Continuity and Physics

**c_323_h** · June 11th 2006, 08:36 AM

How do I know if F is continuous or not?

The gravitational force exerted by the Earth on a unit mass at a distance r from the center of the planet is

$f(x)=\left\{\begin{array}{cc}\frac{GMr}{R^3}, &\mbox{ if }r<R\\\frac{GM}{r^2}, &\mbox { if }r\geq R\end{array}\right$

where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?

BTW, if you can't read it, it is

$\frac{GMr}{R^3}$ and $\frac{GM}{r^2}$

**c_323_h** · June 11th 2006, 08:49 AM

or is everything in nature continuous?

**CaptainBlack** · June 11th 2006, 09:19 AM

Originally Posted by c_323_h

How do I know if F is continuous or not?

The gravitational force exerted by the Earth on a unit mass at a distance r from the center of the planet is

$f(x)=\left\{\begin{array}{cc}\frac{GMr}{R^3}, &\mbox{ if }r<R\\\frac{GM}{r^2}, &\mbox { if }r\geq R\end{array}\right$

You look at what happens to $f(r)$ as $r$ approaches
$R$ form above and below. If both the limits are equal and
finite then $f$ is continuous at $r=R$ .

RonL

**CaptainBlack** · June 11th 2006, 09:20 AM

Originally Posted by c_323_h

or is everything in nature continuous?

In this case the rate of change of force is not continuous.

RonL

**ThePerfectHacker** · June 11th 2006, 10:16 AM

Originally Posted by c_323_h

or is everything in nature continuous?

There is a dictum in physics that everything natural is countinous.

**CaptainBlack** · June 11th 2006, 10:19 AM

Originally Posted by ThePerfectHacker

There is a dictum in physics that everything natural is countinous.

So that proves that my bank balance is unnatural?

RonL

**c_323_h** · June 11th 2006, 11:06 AM

Originally Posted by CaptainBlack

You look at what happens to $f(r)$ as $r$ approaches
$R$ form above and below. If both the limits are equal and
finite then $f$ is continuous at $r=R$ .

RonL

Do I plug in arbitrary numbers to do this? I don't quite understand how I'm supposed to find the limits using letters.

**CaptainBlack** · June 11th 2006, 11:38 AM

Originally Posted by c_323_h

Do I plug in arbitrary numbers to do this? I don't quite understand how I'm supposed to find the limits using letters.

In this particular case it is sufficient to put $r=R$ in both
expressions and find they take the same value.

This is OK because the two forms are both in themselves continuous at
$r=R$ .

RonL

**topsquark** · June 12th 2006, 04:22 AM

Originally Posted by ThePerfectHacker

There is a dictum in physics that everything natural is countinous.

Well, let's not be too hasty about that. Quantum field theory seems to imply that there is a "cut-off" level in the structure of space-time. i.e. It appears that space-time has a non-continuous structure at extremely small scales. (This is, at least, one of the more reasonable ways to remove the ultraviolet divergent integrals.)

For a more basic example, consider an object falling into water. The speed function is certainly continuous, but the acceleration function has a discontinuity at the water-air interface.

-Dan

**mooshazz** · June 16th 2006, 03:30 AM

Originally Posted by topsquark

Well, let's not be too hasty about that. Quantum field theory seems to imply that there is a "cut-off" level in the structure of space-time. i.e. It appears that space-time has a non-continuous structure at extremely small scales. (This is, at least, one of the more reasonable ways to remove the ultraviolet divergent integrals.)

For a more basic example, consider an object falling into water. The speed function is certainly continuous, but the acceleration function has a discontinuity at the water-air interface.

-Dan

the acceleration is contiuens, but her derivation is not
similar to gibbs affect in fourier series

**CaptainBlack** · June 16th 2006, 03:35 AM

Originally Posted by mooshazz

the acceleration is contiuens, but her derivation is not
similar to gibbs affect in fourier series

Gibbs phenomenon - the maximum error of a truncated Fourier series
at a step discontinuity is independent of the number of terms and
proportional to the size of the step?

RonL

**topsquark** · June 16th 2006, 05:01 AM

Originally Posted by mooshazz

the acceleration is contiuens, but her derivation is not
similar to gibbs affect in fourier series

The acceleration cannot be continuous. Before the object hits the water the magnitude of the acceleration is g, but after it is in the water the acceleration is g-B/m, where B is the bouyant force and m is the mass. Thus there is a discrete step in the acceleration function at the interface.

-Dan

Addendum: Apologies!

I was thinking about this again. I had taken the above argument from a problem I had seen worked out several years ago, but it didn't address the following issue: the bouyant force is increasing from zero from the time the object strikes the water until the time it is fully submerged. Thus the acceleration IS in fact continuous as mooshazz stated. It is presumably possible to have an object of such a shape that the bouyant force would be a $C^{\infty}$ function of time, but this would not hold true for a general object. Thus we can state that some derivative of the acceleration function will be discontinous, probably the first derivative for an ordinary object (that is to say an object with the typically random shape found in nature).

-Dan

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