# Distinguishing between solid spheres and hollow spheres (equal mass)

If there are two spheres (hollow and solid) with equal mass and radius and we want to find the hollow sphere without using any equipment.

What's the best way(s) to recognize the hollow sphere and solid sphere?

• I wouldn't call this the "best" method but it made me smile: if the hollow one is filled with gas you can cool them down and hear the condensed gad sloshing around Mar 7, 2015 at 19:26
• I bet if you were to knock on them they would sound different. Not sure one can quantify that enough to make it an answer though. Mar 8, 2015 at 1:31
• Place the two balls on a smooth, solid surface (one is allowed a surface, right?) and spin them with your hands (and one is allowed hands?). The solid ball will spin faster, for a given effort, and will also slow down faster. (Also a good way to tell cooked eggs from raw.) Mar 8, 2015 at 2:42
• I am wondering if bouncing them will produce different results. Mar 8, 2015 at 7:02
• Not sure why this is protected, or I may have posted this as a serious answer: hit them both with a hammer (or rock, if a hammer is equipment.) Their behavior will be quite different! Mar 9, 2015 at 12:34

Let both roll down an inclined plane. The hollow sphere will accelerate slower than the solid one (due to their different moments of inertia).

For a solid sphere, the moment of inertia is $$I = \frac{2}{5}mr^2$$ with mass $m$ and radius $r$.

For a hollow sphere it is $$I = \frac{2}{3}mr^2$$ The hollow sphere therefore has a greater moment of inertia and will accelerate slower under the same torque:

$$M = I \frac{d\omega}{dt}$$ with angular velocity $\omega$ and torque $M$.

• Yes - although this does use "equipment" (a ramp), but this is probably the answer that the questioner had in mind. There is some interesting physics in the rolling/slipping question as well! Mar 7, 2015 at 21:50
• This would really be a cool physics demo. Similar to the inertia wand demo: pasco.com/prodCatalog/ME/ME-9847_rotational-inertia-wands . For the balls, the ratio of acceleration is 3/5, so a dramatic difference in the rate. The challenge would be engineering two balls to meet the equal mass and diameter requirements. But I'll guess that is possible. Mar 7, 2015 at 22:47
• A ramp doesn't need to be equipment. Just take the spheres outside and use any hill. Mar 8, 2015 at 19:00
• I understood that part of the question as "no fancy equipment". Reconsidering this, one might be able to fix the position of one sphere, place the other one on top and let it roll off. Repeating this by switching the spheres might also yield different times in which the spheres roll a certain angle. Mar 8, 2015 at 19:05
• @docscience Yeah a hollow ball is tricky in general. Maybe two stainless steel clamshells welded together with the welds machined over for smoothness. Vs a solid aluminum ball. 3D printing a hollow ball would be easy, but now you need another 3D printable ink which is less dense to make the solid one. Mar 9, 2015 at 0:33

If you are not allowed "any equipment" I suppose that eliminates using a ramp and letting them roll down it (@andynitrox's otherwise good answer).

However, assuming your hands are not "equipment", if you took one ball in each hand and tried rotating your hands back and forth as fast as you could, you would find that the frequency you could achieve with the solid ball would be greater - again because of the smaller moment of inertia.

What you did there was to create a torsion pendulum without using torsion wire (namely, using just your hands). Assuming that your hands have roughly the same strength, you can rotate the object with lower inertia faster. If you are in doubt whether your two hands have the same strength, you can swap hands and repeat.

Another interesting variation on @andynitrox's answer is to see whether the balls will start to slide when they go down the ramp. It turns out that the friction force needed to stop a ball on a ramp from sliding is a function of the moment of inertia. You can see a detailed derivation of (most of) this in this earlier answer I wrote. The implication is that the solid ball will roll without slipping on a steeper slope than the corresponding hollow ball - the critical angle is $\tan^{-1}\left(\frac{7\mu}{2}\right)$ for the solid sphere, and $\tan^{-1}\left(\frac{5\mu}{2}\right)$ for the hollow one.

The advantage of the "skid" method is that it would allow you to do the comparison one at a time. If you have the appropriately angled ramp (with angle between the two critical angles) you can tell whether a single ball is hollow or solid without needing to compare speeds. But it would require "equipment" of sorts.

• The problem with the slide versus non-slide method is that it assumes equal coefficients of friction.... Feb 11, 2021 at 3:36

Andynitrox gives a great answer, but it sounds like you are considering a ramp to be equipment. However, because the moment of inertia describes the rotational motion of each object, you actually don't need the spheres to move translationally at all.

If you just take each sphere and place it on a flat surface and try to spin it about the vertical axis (like you would spin a basketball) you can compare the angular velocity, $\omega$ with which each of the spheres rotate. No need to measure quantitatively either; you just need to know that the hollow sphere will rotate more slowly. This is because the hollow sphere has the same amount of mass as the solid sphere, but because the mass is located further, on average,from the axis of rotation, it will not increase its angular velocity as much, for the same amount of torque (your spin).

Even though there will be some uncertainty in the exact amount of torque you provide each sphere, you can run multiple trials. Additionally, the difference in moment of inertia should be enough that you will still notice an unambiguous difference.

• "Try to spin it about the vertical axis". How would you compare the angular velocity? How do you expect to impart the same angular impulse? Are you looking for the time they take to "spin down" when they start at the same rate? It's not clear from your description. Mar 8, 2015 at 1:02
• You'd compare angular velocity qualitatively. As long as there's some kind of marking on rah sphere, you can observe complete rotations. My idea is that for the same amount of torque on each object, you can compare angular velocities immediately after you have finished applying the torques. Because this is all being done without instrumental help, torques on both bodies will only be approximately equal but this uncertainty is addressed in my answer.
– Sean
Mar 8, 2015 at 1:10

I would use my knuckle, tap each sphere, and listen for the tone. The solid sphere would have a higher pitch tone.

• As they're not made from the same material, why would you think so? Jan 3, 2016 at 21:01
• Yes I know they would not be made of the same material. However, since the hallow sphere is more flexible, it makes a lower pitch sound. Apr 16, 2017 at 1:13
• Why do you believe that the hollow sphere would be more flexible if the materials are different? Feb 11, 2021 at 3:38

No equipment is a problem.
Inclined plane experiments: Does placing two legs of the dining table on a couple of books turn it into "equipment"? Does the drainage slope in the shower or bathtub count as equipment?
What about a bowl of water - spin the balls roughly around a vertical axis as you drop them in: accurate speed match and lack of accurate timing equipment doesn't matter too much as you can wait until they have slowed a factor of ten or so (so long as you can see that they are rotating, of course).
The harmonic sequence of a solid ball is different from a hollow one; you can definietely tell the difference even if the fundamental pitches are in a different order to what you expect - provided you've got the right experience. Is your tapping finger or your ear equipment...

USW