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

[jbmolineux]

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

 

[PeroK]

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.

 

[rcgldr]

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.

 

[A.T.]

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.

 

[rcgldr]

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.

 

[ZapperZ]

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.