Experimentally measuring a sphere's moment of inertia

[More Boosters]

Hello! 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. :huh:

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

[Jeice]

Do you have to *do* the experiment, or just design it? In theory it's pretty simple but it might rely on some quick stopwatch-fu.

The most basic way is to measure its time taken to roll down a ramp. In general, objects with a higher moment of inertia relative to their mass (eg. hollow tubes) will accelerate more slowly down the ramp than objects with a low one (eg. spheres). This is because it takes more torque to make them spin, and the speed of their spin is directly related to the speed they roll down the ramp.

At a first look, I = (m*r^2*t^2/2d)*(g*sin (p)-2*d/t^2)
where r is the radius of the object,
p is the angle of the ramp,
t is the time taken to roll a distance d along the ramp.

You should be able to check this for yourself with a force balance on a ball on a ramp, taking into account a force from the ramp on the sphere pointing up the ramp which induces the angular acceleration. DISCLAIMER I haven't checked this so thoroughly - the principle is right but no guarantees on the math.
EDIT with better maths.

Other experiments to measure this will generally involve the same principle - applying a force which causes the object to both spin and accelerate in a direction, then measuring its linear acceleration and using it to calculate the angular. Edited November 20, 2015 by Jeice

[More Boosters]

I have to perform it as well so stopwatch-fu here I come.

I used the conservation of energy actually, and used the formula mgh = (1/2)(Iw^2 + mv^2) where h is the highest point of the incline and v is the velocity at the bottom. No idea how I will actually measure velocity there, without going into time and slope angle but yeah. Any ideas?

[Jeice]

On the one hand you can try to estimate its speed against a meter ruler at the bottom or some such.

Otherwise I suggest you use V=2d/t (d is distance along ramp, t is time from release to distance d, V is velocity at bottom). This works since the acceleration of the ball down the slope is constant - you should prove this.

[FleshJeb]

[quote name='Jeice']On the one hand you can try to estimate its speed against a meter ruler at the bottom or some such.

Otherwise I suggest you use V=2d/t (d is distance along ramp, t is time from release to distance d, V is velocity at bottom). This works since the acceleration of the ball down the slope is constant - you should prove this.[/QUOTE]

If he puts the low end of the ramp at the edge of a table, he can measure where it hits on the ground (use flour/sand?) and figure out V0 using trajectory equations. All he needs is the angle of the ramp, and the height of the table.

[Jovus]

I recommend making a torus, putting the sphere inside, connecting it to a driven oscillator, and then measuring the time it takes for the sphere to obtain the frequency of the oscillator. (That is, the sphere goes around the hoop in the same time as the oscillator drives the hoop through one sequence.)

But only because I'm mean. Don't do that. You could, but it would be...hard. To say the least.

[K^2]

Simplest way is to time how long it takes to roll down an incline. If you know how to figure final velocity, average velocity will be exactly half. So it is super easy, but not very precise.

If you want a precise measurement, torsion pendulum is the way to go. Also not super complex, but requires more time to set up and math is harder.

[More Boosters]

[quote name='K^2']Simplest way is to time how long it takes to roll down an incline. If you know how to figure final velocity, average velocity will be exactly half. So it is super easy, but not very precise.

If you want a precise measurement, torsion pendulum is the way to go. Also not super complex, but requires more time to set up and math is harder.[/QUOTE]

I don't really want to detail the thing's movement. I just want its moment of inertia so I can divide it by its mass and radius^2 to get the multiplier in front of it so I can do the same thing and approximate Earth's moment of inertia and then compare it to calculations and real life measurement. I'm not sure if assuming that a sphere must have a moment of inertia in terms of constant * mr^2 is sensible to do but I can't mathematically prove why that should be the case (yet, if I'm on the right track it would help a lot so I can mention it's not completely out of my hat) so here we are.

[Yourself]

[quote name='More Boosters']I don't really want to detail the thing's movement. I just want its moment of inertia so I can divide it by its mass and radius^2 to get the multiplier in front of it so I can do the same thing and approximate Earth's moment of inertia and then compare it to calculations and real life measurement. I'm not sure if assuming that a sphere must have a moment of inertia in terms of constant * mr^2 is sensible to do but I can't mathematically prove why that should be the case (yet, if I'm on the right track it would help a lot so I can mention it's not completely out of my hat) so here we are.[/QUOTE]

Oh, well in that case the number you're looking for is 2/5.

[More Boosters]

[quote name='Yourself']Oh, well in that case the number you're looking for is 2/5.[/QUOTE]

I know that, I'm going to measure that number experimentally by finding the I value for my sphere and then dividing it by mr^2