Newton's third law in terms of inertial position vectors for n-body system

In summary, Newton's third law states that for any two objects (named a and b) in a system of more than two objects, the forces exerted on each other are equal in magnitude but opposite in direction. This law applies to individual forces, not the net force on a body, and holds true for gravitational, electric, and magnetic forces.
  • #1
ato
30
0
Assuming
$$\vec{r_{a}}$$ and $$\vec{r_{b}}$$ is calculated from an inertial frame of reference.

then for any two objects (named a and b) in a system of more than two objects,
Is this the Newton's third law,

$$\frac{d^{2}}{dt^{2}}m_{a}\vec{r_{a}}=-\frac{d^{2}}{dt^{2}}m_{b}\vec{r_{b}}$$

i think this can't be right because then this implies

$$\frac{d^{2}}{dt^{2}}m_{i}\vec{r_{i}}=0$$ for every object in that system.

so i think i have misunderstood the law, so my question is can anyone state the law in terms of above variables for n-body system ?

Edit 1 (fix)
fixed a embarrassing mistake d/dt -> d^2/dt^2

thank you
 
Last edited:
Science news on Phys.org
  • #3
No it does not :(

i think the problem i am facing is which frame of reference to use, for example

1. if i use a frame of reference such as its origin and the center of mass of the object from which the position vector is to be calculated, coinsides .so

$$\vec{R_{AB}}$$ is position vector of object A from a frame of reference such as its origin and center of mass of object B coinsides.
and
$$\vec{R_{BA}}$$ is position vector of object B from a frame of reference such as its origin and center of mass of object A coinsides.

and the law would be

$$\frac{d^{2}}{dt^{2}}m_{A}\vec{R_{AB}}=-\frac{d^{2}}{dt^{2}}m_{B}\vec{R_{BA}}$$

but the problem what $$\theta_{AB},\theta_{BA}$$ to choose, certainly the above equation is not true for arbitrary $$\theta_{AB},\theta_{BA}$$ .

2. may be a inertial frame of reference is that frame and following eqaution is the law,

$$\frac{d^{2}}{dt^{2}}m_{A}(\vec{r_{A}}-\vec{r_{B}})=-\frac{d^{2}}{dt^{2}}m_{B}(\vec{r_{B}}-\vec{r_{A}})$$

hence

$$\frac{d^{2}}{dt^{2}}\left(m_{A}-m_{B}\right)\left(\vec{r_{A}}-\vec{r_{B}}\right)=0$$

where $$\vec{r_{A}},\vec{r_{B}}$$ is calculated from an inertial frame of reference .

please tell me which one is correct ? if both not correct please state the law too in terms of $$\vec{r_{A}},\vec{r_{B}}$$ ?

thank you
 
  • #4
it just says
F12=-F21,where F12=m2 dv2/dt(acting on second one).similarly for F21.Don't count others position vector into first one or so.
 
  • #5
Newton's 3rd law simply says:
[tex]\vec{F}_{ab} = - \vec{F}_{ba}[/tex]
When you drag in the acceleration you are really talking about Newton's 2nd law, which involves the net force. No reason to think that the net force on particle a will be equal and opposite to the net force on particle b if other particles exist.
 
  • #6
ok but since $$\vec{F}_{AB}$$ is a vector what is the frame of reference ?

can $$\vec{F}_{AB}$$ be expressed in terms of any positional vectors or any other variables ?
 
  • #7
ANY inertial reference frame will work.
 
  • #8
i think i got it.

$$\vec{F}_{AB}=\left|\vec{F}_{AB}\right|\hat{F}_{AB}$$
where $$\left|\vec{F}_{AB}\right|$$ is all those forces like gravitational etc .

i did not know how to express $$\vec{F}_{AB}$$ . i thought it was same force as defined in 2nd law.

thanks andrien, Doc Al .
 
