Vector addition and Newton's law

In summary, the n-body problem can be complicated for dynamics, but for a static case, the net force on the reference body with mass m can be obtained by adding all the gravitational forces vectorially. This requires the position of all other bodies with respect to the reference body. The system is static if the net force at each body is zero. However, working out the paths of the bodies can become complicated due to the non-linear equations involved. The three-body problem is a prime example of this complexity.
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xaratustra
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I know that n-body problem can be complicated, but that's for the dynamics. What about a static case:

e.g. if I have the distances of several bodies A, B and C etc. and their distance to a reference mass m, can I just use the vector addition of the Newton's gravitational force to add up all of the forces from all those bodies to the reference mass and obtain the resulting vector? or is it more complicated than that?
 
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If you want the net force on the reference body with mass m, just add all the gravitational forces vectorially. Yes, it is that simple. You need the position of all other bodies with respect to the reference body, as the forces of gravity depend on the difference vectors ##\vec r_i-\vec r _0##.
The system is static if the net force at each body is zero (not only at the reference body).
 
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  • #3
xaratustra said:
? or is it more complicated than that?
It doesn't need to be harder to comprehend in principle but you end up with what's effectively a set of simultaneous non linear equations and it gets very complicated if you want to work out the paths of those bodies. Working out the resultant force on each body is only the start. Look at the wiki article on the three body problem (and that's only three.)
 
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What is vector addition?

Vector addition is a mathematical operation in which two or more vectors are combined to form a resulting vector. This operation takes into account both the magnitude and direction of each vector.

How is vector addition related to Newton's law?

In Newton's law of motion, the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This net force can be represented as a vector, and vector addition is used to combine all the individual forces acting on the object to determine the net force.

What is the difference between scalar and vector quantities?

Scalar quantities only have magnitude, while vector quantities have both magnitude and direction. For example, mass is a scalar quantity, while velocity is a vector quantity.

Can vectors be added in any order?

Yes, vector addition is commutative, meaning that the order in which the vectors are added does not affect the result. This is because vectors are independent of each other and can be rearranged without changing their overall effect.

How is the direction of the resulting vector determined in vector addition?

The direction of the resulting vector is determined by the geometric properties of the individual vectors. The resulting vector will be in the same direction as the diagonal of a parallelogram formed by the individual vectors.

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