Determine the weight of a titanium sphere

[Ari Balt]

I need help determining the weight, in pounds, of a volleyball-sized sphere comprised of pure titanium.
I can provide the following facts necessary for the equation:

Assume the circumference of the sphere is 65 centimeters.
Assume the sphere is solid, not hollow.
Titanium has an atomic mass of 47.876 and a density of 4.506 grams.

Any help on this would be appreciated.

NOTE: This is not for homework. If you must know, it is necessary to determine if someone is capable of creating the sphere in question in a fantasy setting role-playing game.

 

[Bandersnatch]

Circumference of a sphere is given by:

d_c=2\pi R

find R and substitute to the equation for the volume of a sphere

V=\frac{4}{3}\pi R^3

and multiply by the density to get mass.

Lastly, covert from pounds to kilograms. 1pound =~0.45 kilograms.

 

[Ari Balt]

Using those formulas, my answer came out to be 46.070 pounds, rounded to the nearest thousandths place assuming I found the correct answer.

For the sake of pretense, let's say a supervillain can expel this 46-lb. titanium sphere at Mach 3, or 1 kilometer per second. He directs this expulsion at a concrete wall from a distance of 300 feet, standing directly perpendicular to the wall. Upon impact, does the sphere penetrate the wall, bounce off, or disintegrate?

Let's assume the concrete wall is 12 inches thick with a hollow area behind it. (It's a bunker, if that matters.)

 

[Bandersnatch]

Now that's a question that can't be answered with simple calculations.

Experiments in materials science, like these:
http://www.sciencedirect.com/science/article/pii/S0734743X02001082
http://www.sciencedirect.com/science/article/pii/S0734743X98000086
(paywalled, but you can read the graphs if you strain your eyes a little)

using 20 and 30 mm diametre steel rods, show penetration of concrete slabs up to 2 metre thick, at speeds close to 1km/s.
Larger projectiles show better penetration.

The projectiles are shown post-test without any noticeable deformation.

The shape of a sphere is less suited for penetration, as it's got a higher cross-section(impact area) to weight ratio than a long cone-tipped rod. Additionally, steel is better suited for the purposes, due to its higher density.
(Titanium is always talked about as being superior to steel in popular media, but that's just considering the aplications that require weight-saving while retaining similar mechanical properties, like with submarines. Your super villain would be more destructive if he could make steel spheres. Or better yet, tungsten of uranium - the denser the better.)

The two articles linked were one of the first that popped up when I googled concrete penetration with projectiles, so it's conceivable that you should be able to find something more definitive with more effort.

You might get somebody in the engineering/materials science section to help you find what you need.Also, consider that an 80kg man shooting a 20kg projectile at 1km/s will be thrown back at 250 m/s due to the conservation of momentum(aka recoil). This might be a very short-lived career of villainy for this guy.

 

[mikeph]

This is a total guess, but at Mach 3 I'd expect it to punch a hole straight through 12 inches of concrete.