  • #9
ato said:
i think i got it.

$$\vec{F}_{AB}=\left|\vec{F}_{AB}\right|\hat{F}_{AB}$$
Well, that would be true for any vector.
where $$\left|\vec{F}_{AB}\right|$$ is all those forces like gravitational etc .
Not sure what you are saying here.

When I wrote Newton's 2nd law as [tex]\vec{F}_{ab} = - \vec{F}_{ba}[/tex]
[itex]\vec{F}_{ab}[/itex] stood for the force on a exerted by b and [itex]\vec{F}_{ba}[/itex] stood for the force on b exerted by a.
 
  • #10
what i mean is that before tackling the third law, it is necessary that the notion $$\vec{F}_{ab}$$ be understood, means how to calculate or express it , which i did not. the point is $$\vec{F}_{ab}$$ is completely different than $$\vec{F}_{a}$$ defined in 2nd law. actually its confusing that both are referred as force.

i think the law is saying that when defining (or at least for all defined) inter-body forces (like gravitational,electric or magnetic), you only need (to define)
$$\vec{F}_{ab-interbody-force}$$
the third law would automatically define,
$$\vec{F}_{ba-interbody-force}$$

Doc Al said:
[tex]\vec{F}_{ab} = - \vec{F}_{ba}[/tex]
[itex]\vec{F}_{ab}[/itex] stood for the force on a exerted by b and [itex]\vec{F}_{ba}[/itex] stood for the force on b exerted by a.

i suppose its for individual interbody force , then that's what i am saying . i am just trying to avoid any personification i can. but if its not for individual forces, then i don't know how to get $$\vec{F}_{ab}$$ if there are more than two forces involved.
my guess would be for two forces gravitational and electric,

$$\left(\vec{F}_{ab-gravitational}+\vec{F}_{ab-electrical}\right)=-\left(\vec{F}_{ba-gravitational}+\vec{F}_{ba-electrical}\right)$$

but then i think Newton's laws would be insufficient to imply this
$$\vec{F}_{ab-gravitational}=-\vec{F}_{ba-gravitational}$$
and
$$\vec{F}_{ab-electrical}=-\vec{F}_{ba-electrical}$$

please tell me if i am wrong .
thanks
 
  • #11
Newton's 3rd law concerns individual forces (interactions between two bodies) not the net force on a body.
 

1. What is Newton's third law in terms of inertial position vectors for n-body system?

Newton's third law states that for every action, there is an equal and opposite reaction. In terms of inertial position vectors for an n-body system, this means that for every force exerted by one body on another, there is an equal and opposite force exerted by the second body on the first.

2. How does Newton's third law apply to an n-body system?

In an n-body system, Newton's third law applies to all bodies within the system. This means that for every force exerted by one body on another, there is an equal and opposite force exerted by the second body on the first. This applies to all bodies in the system, regardless of their size or position.

3. Can Newton's third law be violated in an n-body system?

No, Newton's third law cannot be violated in an n-body system. This law is a fundamental principle of physics and is always observed in nature. Any apparent violation of this law is likely due to an incomplete understanding of the forces at work in the system.

4. How do inertial position vectors play a role in Newton's third law for an n-body system?

Inertial position vectors are used to describe the position and motion of each body in an n-body system. These vectors are essential in determining the forces acting on each body and applying Newton's third law. The direction and magnitude of the forces are determined by the relative positions of the bodies.

5. Can Newton's third law be applied to non-inertial reference frames in an n-body system?

Yes, Newton's third law can be applied to non-inertial reference frames in an n-body system. However, in these cases, apparent forces such as centrifugal and Coriolis forces must also be taken into account. These forces do not violate Newton's third law but must be included in the overall analysis of the system.

Similar threads

Replies
5
Views
865
  • Advanced Physics Homework Help
Replies
11
Views
972
Replies
17
Views
869
Replies
2
Views
1K
Replies
35
Views
3K
  • Introductory Physics Homework Help
Replies
1
Views
650
Replies
17
Views
1K
Replies
0
Views
1K
  • Thermodynamics
Replies
4
Views
1K
Replies
7
Views
5K
Back
Top