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You are given a blob of Play-Doh (with a fixed mass and uniform density) that you can shape however you choose. How can you shape it to maximize the gravitational force at a given point

*P*on the surface of the Play-Doh?Solution

In cylindrical coordinates, where the point P is taken to be the origin, the z-component of the gravitational field due to any point (s, z) felt at the origin, is proportional to

\[\frac{1}{s^{2}+z^{2}}\frac{z}{\sqrt{s^{2}+z^{2}}},\]

where the second factor is necessary to take the z-component. If we have azimuthal symmetry, the magnitude of the gravitational field is given by the sum of all the z-components of the forces due to all points in the planet. For a given contour

\[\frac{z}{\left(s^{2}+z^{2}\right)^{3/2}}=C,\]

for some constant C, each point on the interior contributes more to the total gravitational field than each point on the exterior. Thus, the gravitational field is maximized if the surface of the planet takes the shape of one of these contours. Solving for s(z), we get\[s\left(z\right)=\sqrt{\left(\frac{z}{C}\right)^{2/3}-z^{2}}.\] If the length of the planet in the z-direction is \[z_0,\] we can replace C in the above expression to get

\[s\left(z\right)=\sqrt{\left(z_{0}^{4}z^{2}\right)^{1/3}-z^{2}}.\]

As can be seen by the plot above, this shape looks a lot like a sphere, but slightly smushed toward the point of interest P. We can rigorously compare the field of the maximal gravity solid to that of a sphere with the same volume. The volume of the maximal solid is given by

\[V=\pi\int_{0}^{z_{0}}s^{2}\left(z\right)dz=\pi\int_{0}^{z_{0}}\left[\left(z_{0}^{4}z^{2}\right)^{1/3}-z^{2}\right]dz=\frac{4\pi}{15}z_{0}^{3}\]

The volume of a sphere of radius r is of course \[V=\frac{4}{3}\pi r^{3},\] so in order for the volumes to be the same, the sphere must have a radius of \[r=z_{0}/5^{1/3}.\]

The acceleration due to the maximal solid is proportional to

\[a_{max}=\int dVa_{z}=2\pi G\rho\int_{0}^{z_{0}}dz\int_{0}^{s\left(z\right)}sds\frac{z}{\left(s^{2}+z^{2}\right)^{3/2}}=\frac{4\pi G\rho z_{0}}{5},\]

while the acceleration due to the sphere is just \[a_{sphere}=\frac{G\rho\frac{4}{3}\pi r^{3}}{r^{2}}=\frac{4\pi G\rho z_{0}}{3\cdot5^{1/3}}.\]

Thus, we have

\[\frac{F_{max}}{F_{sphere}}=\frac{4\pi G\rho z_{0}/5}{4\pi G\rho z_{0}/\left(3\cdot5^{1/3}\right)}=\frac{3}{5^{2/3}}\approx1.026.\]

So the solid that gives the maximum gravitational field at a point is only about 3% better than a sphere.

For a more detailed discussion, see Alemi's solution.

EXTERIOR surface?

ReplyDeleteThe problem is only interested in maximizing the gravitational force at a single point on the surface of the blob. Other points on the surface we aren't concerned with. Does that answer your question?

ReplyDeleteInteresting problem! A quick thought experiment and use of WolframAlpha's equation solver leads me to the following:

ReplyDeleteIf we define the z-axis as the normal to the surface at the point of contact (where 0 is the point of contact, and positive numbers are into the mass):

A cylindrically-symmetric solid of radius

r = Sqrt [ (z/c)^(2/3) - z^2 ]

Where c is some constant having to do with how big the blob is.

Looking at a plot of this sucker, it's not too different from a sphere, but the "bulge" gets shifted a little bit closer to the point of maximum gravity and the far side becomes a point, rather than smooth. The bulge getting shifted closer to the point makes qualitative sense to me.

Hey! It's "next" Friday! Did I get the answer right? The suspense is killing me!

ReplyDelete