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Gravitational Potential: Real-Life Results Abstract the Equilibrium Surface of a Large Body of Water Is a Gravitational Printed in the UK Equipotential

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Gravitational Potential: Real-Life Results Abstract the Equilibrium Surface of a Large Body of Water Is a Gravitational Printed in the UK Equipotential IOP Physics Education Phys. Educ. 51 L ETTERS Phys. Educ. 51 (2016) 016501 (3pp) iopscience.org/ped 2016 Gravitational potential: real-life © 2016 IOP Publishing Ltd results PHEDA7 Carl E Mungan 016501 Physics Department, US Naval Academy, Annapolis, MD 21402-1363, USA C E Mungan E-mail: [email protected] Gravitational potential: real-life results Abstract The equilibrium surface of a large body of water is a gravitational Printed in the UK equipotential. However, that does not imply that in a local region of increased gravitational field g the height h of the water surface will be lower to keep the potential gh constant, in contrast to a recent claim to that effect. PED In the July 2015 issue of Physics Education, Bill is the density of water, taken to be incompress- 10.1088/0031-9120/51/1/016501 Baird proposes a thought experiment in which ible) has to have the same value everywhere at the the gravitational field strength g increases from floor of the classroom. But the floor can exert dif- 5 N kg−1 at the left wall of a classroom to 15 N fering local normal forces at different locations to 0031-9120 kg−1 at the right wall [1]. He then suggests that sustain variations in the water pressure along the if the room were partly filled with water, the floor. Note too that water can be thought of as a Published liquid surface would adopt an equilibrium shape set of molecular bowling balls. Thus the center of in which the water height h above the floor of the mass of the water will displace in the same direc- room is largest at the left wall and smallest at the tion as a bowling ball would, in contrast to what January right wall. His reasoning is that the surface of the Baird claims. water must be an equipotential, so that gh must To expand on the argument in [2], consider 1 be constant on the surface and thus locations of a spherical nonrotating planet of mass M and larger g correspond to smaller h. radius R. The density of the planet can vary radi- However, that would not happen in real life. ally, but assume it does not vary in the angular If the earth were covered with water and one directions, so that one can apply the standard ignored time-varying effects (such as winds and shell theorems to calculate the gravitational tides), the ocean would mound up above the points field. Now embed a point mass M just under the of largest values of g known as geoid anomalies. north pole, say at a depth of 0.1R, which means The source of such increased values of g is higher it is 0.9R from the center of the planet, as illus- density material in the crust of the earth. (This trated in figure 1. idea is used in gravimetric prospecting on land It can be shown that a point in space above to locate petroleum or ore deposits by measuring the north pole needs to be at higher altitude than local changes in the value of g.) A blob of higher a point in space radially above the south pole if density material in the crust would gravitationally those two points are to be at the same gravita- attract extra water to its vicinity, creating a ‘hill’ tional potential V. The math works out easily if of water above it [2] even if the surface of the we choose the southern point to be 0.1R radially earth remains flat. Implicitly Baird is assuming above the pole, so that it is distance r1 = 1.1R away that the hydrostatic gauge pressure ρgh (where ρ from the center of the planet. The gravitational 0031-9120/16/016501+3$33.00 1 © 2016 IOP Publishing Ltd C E Mungan potential at this southern point is then the sum of that due to the planet and the added mass, GM GM GM GM V1 =− − =− − . (1) r11rR+ 0.91.1R 2R Next we want to find the point above the north pole at distance r2 from the center of the planet that has this same potential. We again do a sum, but this time the added mass is on the same side of the planet’s center as the field point of interest, so that the plus sign in the second denominator becomes a minus sign, GM GM V2 =− − . (2) r22rR− 0.9 If we set the right-hand sides of equations (1) and (2) equal to each other and solve for r2, we get a quadratic equation. Only one root is larger than R, which is the one we want. By direct inspection of the two equations, we see that the answer is r2 = 2R. Therefore, if we were to cover the sur- face of the planet with water such that the