# Experimentally measuring a sphere's moment of inertia

I'm a freshman in Physics and I'd like to design an experiment to measure a sphere's moment of inertia except, well, I'm not very familiar with the concept beyond the formulas it comes with as it was not covered in class yes and in typical student fashion I didn't look into it earlier.

Can anyone give some ideas on how I could go about it?

• Welcome to physics stack exchange. Please show some idea you have regarding how this might work. – AGML Nov 20 '15 at 20:44
• Measure the speed of a ball rolling (not slipping) down a smooth, flat slope (incline) gives you the conversion of potential to kinetic energy and would allow to calculate $I$. – Gert Nov 20 '15 at 20:45
• Well I had the idea of rolling a ball down a slope but wasn't sure how I could physically measure its velocity at the bottom of the track. I can only calculate it with (1/2)Iω^2 = mgh I also intend to extrapolate from this the moment of inertia of earth, and my claim is that since our test sphere is a sphere we know its I value has to be in terms of kmr^2 where k is a constant. Admittedly this isn't from any math knowledge but by just looking at the original I value for a solid sphere. – wololo Nov 20 '15 at 20:52
• Edit: Whoops, completely forgot the linear component! Okay, resorted the formula as (1/2)Iw^2 + (1/2)mv^2 = mgh – wololo Nov 20 '15 at 21:15
• to measure the speed of the ball just film it in video at the maximum available frame rate. Then stop the video and measure the position of the ball at two difference frames when the ball is at the bottom of the track an calculate the speed as the difference in position divided by the time between those frames – user83548 Nov 21 '15 at 1:39

As you've already figured out, roll the sphere down a frictionless hill with zero slipping and use $$\frac{1}{2}I\omega^2 + \frac{1}{2}m v^2 = mgh.$$

To use this equation, you will need to understand the relation between $v$ and $\omega$ and then you will need some way to measure either $v$ or $\omega$. How you measure the velocity will depend on the equipment at your disposal. A smartphone camera and a carefully placed meter stick along with some simple image analysis software should do the trick. If you actually have to perform this experiment, you could try using ImageJ (free, cross platform). There are some (paid) smartphone apps that allow you to measure distances from videos.

• Thanks I will certainly get back to this once it is time to actually do the experiment. And stuff like "it wasn't a frictionless hill with zero slipping" are perfect fodder for my lab report and since I'm a freshman all they really expect is for us to understand what went wrong if something doesn't add up. Edit: I accidentally edited my response into your post (cool for when someone gives an incomplete answer and isn't around to fix it I guess), and it says 6chars when I try to delete it. Only now I realize that enter means send text here. – wololo Nov 20 '15 at 21:51
• Do you have access to logger pro equipment and software? If so, that would make it extremely easy. You could use the sonar sensor or the camera. – dhudsmith Nov 20 '15 at 22:02
• I'm not sure what you are talking about. So probably not.:P – wololo Nov 20 '15 at 22:26

From dhudsmith's answer you know that $\frac{1}{2}I\omega^2+\frac{1}{2}mv^2=mgh$ and if slipping is to be zero you can say, that $v=\omega r$ (rolling condition). To get the velocity you could measure the time $t$ it takes to cover the distance $s$, then you could derive the velocity. $v_{avg}=\frac{s}{t} \Rightarrow v=2v_{avg}=2\frac{s}{t}$

Experimentally measuring a sphere's moment of inertia

Once you build the platform and determine its moment of inertia you put a sphere on the platform and determine the moment of inertia for the system sphere - platform and then by subtraction the moment of the sphere alone.