Wikipedia:Reference desk/Archives/Mathematics/2012 December 1

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December 1

Function defining a gravity well?

I've been thinking about this question recently, yet I can't quite figure it out. How would we define the graph of a gravity well in terms of G and the mass of the central object?

The force of gravity on an object on an inclined plane is F = m g sin θ, and at the same time θ = arctan(dy / dx), so if I set the force due to incline to equal the force due to "gravity" (the force we want to simulate), I think I get

$mg\sin \left(\arctan \left({\frac {dy}{dx}}\right)\right)={\frac {GMm}{x^{2}}}$

Where m is my little mass (the planet or marble), M is the big mass (the "sun", I suppose, except it's a point mass), and x is how far the little mass is from the point mass. However, I don't know if I'm right or if that equation even has a clear-cut solution.

I hope my explanation of my question is understandable. Thanks. 66.27.77.225 (talk) 02:56, 1 December 2012 (UTC)

To me it's not obvious what the problem you are discussing is. We the equation for the force on a mass m on an inclined plane at the surface of the earth F = m g sin θ, the relation of that angle to the gradient θ = arctan(dy / dx), but then we equate this to Newtons law of gravitation? It seems what you have done is tried to equate two different models of gravitation, the Newtonian inverse square law ( F = G M m / r²) , and an approximation for constant acceleration due to gravity at the earth's surface (F = m g). Overlooking other mistakes in this process, this is at least uninformative.

To elucidate the dynamics of a marble on a slope (if that is the problem) we need an equation of force F_y = m g sin θ, and to know how an object responds to a force, for which we use Newtons second law F = m a. Then we have a_y = g sin θ with a_y = d²y / dt². This is the dynamical equation of motion for your marble.

An interesting link if you enjoy thinking about such things might be Brachistochrone curve. — Preceding unsigned comment added by 182.55.244.11 (talk) 03:41, 1 December 2012 (UTC)

I believe the OP is thinking about a popular visualization for gravity wells, where the behavior of an object in a gravitational potential is analogous to the behavior of a ball rolling on a surface with the shape of the potential graph.

However, this is only an inaccurate visual analogy - there is no physical truth behind it. The linked article describes some of the flaws.

The "real" gravity well - the gravitational potential energy at every point - is given simply by $U={\frac {-GM}{r}}$. -- Meni Rosenfeld (talk) 20:33, 1 December 2012 (UTC)

There's really no reason to bring trig into your original formula... The purpose of the sin() is to calculate the vertical component of the ramp, which can be done directly from your x and y without an arctan. sin(arctan(y/x)) reduces to y/sqrt(x^2+y^2) assuming positive x and y, which makes perfect sense when you look at it graphically. A physical model of a gravity well will have a slope proportional to the gravitational potential energy at any given point, so what you really want to do is integrate GM/|x|, which gives GM*log(x) for positive x, and the other side will be symetric. Basically, you end up with a logarithmicly curved ramp, rotated around the y-axis. It falls off infinitely in the center, but you can cut it off at the point where you would hit the surface of the body. I just threw this together without double-checking any assumptions or doing much research, so hopefully someone else around here can confirm that my thinking is correct. 209.131.76.183 (talk) 18:59, 7 December 2012 (UTC)

Confidence interval with detected sampling bias

According to this dining-hall survey (p. 16, Fig. 1b), 47% of MIT students (n=236) were virgins as of fall 2001. The rates were 70%, 32%, 28% and 33% for undergrad years 1, 2, 3 and 4-5 respectively; 35% for grads; and 49% for undergrads combined. But according to this report from the MIT Office of the Registrar, the enrolments by year were 1033, 1039, 1037 and 1104 undergrads and 5984 grads. By my calculation, the stratified percentage for undergrads would have been 40.6%, which clearly indicates a sampling bias. Can confidence intervals still be estimated, without knowing the number of students surveyed at each level? NeonMerlin 14:34, 1 December 2012 (UTC)

If you mean confidence intervals for the true percentages, no, not really, not without additional information. Looie496 (talk) 17:12, 1 December 2012 (UTC)

The solution of the equation: x^y=y^x , for x≠y, {x,y}≠{2,4} , in natural numbers.

Any suggestions? HOOTmag (talk) 20:51, 1 December 2012 (UTC)

How about -2 and -4 ? StuRat (talk) 22:16, 1 December 2012 (UTC)

Oops, you did say natural numbers. Are we allowing zero ? StuRat (talk) 22:18, 1 December 2012 (UTC)

Taking natural logs, y ln(x) = x ln(y) so ln(x) / x = ln(y) / y. Plot ln(z) / z and you get it increasing in the range (0, e), a single max at z=e, and decreasing for z > e. So the only possible solutions have x < e, and since x=1 doesn't work, x = 2 is the only solution in natural numbers (with y=4 giving the same value of the function). Duoduoduo (talk) 22:36, 1 December 2012 (UTC)

I ran a simulation, which agrees that there are no other solutions, in the range tested (up to 1600). StuRat (talk) 23:21, 1 December 2012 (UTC)

Yes, you can also prove this for all natural numbers, as Duoduoduo did (Note that one can prove that the function ln(x)/x increases until it reaches the argument e, and then decreases without increasing any more). HOOTmag (talk) 09:43, 5 December 2012 (UTC)