Chapter 17: Gravitational Forces and Fields

Chapter 17: Gravitational Forces and Fields 1, 2, 3, 4

 

Equations Introduced and Used for this Topic:

 

[latex]F_{g}=\dfrac{Gm_{1}m_{2}}{d^{2}}[/latex]     [latex]a_{g} or g =\dfrac{Gm}{d^{2}}[/latex]      [latex]\Delta{V} = - G\dfrac{m}{d}[/latex]

 

G  =  6.67408(31) x 10^-11 Nm²/kg²  (round to needed accuracy)

Where 

F_g   is the Gravitational Force of attraction, measured in Newtons (N)

G is Newton’s Gravitational Constant, currently estimated to be 6.67408(31) x 10^-11 Nm²/kg²

m₁ & m₂ are the Masses of the two interacting bodies, measured in Kilograms (kg)

a_g or g   is the Gravitational Field Strength of a body, measured in Metres/Second squared (m/s2) or Newtons per Kilogram (N/kg) ⁵

d   is the Distance away from the Mass Center of a Body (gravitational field) or the Distance between Mass Centers of Two Bodies (gravitational fields), measured in Metres (m)

∆V  is the potential difference measured in J/kg

 

Solar System Data ⁶

BODY	MASS (kg)	SIZE (radium ... m)	ORBIT (radius ... m)
Sun	1.9891 x 1030	6.980 x 108	n/a
Mercury	3.302 x 1023	2.439 x 106	5.791 x 1010
Venus	4.869 x 1024	6.052 x 106	1.082 x 1011
Earth	5.974xx 1024	6.371 x 106	1.496 x 1011
Mars	6.419 x 1023	3.390 x 106	2.279 x 1011
Jupiter	1.899 x 1027	6.991 x 107	7.783x 1011
Saturn7	5.685 x 1026	5.823 x 107	1.429 x 1012
Uranus	8.685 x 1025	2.536 x 107	2.871 x 1012
Neptune	1.028 x 1026	2.462 x 107	4.504 x 1012
Moom	7.349 x 1022	1.737 x 106	*3.844 x 108

 

Notes: The radius of orbits are given around the sun, *except the moon, which is given as its orbit around the earth.

 

Solar Mass (m_s)is the mass of the Sun   1 Solar mass = 1.98855 ± 0.00025 x 10³⁰ kg

 

Astronomical Unit (AU) is the distance from Earth to the Sun 1 AU = 1.49597870700 x 10¹¹ m

 

17.1 Gravitational Forces 8, 9, 10, 11, 12, 13, 14

 

Equations Introduced:

[latex]F_{g} =\dfrac{Gm_{1}m_{2}}{d^{2}}[/latex]

 

Conceptions of Gravity and Astronomical Motion

 

Recorded concepts of gravity in antiquity center around the Greek philosopher Aristotle (384-322 BC). Aristotle’s concept of gravity emerged from his understanding of natural motion, in that all matter will rise or fall to its natural place in the universe. He contrasted this with forced or violent motion, where an object could be made to move until the agent causing it to move stopped.  In Aristotle’s paradigm, gravity was a manifestation of objects moving to simply reach their natural place in the universe.

 

Aryabhata (476-550) held the view that the Earth rotates daily on its axis, but held onto the heliocentric model of the universe rotating around the Earth, as taught by the earlier pre-Ptolomaic Greek astronomers. While a great deal of Aryabhata’s work is lost, remnants indicate that he might have believed the planets move in elliptical orbits, a view that took nearly a thousand years to resurface in the works of Nicolaus Copernicus (1473-1543). There is also evidence that Aryabhata considered gravity to be an attractive force that allowed people to stand upright no matter where they were positioned on the Earth.

 

Galileo (1564-1642) was the first to record counter arguments to Aristotelian views on the movement of bodies to their natural place in the universe, through his experiments with the acceleration of falling bodies. Galileo’s concepts of acceleration were later confirmed by the works of Francisco Maria Grimaldi (1618-1663) and Giovanni Battista Riccioli (1589-1671). It was Riccioli¹⁵ who, through his analysis of free-fall motion in his attempts to find errors in Galileo’s theories, ended up finding the first measure of the acceleration of gravity. He estimated this to be 9.36 ± 0.22 m/s², which is within a 5% error of today’s accepted value of 9.8 m/s².

