notes

gravitational force

Sparisoma Viridi
3 mins read ·

Force between two masses, also between mass and field, considered in butiran.

When we talk about force between two masses at distance some terms come up, such gravity, gravitational force, and gravitation. Gravity is a real force but not in the traditional sense, because at fundamental level there are not any action-at-distance forces, since it just a mass creating gravitational field and then the other mass moving and interacting with this field 1. It is better to use the term gravitational force instead of gravity about the force between to masses that are not in contact 2. Newton’s law of universal gravitation usually present the force in scalar form 3, which is not easy to use for interaction more than to masses. The vector form 4 will be used here since it is more general and easier to use in calculating force superposition. There is also gravitional force on a mass in a gravitational field as weight of a mass is calculated near earth surface 5.

Mass mim_i will experience gravitational force

FG,i=mig(1)\tag{1} \vec{F} _{G,i} = m_i \vec{g}

in gravitational field g\vec{g}, where source of gravitational field is atually another mass or mass distribution.

Mass mim_i will experience gravitational force

FG,ij=Gmimjrij2r^ij(2)\tag{2} \vec{F} _{G,ij} = -G \frac{m_i m_j}{r _{ij}^2} \hat{r} _{ij}

due to existence of other mass mjm_j, where position of mass mim_i is ri\vec{r}_i and position of mass mjm_j is rj\vec{r}_j.

Relative position of mass mim_i from mass mjm_j is

rij=rirj,(3)\tag{3} \vec{r} _{ij} = \vec{r}_i - \vec{r}_j,

distance between two masses is

rij=rij=rijrij,(4)\tag{4} r_{ij} = |\vec{r} _{ij}| = \sqrt{\vec{r} _{ij} \cdot \vec{r} _{ij}},

and

r^ij=rijrij(5)\tag{5} \hat{r} _{ij} = \frac{\vec{r} _{ij}}{r _{ij}}

is the unit vector.

From Eqns (1) and (2) it can be obtained

gj(r)=Gmjrrj3(rrj)(6)\tag{6} \vec{g}_j(\vec{r}) = -G \frac{m_j}{|\vec{r} - \vec{r}_j|^3} (\vec{r} - \vec{r}_j)

as gravitational field produced by mass mjm_j experienced by another mass in the surrounding of mass mjm_j.

Challenge 1. Show the steps to obtained Eqn (6) from Eqns (1) and (2). If necessary use also Eqns (3) and (5).


  1. Christopher S. Baird, “Why is gravity not a real force?”, Science Questions with Surprising Answers, 5 Aug 2022, url https://www.wtamu.edu/~cbaird/sq/2022/08/05/why-is-gravity-not-a-real-force/ [20241013]. ↩︎

  2. GPT-4o, “Gravity vs Gravitational Force”, ChatGPT, 13 Oct 2024, url https://chatgpt.com/share/670b17e8-43d4-800a-a374-13fe63072334 [20241013]. ↩︎

  3. Tom Henderson, Trevor Fayas, “Newton’s Law of Universal Gravitation”, The Physics Class Room, Jul 2022, url https://www.physicsclassroom.com/class/circles/lesson-3/newton-s-law-of-universal-gravitation [20241013]. ↩︎

  4. Simon Bridge, “Why is there a minus sign in front of the vector form of Newton’s law of universal gravitation?”, Quora, 27 Sep 2017, url https://qr.ae/p2UWj1 [20241013]. ↩︎

  5. -, “Weight Equation”, Gelnn Research Center, NASA, 22 Jul 2022, url https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/weight-gravitational-force/ [20241013]. ↩︎

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