# Reciprocity

claimed by Nicole Romer (Spring 2017)

edited by Laurentino Castro (Spring 2018)

This section covers the top of reciprocity which explains why forces, such as gravitational and electric forces, act upon each other with equal magnitudes.

## The Main Idea

Reciprocity is the idea that the force object 1 exerts on object 2 is the same as the force object 2 exerts on object 1. This idea comes from Newton's Third Law of Motion, which states that every action has an equal and opposite reaction. For example, if I put my hand on a table, I am exerting a contact force on the table. Because my hand doesn't go straight through the table but instead rests on top of it, the table must also be exerting a force on me that is equivalent to the force I am placing upon it. Though it seems like I am putting in more effort, the forces are the same.

Newton's 3rd law implies that all forces come in pairs. The two forces are called "action" and "reaction" pairs. When forces are in these pairs, the magnitude of the two forces equal each other. However, in vector form, the two forces would be in opposite directions of each other, so one force would have a negative sign on it. This applies to both the gravitational and electric forces. Contact forces that we observe (such as the normal force) are actually due to electrical forces at the atomic level.

Because these forces are equal, so are the changes in momentum of the objects the forces are acting upon (remember, [math]\displaystyle{ |\vec{\mathbf{F}}|=|\frac{d\vec{\mathbf{p}}}{dt}| }[/math]). Two planets exerting attractive forces on each other will have the same change in momentum over the same amount of time. However if one planet is much larger it will have a much smaller change in velocity then the smaller planet. This is because p = mv and thus [math]\displaystyle{ {\Delta}v={\Delta}p/m }[/math]. So even though the forces will be the same, they may appear unequal due to a smaller change in velocity in a larger object.

The following image showcases the principle of reciprocity as the two people pulling exert an equal and opposite force on each other through tension. Also the object laying motionless on a surface has a net force of zero because the normal force of the table is equal to that of gravity on the object.

### A Mathematical Model

The force due to gravity is:

- [math]\displaystyle{ |\vec{\mathbf{F_g}}|=\frac{Gm_{1}m_{2}}{r^2} }[/math]

Although the force of gravity is dependent on the mass of both objects, both will experience the same magnitude of gravitational force. This can be clearly seen from the above equation.

The force due to electric interactions is:

- [math]\displaystyle{ |\vec{\mathbf{F_{elec}}}|=\frac{1}{4{\pi}{\epsilon}_0}\frac{q_{1}q_{2}}{r^2} }[/math]

Electric charges that are of different sign will attract each other with the same magnitude of force. Furthermore, two like charges will repel each other, and the same magnitude of repelling force will act on each of the charges.

A more formulaic representation of the principle of reciprocity is as follows:

- [math]\displaystyle{ \vec{\mathbf{F_{2 on 1}}}=\vec{\mathbf{F_{1 on 2}}} }[/math]

This can be represented in vector form as follows:

- [math]\displaystyle{ \vec{\mathbf{F_{2on1}}} }[/math] =
*<F,0,0>* - [math]\displaystyle{ \vec{\mathbf{F_{1on2}}} }[/math] =
*<-F,0,0>*

- [math]\displaystyle{ \vec{\mathbf{F_{2on1}}} }[/math] =

### A Computational Model

## Examples

Be sure to show all steps in your solution and include diagrams whenever possible

### Simple

*If you exert 20 N on the table, what would be the normal force of the table on you?*

Since you are exerting 20 Newtons, due to reciprocity the table will be exerting a normal force of 20 Newtons.

*If you push a box to the right with a force of 15 N, what is the magnitude and direction of the reciprocal force from the box on you?*

The box exerts a force of 15 N on you towards the left.

### Middling

*A 60 kilogram man stands on the surface of the Earth. What is the force Earth exerts on the man? What is the force the man exerts on the Earth?*
Recall, due to reciprocity, the force that the Earth exerts on the man should be the same as the force that the man exerts on the Earth. We know how much the Man exerts on the Earth because he is standing on the Earth with a force equal to his weight.

- [math]\displaystyle{ \mathbf{F_g}=mg=(60kg)(9.8\frac{m}{s^2})=588N }[/math]

Thus 588 N is the force that the man exerts on the Earth in the downwards direction. Because of reciprocity, the Earth must also be exerting 588 N on the man, but in the upwards direction.