 

In working to quantify Robert Hooke’s¹⁶ (1635-1703) suggestion that gravitational force was an inverse square law relation¹⁷, Isaac Newton (1643-1727) came up with the proportionality relation:

F_g ∝  m₁m₂

      d²

The first measured value of the constant G needed for Newton’s equation came from the 1797 experiments by Henry Cavendish (1731-1810)¹⁸.  This was accomplished by Cavendish’s measure of the density of the Earth, which he calculated to be 5.48 times that of water.  Cavendish’s work led other natural philosophers to estimate a value for G to be 6.754 × 10⁻¹¹ N-m²/kg², close to the current accepted value of 6.67428 × 10−11 N-m²/kg².  The Cavendish experiments19 are considered to be highlights of accuracy in laboratory science; his work is taught as representing a gold standard in experimental research.

 

Shown below is a sketch of the torsion apparatus that Cavendish used to measure the gravitational constant G.²⁰

The following examples illustrate how to solve Newton’s Gravitational Force Equation.

 

 

 

17.2 Gravitational Field Strength ²²

 

Equations Introduced:

[latex]a_{g}\text{  or  g }=\dfrac{Gm}{d^{2}}[/latex]

 

 

In Newtonian mechanics or Classical Mechanics (non-relativistic) an equation quantifying gravitational strength of any body can be found by equating Newton’s Second Law and Newton’s Law of Gravitational Force.  This can be shown as follows:

 

F_net = ma  or  F_g = mg      is equated to     [latex]F_{g} =\dfrac{Gm_{1}m_{2}}{d^{2}}[/latex]

 

This means that

[latex]mg =\dfrac{Gm_{1}m_{2}}{d^{2}}[/latex]

 

cancelling out the common mass leaves us with

 

[latex]g =\dfrac{Gm_{1}}{d^{2}}[/latex]

 

Applying the conventional negative direction to this derivation we are left with

 

[latex]g =\dfrac{Gm_{1}}{d^{2}}[/latex]

 

 

In most classical mechanics questions this negative value is left out and the strength of gravity is just given as a scalar value.

 

The units of gravitational field strength can be either m/s² or N/kg.   This equation relates to the gravity at any distance away from a point mass and is an inverse square law relationship.

 

g ∝ 1/d²

 

When we look at the gravitational field strength as we move towards the center of the mass, the gravitational field strength drops inversely to zero. Beneath the planet’s surface gravity is an inverse linear relationship.

 

g ∝ 1/d

 

17.3 Gravitational Potential ^{24, 25, 26, 27}

 

Equations Introduced and Used for this Section:

[latex]\Delta{V} = -G\dfrac{M_{1}}{d}[/latex]

 

In classical mechanics, the gravitational potential at any location equals the work per unit of mass needed and moves the object from a fixed reference position to a final position within or outside a gravitational field. Typically, the gravitational potential refers to some final position infinitely far away from the gravitational source. As a result, at an infinite distance away the potential energy of any object in relation to its source position is zero. (E_pf = 0).  Also note that the movement that changes an object’s potential energy must be either toward or away from the gravitational source.

 

If we look at the work needed to move an object away from a gravitational source, we start by using the conservation of mechanical energy, specifically²⁸:

 

W = ∆ Energy  Since the ∆ E is a change in potential energy   W =  E_pf – E_pi

 

This can be simplified for cases where the object is moved to an infinite distance away to

 

W =  – E_pi   since   E_pf = 0

 

We can expand  W =  – E_pi   to    W = – mgh_i   and replace gravity with  g = Gm₁

                                                                                                                                         d²

This yields:  W = – m Gm₁  h_i … And since h_i and  d²  are measuring the same distance, we can

d²

 

cancel common terms.

 

 

This leaves us with          W = – G[latex]\dfrac{m_{1}m_{2}}{d}[/latex]

 

The derivation of gravitational potential is almost complete. What remains is to isolate the second mass m₂.

 

Isolating m₂ from the main equation leaves us with:  W = – G[latex]\dfrac{m_{1}(m_{2})}{d}[/latex]

 

The symbol we will use for gravitational potential is the same as the one we use for electric potential.

gravitational potential (∆V)   where    ∆V= – G[latex]\dfrac{m_{1}}{d}[/latex]

 

In this case, the work done (Joules) to move an object infinitely away from a measurable gravitational field is found by the equation:

 

W = – G[latex]\dfrac{m_{1}(m_{2})}{d}[/latex]or     W = – ∆V (m₂)

Likewise, the gravitational potential energy of an object removed from the influence of a planetary or sun’s gravitational field is also found by:

 

          ∆E_p =- G[latex]\dfrac{m_{1}(m_{2})}{d}[/latex]or     ∆E_p = – ∆V (m₂)

 

How do we understand the negative value that shows up in these equations?

