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Newtonian mechanics

Why do we equate “centripetal acceleration” and “gravitational acceleration”?

[gemini]

I know that in Newtonian mechanics they equate “centripetal acceleration” Acent = 4pi2 R/T2 with “gravitational acceleration” Agrav = GM/R2. Can you explain the justification for this equality?

You’re touching upon a fundamental concept in Newtonian mechanics: how we understand orbits. Let’s break down the justification for equating centripetal acceleration and gravitational acceleration in the context of circular orbits:

Centripetal Acceleration (\(A_{cent}\)):

Definition:

Centripetal acceleration is the acceleration required to keep an object moving in a circular path.

It’s always directed toward the center of the circle.

Formula:

\[A_{cent} = \frac{4\pi^2 R}{T^2}\] where:

  • \(R\) is the radius of the circular path.
  • \(T\) is the period (the time it takes for one complete orbit).
  • This formula can also be expressed as \(A_{cent} = \frac{v^2}{R}\) where \(v\) is the orbital velocity.
Physical Meaning:

This acceleration is not causing the object to speed up or slow down, but rather to constantly change its direction, keeping it on the circular path.

Gravitational Acceleration (\(A_{grav}\)):

Definition:

Gravitational acceleration is the acceleration experienced by an object due to the gravitational force of another object (like a planet or star).

Formula:
\[A_{grav} = \frac{GM}{R^2}\] where:

  • \(G\) is the gravitational constant.
  • \(M\) is the mass of the central object (the object creating the gravitational field).
  • \(R\) is the distance from the center of the central object to the orbiting object.
Physical Meaning:

This acceleration is the force of gravity pulling the orbiting object toward the central object.

Equating the Accelerations:

The Key Idea:

For an object to maintain a stable circular orbit, the centripetal acceleration required to keep it moving in a circle must be provided by the gravitational acceleration.

In other words, gravity is the force that “bends” the object’s path into a circle.

  • Setting the Equations Equal:

    Therefore, we set \[A_{cent} = A_{grav}\]:

    \[\frac{4\pi^2 R}{T^2} = \frac{GM}{R^2}\]

Deriving Kepler’s Third Law:

By rearranging [?] this equation, we can derive [can we derive anything new by rearraging?] a form of Kepler’s Third Law, which relates the period of an orbit to its radius:
\[T^2 = \frac{4\pi^2}{GM}R^3\]

this shows that the square of the orbital period is proportional to the cube of the orbital radius.

Physical Significance of the equality:

This equality demonstrates that the gravitational force is precisely the force that provides the necessary centripetal acceleration for circular orbits.

It allows us to relate the orbital parameters (period, radius) to the mass of the central object.

It is the foundation for calculating the masses of celestial objects.

In summary:

The equality between centripetal and gravitational acceleration is not an arbitrary assignment. It’s a fundamental consequence of Newton’s laws of motion and gravitation. It reflects the fact that gravity is the force that dictates the motion of objects in orbit, providing the necessary “inward pull” to maintain a circular path.

The equivalence principle

I vaguely remember that this equality is somehow related to what is known as the “equivalence principle”. If so can you explain?

  1. You’re right to connect the equality of centripetal and gravitational acceleration with the equivalence principle.
  2. While the direct equation of those accelerations is a Newtonian concept, the equivalence principle, primarily from Einstein’s general relativity, provides a deeper understanding of why this connection is so fundamental.

Here’s how the equivalence principle relates:

The Equivalence Principle:

  1. In its simplest form, the equivalence principle states that the effects of gravity are indistinguishable from the effects of acceleration.
  2. There are a few forms of the equivalence principle, but the core idea is that, locally, you cannot tell the difference between being in a gravitational field and being in an accelerating frame of reference.
  3. This principle is what led Einstein to develop his theory of general relativity, where gravity is understood as the curvature of spacetime.

Connection to Centripetal and Gravitational Acceleration:

Inertial vs. Gravitational Mass:
  1. A key aspect of the equivalence principle is the equality of inertial mass (resistance to acceleration) and gravitational mass (response to gravity). This equality is what allows us to equate the effects of acceleration and gravity.
  2. When we equate centripetal and gravitational acceleration, we’re implicitly relying on this equality.
Frames of Reference:
  1. Consider an astronaut in a spacecraft orbiting Earth. They experience a sensation of weightlessness. However, they are constantly accelerating toward Earth (centripetal acceleration).
    • The equivalence principle tells us that this situation is analogous to being in an accelerating spacecraft in empty space. The astronaut would experience a similar “force” pushing them outward, even though there’s no gravity.
    • Essentially, the centripetal acceleration needed to maintain the orbit, can be thought of as a result of the “curved” nature of spacetime around the earth, and that curvature is what we interpret as gravity.
General Relativity:
  1. General relativity extends this idea by describing gravity as the curvature of spacetime caused by mass and energy.
  2. In this view, objects in orbit are not “forced” to curve by gravity, but rather they are following the “straightest” possible paths (geodesics) through curved spacetime. [explanation of the simple with more complicated]
  3. The centripetal acceleration is then the manifestation of following those geodesics.
In essence:
  1. The equivalence principle provides a deeper understanding of why gravitational acceleration can be treated as equivalent to other forms of acceleration, like centripetal acceleration.
  2. It bridges the gap between Newtonian mechanics and Einstein’s general relativity.
  3. It highlights that gravity is not simply a force, but a manifestation of the curvature of spacetime.

Created: 2025-03-03 Mon 17:15

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