Why do we integrate along the whole length while finding the gravitational force between a object of mass (m) and a rod of length L and mass M? Can't we simply use F = G m M r 2 ??

Question

Why do we integrate along the whole length while finding the gravitational force between a object of mass (m) and a rod of length L and mass M? Can&#039;t we simply use   F  =            G      m      M              r      2          ??

empatteMattmkezo · Accepted Answer

We have to consider the contribution of each little piece of mass       d    M  =  ρ      d    V and how far it is from the other mass, m resulting in a small force       d    F. If you have uniform distribution of mass,   ρ  =  M      /    (  L  A  ), where A is the cross sectional area of the rod, and we probably don&#039;t integrate over that because it&#039;s a small diameter rod. So       d    V  =  A        d    x. Then we have a   1      /        r    2   force behavior where r is the distance from each       d    M      d    F  =            −      G      m              r      2            d    M  =            −      G      m              r      2            M    L        d    xTo find r, you have to give a specific geometry, and let x be the variable of integration that runs along the length and integrate from 0 to   L.   r will be a function of x. The geometry dictates what that function is. Because of the nature of   1      /        r    2  , the force does not change linearly, but the distance from rod to mass m might (again, the geometry is important).In your comment, you mentioned the sphere situation. That means that for a sphere of mass   M and radius       R    0        d    M  =  ρ        d    V  =      M          V              s        p                  R    2    sin  ⁡  θ        d    R        d    θ        d    ϕ  0  &amp;lt;  R  &amp;lt;      R    0  where R is a spherical variable of integration, along with   θ and   ϕ. The distance r will be a function of R,   θ, and   ϕ. Without going into the details, an examination of the form of       d    M shows that the mass is not linearly distributed, so there&#039;s a chance, at least, that a   1      /        r    2   force will reduce to something simple.

Why do we integrate along the whole length while finding the gravitational force between a object of

Answered question

Answer & Explanation

New Questions in Force, Motion and Energy