*A box is sitting on Earth's surface. The Earth exerts a force of 196 N on the box. What is the force on the Earth? What is the box's mass?*
Here, we have the force. Because of reciprocity, the force of the box on the Earth must also be equal to 196 N. Then, finding the mass of the box is relatively straightforward, as is seen below:

- [math]\displaystyle{ m = F/g = 196N/9.8 = 20 kg }[/math]

### Difficult

*Two blocks of mass [math]\displaystyle{ m_1 }[/math] (under rod) and [math]\displaystyle{ m_3 }[/math] (above rod) are connected by a rod of mass [math]\displaystyle{ m_2 }[/math]. A constant unknown force F pulls upward on the top block while both blocks and the rod move upward at a constant velocity v near the surface of the Earth. The direction of the gravitational force on each block points down. Find [math]\displaystyle{ F_{1on2} }[/math], the magnitude of the force exerted by the bottom block on the rod.*

See the diagram to the side for a depiction of the situation. Because the velocity is constant, the net force on the system, and thus on block 1, is equal to zero.

- [math]\displaystyle{ F_{net}=0=F_{2on1}+F_g }[/math]
- [math]\displaystyle{ \vec{\mathbf{F_{2on1}}}-m_{1}g\hat{\mathbf{y}}=0 }[/math]
- [math]\displaystyle{ \vec{\mathbf{F_{2on1}}}=m_{1}g\hat{\mathbf{y}} }[/math]

Because [math]\displaystyle{ \vec{\mathbf{F_{1on2}}}=-\vec{\mathbf{F_{2on1}}} }[/math] it stands that:

- [math]\displaystyle{ \vec{\mathbf{F_{1on2}}}=-m_{1}g\hat{\mathbf{y}} }[/math]

*You are pulling a rope with a force of 15 N. The rope is connected to a box on a frictionless surface. The rope and box are moving at a constant rate. Find the force of the rope of the block, the block on the rope, and the rope on you.*

Because the velocity of the system is constant, the net force on the system is equal to zero ([math]\displaystyle{ F_{net} = 0 }[/math]).

- The Force of the rope on you is -15 N (equal and opposite to the force you are exerting on the rope)
- The Force of the rope on the box is 15 N (this must be true in order to make the net force on the rope equal to zero)
- The Force of the box on the rope is -15 N

## Connectedness

- The first physics I ever learned was Newtons laws. Before heading into any science class, I always thought, every reaction gets an equal and opposite reaction. I didnt really understand it. That is a fundamental principle that we use in almost all physics problems. It has been test questions and homework questions. The thing that intrigues me the most is how an ant can be pushing against a rhino and though the rhino is so much bigger, they are still exerting the same force.
- I am an industrial engineering major and though there is very minimal use of physics in that field, I do believe it is something that will help us go about our days knowing that force isn't how much effort you put in but about the action reaction pairs.
- Forces are something we deal with everyday. Everything we touch, me typing this page right now is all the result of forces. An important industry that deals with this is the automobile industry. If we understand the forces of the wheels on the road, we will know how to make wheels that best suit an automobile.
- This topics is one of the fundamental underlying principles of physics. It applies to pretty much everything in physics.
- Physics applies to almost any major. As an Environmental Engineer reciprocity is important to remember reciprocity, specifically in waste water management. Water pressure (force from water on pipe and force from pipe on water) is important when designing water infrastructure.
- Reciprocity is important to remember in buildings. When creating a structure that can sustain a storm, you have to think about how high winds will change how parts of the structure interact with each other.
- I remember back in elementary school the example that my teacher used was: when you kick a can, which experiences a greater force, the can or your foot? I was astonished to find out that both experienced a equal force due to reciprocity. I spent weeks kicking cans trying to understand this phenomenon until I learned for myself the forces were experienced relative to mass. This was my first real exposure to seeking to understand the world through the lens of physics.

## History

Isaac Newton was born in Woolsthorpe, England. When he was a child, one day he was resting under an apple tree when suddenly an apple fell on his head. He thought about why things fall down and not fall back up. He spent years figuring out the phenomenon. After all this, he came up with three laws of motion. This is when he discovered gravitation as a force. Where Newton's Law comes into play is that the Earth is exerting a force on us to stay with it since closer objects exert stronger forces on each other. We are also exerting a force on Earth so that we stay on the ground and don't go flying off. The date of this story is not known, and some even believe it to be a myth. However William Stukeley, author of *Memoirs of Sir Isaac Newton's Life* noted that he had a conversation with Newton and Newton talked about why an apple falls to the ground due to gravitational interaction.

## See also

### Further reading

**2:**Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3.4

### External links

https://www.youtube.com/watch?v=NfuKfbpkIrQ

## References

**5:**Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3.4