Is Hooke's law related to the time-squared rate of acceleration?

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Discussion Overview

The discussion revolves around the potential connections between Hooke's law, the inverse square law of gravity, and the time-squared rate of acceleration of falling bodies. It explores theoretical implications and relationships among these concepts, focusing on gravitational acceleration and oscillatory motion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question whether there is a connection between the inverse square law of gravity and the time-squared rate of acceleration of falling bodies.
  • One participant argues that acceleration is defined as distance over time squared, suggesting that the relationship between force and distance does not directly affect acceleration.
  • Another participant explains that for gravitational acceleration, the inverse square law implies that acceleration is a function of distance, and discusses the use of the chain rule to relate time-based acceleration to position-based acceleration.
  • It is noted that the rate of acceleration of falling bodies corresponds to the local strength of the gravitational field, which is influenced by the masses involved.
  • A participant challenges the universality of the connection by referencing Hooke's law, stating that it describes a different relationship where force is proportional to displacement, yet still results in acceleration that can be expressed in the same dimensional form.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between Hooke's law and gravitational acceleration, with some asserting a lack of connection while others explore the implications of each law. The discussion remains unresolved regarding the nature of these relationships.

Contextual Notes

Participants highlight the complexity of integrating equations related to gravitational acceleration and the challenges in converting between different forms of acceleration. There are also discussions about the definitions and implications of force and distance in different contexts.

jbmolineux
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Is there a connection between the inverse square law of gravity and the time-squared rate that bodies fall (i.e. (32ft/second)/second))?
 
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jbmolineux said:
Is there a connection between the inverse square law of gravity and the time-squared rate that bodies fall (i.e. (32ft/second)/second))?

Not reallly. It doesn't matter how a force is related to distance between the objects, acceleration (by definition) is (distance/second)/second.
 
For gravity, inverse square law meas that acceleration is a function of distance, so typically chain rule is used to convert time based acceleration into position based acceleration. Define the two masses as m1 and m2 :

a = dv/dt = v dv/dr = -G (m1 + m2) / r2

For an initial distance r0 and final distance r1, the above equation is integrated twice. The first integration isn't too bad, but the second one is complicated, and you end up with time as a function of initial and final distance, which can't be converted into distance as a function of time.
 
jbmolineux said:
Is there a connection between the inverse square law of gravity and the time-squared rate that bodies fall (i.e. (32ft/second)/second))?
The rate at which falling bodies accelerate is the local strength of the gravitational field.
 
A.T. said:
The rate at which falling bodies accelerate is the local strength of the gravitational field.
For a two body system, this would be the rate of acceleration towards a common center of mass for the two body system (use the common center of mass as the source for a reference frame). Each mass accelerates towards the common center of mass based on the gravitational field of the "other" mass.
 
jbmolineux said:
Is there a connection between the inverse square law of gravity and the time-squared rate that bodies fall (i.e. (32ft/second)/second))?

I can falsify your notion of the universality of such a connection. Consider Hooke's law, where the force is proportional to the displacement from equilibrium, i.e. not an inverse square law. Yet, the acceleration of the oscillating mass is still L/T2.

One has nothing to do with the other. One is the relationship between force and distance from the source of that force. The other is the dimension of a quantity.

Zz.
 

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