water Figure 1. Two points (indicated by asterisks) above the south pole were 0.1R deep, then the radially above the north and south poles at which the water above the north pole would be R deep. The gravitational potentials V1 and V2 are equal, due to the net gravitational field produced by an off-center point water is higher where the total gravitational field object and the extended planet, both of mass M. is stronger. The key point is that it is not possible for a ends to two vertical pipes. Now place a point static gravitational field to point everywhere ver- mass directly underneath one end of the U-tube tically while having a strength that varies hori- and fill it with water. The resulting gravitational field will cause the water to move as close to the zontally. Say it has the form −kx ^j where k is a constant. Such a field would not be irrotational point mass as it can. Hence there will be more (curl-free). The line integral of g around the black water in the vertical tube under which the point loop indicating the ceiling, right and left walls, mass is located (where the gravitational field is and floor in figure 1 of [1] would be nonzero, so stronger) than in the other, more distant vertical that the field could not be the gradient of a poten- tube. The equipotentials are concentric spheres tial. In reality, the field must also have a horizontal centered on the point mass. In like fashion, the component, and consequently there is a horizon- water surface in Baird’s room must curve upward tal component of the gravitational force on any to the right and be concave downward, unlike cubic element of fluid which balances the pres- what is drawn in figure 1 of [1]. sure difference on the left and right faces of the element, preventing that bit of water from being Acknowledgments ‘squeezed’ into lower g regions. A similar expla- I thank Joel Helton and Bill Baird for helpful nation has recently been presented to explain the discussions. tidal bulge for a water-covered planet [3]. Received 13 July 2015, in final form 1 October 2015 Imagine an isolated U-tube in outer space, Accepted for publication 8 October 2015 consisting of a horizontal pipe connected at its doi:10.1088/0031-9120/51/1/016501 January 2016 2 Physics Education Gravitational potential: real-life results References Carl Mungan is an associate professor of physics with research interests in [1] Baird W H 2015 Gravitational potential: a thought laser optics and solidstate spectroscopy. experiment Phys. Educ. 50 397–8 [2] http://principles.ou.edu/earth_figure_gravity/ geoid/ [3] Ng C-K 2015 How tidal forces cause ocean tides in the equilibrium theory Phys. Educ. 50 159–64 January 2016 3 Physics Education IOP Physics Education Phys. Educ. 50 F RONTLINE iopscience.org/ped 2015 Gravitational potential: a thought © 2015 IOP Publishing Ltd experiment PED William H Baird 397 Department of Chemistry & Physics, Armstrong State University, Savannah, GA 31419, USA W H Baird E-mail: [email protected] Gravitational potential: a thought experiment Abstract The electrostatic potential is a key element of the second semester of Printed in the UK introductory physics. Teaching about its gravitational analog in the first semester allows students to make more connections between the two courses. The use of a simple thought experiment with an unexpected outcome provides PED a method for introducing gravitational potential in a way students may remember. 10.1088/0031-9120/50/4/397 Introduction of voltage, known as the gravitational potential, 0031-9120 When teaching the second semester of introduc- is a commonly discussed topic in physics courses tory physics, the parallels between electrostatics in the UK. Examination of popular textbooks and gravitation are too numerous to ignore [1]. used in the USA, however, yields a different pic- Published Since gravity is something students have always ture. Reviewing both calculus-based [2–4] and algebra-based [5 7] texts shows full chapters on had both a figurative and literal‘ feel’ for, connect- – electrostatic potential (except for reference [7], July ing that knowledge to the new world of electric charges (where physical intuition is in compara- where electrostatic potential shares a chapter with tively short supply) can be expected to ease their the electric field), but no mention of gravitational 4 potential, even in the indices. discomfort. Coulomb’s law and Newton’s law of universal gravitation are obvious partners, and the The decision to omit the gravitational poten- respective fields produced by point charges and tial may be forgiven in light of the material to be point masses are similarly related.
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