 

• As you move away from a gravitational field, you will need energy to escape. You can consider this energy to be lost.

• As you move into a gravitational field, you will be gaining energy, which is added to your mass.

 

Equation Summary: To move any mass away from a gravitational source, use:

 

  W = – G[latex]\dfrac{m_{1}(m_{2})}{d}[/latex]or     ∆E_p = – G[latex]\dfrac{m_{1}(m_{2})}{d}[/latex]

 

Using    ∆V = – G[latex]\dfrac{m_{1}}{d}[/latex]  the above equations shorten to:

 

                              W  = – ∆V m₂   or   ∆E_p  = – ∆V m₂

 

 

 

 

 

17.4.1 Earth-Moon Gravity Neutral Point ²⁹

Between the Earth and the Moon, there exists a “fluctuating” position where the gravitational fields of the Earth and Moon balance each other out. This position is defined as the Earth-Moon Gravity Neutral Point.  Described by an equation, this position can be found from:

 

g_Earth  =  g_Moon

 

where    g  =[latex]\dfrac{G M}{d_{2}}[/latex]

 

Technically, this position is where the gravitational fields from both the Moon and the Earth balance each other out, meaning that any object in this position having no velocity relative to the Earth and Moon should stay in that position without being pulled to the Earth or the Moon. Should any spaceship reach this position headed to Earth or the Moon, it would then be able to fall to either side of this position without using any more fuel for thrust. This can be compared to a moving vehicle at the top of a hill. If the vehicle is moving in any direction down the hill when at the top, then the vehicle will be able to roll down hill without using any fuel. If the vehicle is stopped at the top of the hill, then it should be able to remain stationary unless some external unbalanced force acts upon it, which would then act to nudge the vehicle downhill in one of the two directions.  This position is termed fluctuating because of a number of factors which are described by NASA’s David Ellis³⁰ as follows:

“…the distance is given as an approximate value. Newton’s work assumes point masses or objects that are perfectly spherical and have a uniform density. As a quick glance around the Earth and up into the sky can tell you, neither the Earth nor the Moon are points or perfectly smooth spheres. Furthermore, the density of both bodies varies with the type of rock that predominates within a region. Also, the orbit of the Moon is not circular but elliptical. As the distance varies, the attractive forces acting upon a spacecraft also vary. Under those conditions the neutral point actually moves constantly around a mean value. Seemingly meaningless things, such as the direction the Himalayas are pointing, actually become important.”

 

The gravity neutral position is visually shown by mapping the variable strength of the Earth’s gravitational field as one heads to the Moon, overlaid by the variable strength of the Moon’s gravitational field as one heads back to the Earth. The intersection of these two gravitational fields’ lines marks the approximate position. Due to fluctuations in the location of this position, any spaceship parked there would need corrections by booster rockets to remain in the region of neutral gravity.

1. Mechanical Universe – Episode 8 – Apple and the Moon: https://www.youtube.com/watch?v=f7MTOb8GUwk

2. Image is from the European Space Agency (ESA) GOCE Geoid, showing the variation of gravity over the Earth’s Surface(25-45% of this variation is due to melted glaciers that used to cover the region and the rest is thought to be caused by downward drag of the Earth’s magma): https://www.youtube.com/watch?v=qu-o75pe5GY

3. A-Level Physics Tutor: http://www.a-levelphysicstutor.com/index-field.php

4. Extra Help – Gravity is More Than a Name: https://www.physicsclassroom.com/class/circles/Lesson-3/Gravity-isMore-Than-a-Name

5. Article to Read – University of Otago (2018) Scientists invent easier, cheaper way to measure gravity: https://phys.org/news/2018-12-scientists-easier-cheaper-gravity.html

6. Article to Read – Carnegie Institution for Science (2018) Outer solar system experts find ‘far out there’ dwarf planet: https://phys.org/news/2018-12-most-distant-solar.html

