Agent Object
A. Gravity 600 N Down (inwards) Earth Person
B. Normal Force 600 N Up (outwards) Floor Person
C. Gravity 600 N Outwards Person Earth
D. 600 N Normal Force Down (inwards) Person Floor

In this table, A and C are an interaction pair, and B and D are another interaction pair.

### The Reaction Force to Weight

Every second of your life, the force of gravity has been pulling you inward, towards the center of the earth. But that also means that, according to Newton's Third Law, every second, you have been pulling the rest of the earth outwards, towards you, with exactly the same force! This concept is clearly very counterintuitive. How could it be that you pull the earth just as strongly as the earth pulls you? Consider two concepts:

##### You versus earth:

Imagine that you have a mass of 61.22 kg, and you are in free-fall on planet earth:

The magnitude of this force is 600 N, and because it is the only force on you, therefore, it is equal to the net force acting on you. According to Newton's Second Law, this force will cause you to accelerate towards the center of the earth: $$a = \frac{\Sigma F}{m} = \frac{600}{61.22} = 9.8 \text{m/s/s}$$

According to Newton's Third Law, the same force of 600 N acts on the earth, pulling it towards you. As a result the earth will accelerate towards you. But will it cause any genuine motion? The mass of the earth is $$5.97 \cdot 10^{24} \text{kg}$$. Thereforce, the acceleration of the earth is: $$a_{\text{earth}} = \frac{\Sigma F}{m} = \frac{600}{5.97 \cdot 10^{24}} = 1.0 \cdot 10 ^{-22} \text{m/s/s}$$

You and the earth are pulling each other with the same force. However, your acceleration towards the earth is 9.8 m/s/s. The earth's acceleration towards you is 0.0000000000000000000001 m/s/s. So, no, the earth does not move towards you. In fact, the reaction force to weight has a negligible effect on the earth.

##### You on the earth:

Here is another way to think about the problem above: You are standing on the floor, but you can instead consider the floor to be a part of the earth. I don't usually frame problems this way, because I think it can confuse gravity, which is caused by the entire earth, and the normal force, which is caused only by the tiny portion of earth just beneath your feet. However, if you consider it this way, the table above looks like this:

Force Magnitude Direction Agent Object
A. Gravity 600 N Down (inwards) Earth Person
B. Normal Force 600 N Up (outwards) Earth Person
C. Gravity 600 N Outwards Person Earth
D. 600 N Normal Force Down (inwards) Person Earth

In this framework, the net force on you is equal to zero, and the net force on the earth is equal to zero as well. So, even if the force of gravity of you pulling on the earth was significant, it is negated by the equally strong normal force of your pressing on the earth.

##### The earth-you system:

Here is one final way to think about yourself and the earth: you can consider you and the rest of the earth as part of the same system. (In fact, you do this without knowing every time you solve a problem involving the whole earth: consider everything on the earth to be a part of the system).

When two objects are a part of the same system, they are combined and considered to be one point. So, even though there is a force between you and the earth, it does not influence the motion of the system. Therefore, the gravity between you and earth does not affect the motion of the earth-you system.

This is an important concept to consider when considering something much larger than you: the moon. The earth pulls on the moon, and the moon pulls on the earth with equal magnitude, but these two forces do not affect the motion of the earth-moon system.

#### Questions:

1. The following free-body diagram shows the forces acting on an object at rest on a flat surface. Explain why the two forces in the free-body diagram are not an interaction pair.
2. Here is some Dude, standing on the earth (diagram not to scale):

1. Draw a free-body diagram of the Dude. The free-body diagram should be in relation to the entire earth, which means it should not show two forces going up and down.
2. The blank table below has space for four forces. Add the two forces in your free-body diagram. Make these forces A and B.
3. Fill in the other two rows of the table with the interaction pairs of the two forces you have already identified. Logically deduce the information on these forces by applying Newton's Third Law: Each force is in a pair. The forces in the pair have equal magnitude, opposite direction, and agent and object flipped.
4. Force Magnitude Direction Agent Object
A.
B.
C.
D.
5. The four forces in your table include two pairs of interaction forces. Identify these pairs.
6. Draw a free-body diagram of the earth. Your free-body diagram should include only one force, because it contains only forces relevant to this particular situation. (Even though many forces act on the earth).
7. Draw a free-body diagram of the ground on which the Dude is standing. Your free-body diagram should include only one force, because it contains only forces relevant to this particular situation. (Even though many forces act on the earth).
8. Of the four forces in your table, identify one of them as the weight of the Dude. Weight is defined as the force of gravity acting on a person while they are on a planet.
9. Of the four forces in your table, two of them are a balanced force pair, which are two forces of equal magnitude and opposite direction acting on the same object. Identify them.
10. Even though balanced force pairs have equal magnitude and opposite direction, they are not a Newton's Third Law pair. Explain why.
##### Table for #2
Force Magnitude Direction Agent Object
A. Gravity 500 N earth person
B. Normal Force 500 N ground person
C. Gravity 500 N person earth
D. Normal Force 500 N person ground