Gravity (truly) Explained

(The following assumes familiarity with high school calculus and physics.) 

For over two hundred years before the advent of propulsion systems and heavier-than-air flight, “escape velocity” only pertained to projectiles trying to flee Earth’s gravity. While the physics hasn’t changed, the implied meaning of “escape” unfortunately has. Although spaceships do exceed escape velocity for reasons I’ll explain, this has cultivated the misimpression that a hypothetical vessel must. 

Newton’s quantification of gravity wasn’t just an achievement in itself; it also showed that gravity is merely a force, one not altogether different than the force that projected the cannonballs in his thought experiments. Since gravity is just a force, any propulsion system with a sufficient stored energy to mass ratio could escape going an inch a second. Relative to what’s hypothetically possible, however, modern propulsion ships are primitive and easily weighed down--requiring high accelerations where fuel efficiency is optimal. Moreover, vessels are often required to exceed escape velocity to orbit the desired altitudes (note that astronauts in orbit are in freefall, not at zero G).

 

This post will elaborate on the above and provide a context—and excuse—to discuss issues not covered in standard texts. It will also present a non-vectorial derivation of centripetal force, necessary for understanding orbits.

 

 

Having yet to officially publish his work on the calculus, Newton used classical geometry to prove the inverse square law in the Principia. Calculus would later be used, as would other techniques. For the beginner, however, an intuitive argument should suffice. 

 

Gravity is a force that extends infinity and equally in all directions, as if to fill an arbitrarily large sphere, and it gets weaker with distance. In the case of a sphere with an approximately radially symmetric mass distribution (which most celestial bodies have), one can calculate the gravity a body exerts at a point from its center as if all its mass is at its center of mass, which is at its center (however, gravity is zero near the center itself—and apologies for all the uses of the word center).

 

Imagine a small sphere within a larger one; their areas, like all spheres, are 4πR^2. Imagine a nearly infinite number of “lines of force,” each with the same magnitude of force, extending from the core of the inner sphere to its surface, such that the sphere now appears fully colored. Then imagine those same lines extending into the larger sphere onto its surface. Although you couldn’t see it, those same number of force lines are dispersed over an area R2^2/R1^2 larger, making field line density the inverse of that ratio smaller. The lower field line density means a proportionally smaller force on the surface area, since less field lines strike the same point on the surface. Thus, force at a point is inversely proportional to the area of a sphere’s surface.       

 

But gravity also gets stronger with mass. In the case above, assuming the spheres had equal mass densities, they would have equal intrinsic gravity at their surfaces, since the larger sphere would have proportionally more mass. Now imagine two isolated spheres some distance from each other, such as a planet and a sun. Firstly, the bodies exert forces upon each other that draw them together. Secondly, Newton’s third law says the force the two bodies exert on each other must be equal. Finally, according to his second law, acceleration must be proportional to mass; the same force must impart the smaller body with a greater acceleration in accordance with the ratio of the masses—e.g., a body half the mass must accelerate twice as much.  

 

Therefore, the law of gravity is expressed as the product of the masses and a constant (which can equal one if it proves unnecessary) divided by 4πR^2, or CM1M2/4πR^2. Since a constant divided by 4π is also a constant, C/4π becomes G, the universal gravitational constant. 

 

Unlike the similar static electric force, gravity can’t be shielded. If it could, any radially symmetric body would have zero gravity within. 

 

Calculating escape velocity is very easy. Gravity does work upon an object over a distance or range of position, R; that kinetic energy must equal the integral of GM1M2dr/Rsurface of Earth^2 to infinity, which approaches the former as the position approaches infinity. In other words, the object must have enough kinetic energy to outwork Earth’s gravity well. This distance can be found by using the gravitational-KE equation to find R when the escape velocity is, say, an inch per second. In the case of the Earth, this point is roughly equal to the distance of the moon, a quarter million miles (reminder: astronauts in orbit are in freefall). Alternatively, you could set the gravitational force to some arbitrarily low value and solve for R.  

 

However, as mentioned, since gravity is just a force, a high energy-to-mass propulsion system could escape going an inch a second. Because the air becomes thin in the upper atmosphere, this eliminates all air-aided animal flight and craft, including jets, which require air for combustion. With modern technology, this leaves space programs to rocket ships that carry liquid oxygen, decreasing effective fuel efficiency; and heat engines don’t tend to be very efficient, anyhow. Now a longer trip will tend to require less fuel per unit time, but the trip also TAKES more time, burning fuel for LONGER. It turns out that fuel efficiency peaks at high acceleration, necessitating fast takeoffs. The force of gravity isn’t terribly high; exceeding it, especially for a stored-energy efficient vessel, isn’t terribly difficult. The situation would be no different if the force came from a controlled electromagnet.         

 

Understanding Centripetal Force and Orbits

 

Suppose something travels in a circle. 

 

Force is required for change in direction. Sharper turn means bigger force, which means bigger acceleration. Change in velocity is just the velocity in the new direction, and velocity divided by time is acceleration (don’t confuse simply CHANGE IN VELOCITY with change in V with respect to TIME, acceleration). Thus, a sharper turn means bigger velocity. Circular, including elliptical, motion is no exception, and there must be an appropriate force acting toward the center. Increasing velocity and/or decreasing radius requires more force. If you’re running in a circle and want to increase speed without changing your path, you must bend your knees and tilt your body to push off the ground harder to increase force and move faster. In the case of a space vessel, the ship would have to tilt their thrusters and increase thrust to maintain path. But if you move in a circle in the case where gravity is acting upon you, don’t you fall toward the center? You do: You’re falling toward the center along the path of the curvature. 

 

A simple, Non-vectorial Derivation of Centripetal force

 

The object is changing direction and must exert a force toward the center. Finding acceleration is just a matter of multiplying V, which is a constant, by the angle of the directional change when dt approaches zero and diving by dt. When dt approaches zero, change in position, Vdt, approaches a perfectly straight line; and the angle approaches the change in position divided by radius, R (this can be imagined here if you scroll down to the circle in Step 2). Change in velocity is V(Vdt/R), equaling V^2dt/R. Diving by dt to get acceleration, you get V^2dt/R. Thus, centripetal force is MV^2/R.

 

Although centripetal force and acceleration aren’t constant in the case of the ellipse and calculation is a bit more difficult, the same general concepts apply. In the case of an orbit, the centripetal force varies in accordance with the distance from the body being orbited. Of course, depending on its mass and initial velocity, an object passing a large celestial body is unlikely to stay in a fixed, stable orbit—or even orbit at all. If it draws close enough to the body and it gets into an orbit, decaying orbits are more likely--where an object slowly circles the body, often multiple times, drawing ever closer until it crashes into it. The formation of a solar system is the result of huge numbers of massive objects colliding and sometimes sticking together until a system of bodies orbits a sun(s). There may have been thousands of passing planetoidal bodies in giant collections of space debris that were candidates for planets and moons that failed to settle into stable orbits. In other cases, stable solar systems might fail to form all together.    

 

 

Fast rocket liftoffs obfuscate that gravity is just force; explaining the actual need for fast liftoffs remedies that confusion. Newtonian mechanics contrasts with other branches of science and physics like biology, geology, electromagnetism, and quantum mechanics in that there really aren’t many new concepts: It’s the “physics of intuition” and little more than an extension of kinematics. The mindset one brings into an activity affects how he/she perceive it and, therefore, how well they learn it. It should be taught in a manner that cultivates this outlook.