7. Article to Read – NASA’s Goddard Space Flight Center (2018) NASA research reveals Saturn is losing its rings at ‘worst-case-scenario’ rate: https://phys.org/news/2018-12-nasa-reveals-saturn-worst-case-scenario.html

8. Article to Read – Physicists set limits on size of neutron stars: https://phys.org/print449228257.html

9. Article to Read – Astronomers Detect Giant Rouge Planet Floating Near Our Solar System: https://
www.newsweek.com/astronomers-detect-giant-rogue-planet-floating-near-our-solar-system-1058270?spMailingID=3887126&spUserID=MTI0NzM2NzgwNjYS1&spJobID=1090235283&spReportId=MTA5MDIzNTI4MwS2  LINK DOES NOT WORK

10. Article to Read – Boyle, R. (2018) These dusty young stars are changing the rules of planet-building: https://www.nature.com/articles/d41586-018-07591-8?utm_source=briefingdy&utm_medium=email&utm_campaign=briefing&utm_content=20181205

11. Article to Read – The Guardian (2019) Nearby Galaxy set to collide with Milky Way, say scientists: https://www.theguardian.com/science/2019/jan/04/nearby-galaxy-large-magellanic-cloud-set-to-collide-with-milky-way

12. Article to Read – Where is the universe hiding its missing mass?: https://phys.org/news/2019-02-universemass.html

13. Article to Read – GFZ GeoForschungsZentrum Potsdam (2019) What Earth’s gravity reveals about climate change: https://www.sciencedaily.com/releases/2019/04/190416132155.htm

14. Article to Read – Mass anomaly detected under the moon’s largest crater: https://phys.org/news/2019-06-massanomaly-moon-largest-crater.html

15. Article to Read – Anatomy of a fall: Giovanni Battista Riccioli and the story of g: https://
physicstoday.scitation.org/doi/full/10.1063/PT.3.1716

16. Hooke’s personality – Hooke is described as irascible, proud and prone to take umbrage with intellectual competitors in his later years. Of note is his arguments with Isaac Newton on who should take credit over the discoveries in gravitation.

17. Extra Help – The Apple, the Moon and the Inverse Square Law: https://www.physicsclassroom.com/class/circles/Lesson-3/The-Apple,-the-Moon,-and-the-Inverse-Square-Law

18. Extra Help – Cavendish and the Value of G: https://www.physicsclassroom.com/class/circles/Lesson-3/Cavendish-and-the-Value-of-G

19. Article to Read – Cavendish Experiment: https://sciencedemonstrations.fas.harvard.edu/presentations/cavendishexperiment

20. Blog – Article to Read – Is string theory still in the running for a theory of everything?: https://www.quora.com/Is-string-theory-still-in-the-running-for-a-theory-of-Everything

2.1 One nanometre (nm) = 1 x 10-9 m

22. Extra Help – The Value of g: https://www.physicsclassroom.com/class/circles/Lesson-3/The-Value-of-g

23. The Earth is losing 50,000 tonnes of mass every single year.
The Earth gains 40,000 tonnes of space dust and asteroid.
The Earth loses 16 tonnes due to the fission in the Earth’s core.
The Earth loses 95,000 tonnes of escaped hydrogen.
The Earth loses 1,600 tonnes of escaped helium.
The net loss of Earth Mass is about 0.000000000000001% each year
Source – Earth loses 50,000 Tonnes of Mass Every Year: https://scitechdaily.com/earth-loses-50000-tonnes-of-massevery-year/

24. Gravitational Potential Image Source: https://commons.wikimedia.org/wiki/File:GravityPotential.jpg

25. In mathematics, the Gravitational Potential is also known as the Newtonian Potential and is fundamental in the study of Potential Theory.

26. The concept of Gravitational Potential will be repeated in your study of Electricity as the Electric Potential which is more commonly known as Voltage.

27. Article to Read – Gravitational Waves Reveal the Hearts of Neutron Stars https://www.scientificamerican.com/article/gravitational-waves-reveal-the-hearts-of-neutron-stars1/

28. Gravitational Potential and Gravitational Potential Energy, explained: https://www.youtube.com/watch?v=dngQetgjXBk

29. Infographic poster of a proposed future Moon Base: https://drive.google.com/file/d/1_gG4VE4_0D8C4-ggPIXtgNM2MCrhp-bm/view

30. Source: http://www.madsci.org/posts/archives/1998-08/899933769.As.r.html