# Determination of moment of inertia of a hollow cylinder

Alettix
Hi!
I got the task to determine the moment of inertia of a hollow cylinder, however it's not about just measuring the mass and the inner and outer radius and putting it into the right formula, instead I should roll it down an inclined plane.

1. Homework Statement

I'm only allowed to use the following tools:
• An inclined plane
• Ruler and measuring-tape
• Vernier caliper
• Stopwatch
• A hollow cynlinder
The problem states that the friction can be neglected and that the kinetic energy of the cynlinder will be the sum of it's translational and rotationan energy.

## Homework Equations

Translational energy: Et = 1/2 * m v2
Rotational energy: Er = 1/2 * I ω2
Potential energy Ep = mhg

## The Attempt at a Solution

My idea was to measure a short distance infront of the inclined plane, roll the cylinder down and use the stopwatch to determine the cylinders approximate velocity when leaving the plane. The angular velocity could then be calculated with the help of a measurment of the cylinders circumfence. Then, using the law of conservation of energy we get:
I = 2m (hg - v2)/ ω2
The problem is that this expression (just as the others I tried to derive) contains the mass of the cylinder...

Do you have any suggestion on how the measurment and the calculations should be done?

Thank you!

Gold Member
What formulas do you know that contain I?

Homework Helper
Gold Member
2021 Award
Maybe you've got the answer. The motion of any object under gravity is independent of its mass. So, unless you weigh the cylinder I'm not sure how you would calculate its mass.

Homework Helper
So this way you measure I/m . Should count as a determination of the moment of inertia of a cylindrical shell. If the exercise wants more (such as a formula for thick-walled cylinders), you'll need a bunch of cylinders. And a scale.

Alettix
What formulas do you know that contain I?

Well, not much. Bascially L = I * ω and τ = I * α

Maybe you've got the answer. The motion of any object under gravity is independent of its mass. So, unless you weigh the cylinder I'm not sure how you would calculate its mass.

Yes, I too came to the conclusion that I could not determine its mass with the listed tools. But the task explicitly asks me to calculate the moment of inertia, so I guess there should be a way to do so without knowing the weight, I just can't find it...

Homework Helper
Gold Member
2021 Award
##\sqrt{\frac{I}{M}}## is the radius of gyration. To find M you'll need a force other than gravity.

Mentor
Do you know the density of the material comprising the cylinder?

Homework Helper
MHB
Hi Alettix! ;)

It should be:
$$I = \frac{m(2gh-v^2)}{\omega^2}$$
With ##\omega=\frac v r##, it becomes:
$$I=mr^2\left(\frac{2gh}{v^2} - 1\right)$$

If the cylinder would have double the density, both its mass and its moment of inertia would double, but all measurements would remain the same.
So as stated, you either need to measure the mass with different means, or you can only give the ratio ##I/m##.

To measure the mass, you typically need a reference mass (and scales), or you need a reference force (spring or weighing equipment), neither of which is apparently available.
So I suspect you're supposed to determine ##I## in the form ##I=cm## or ##I=cmr^2##, where ##c## is the constant you need to find.

• Alettix
Mentor
If you knew the density, you could determine the mass from the dimensions of the cylinder. Then you could determine I from the experiments.

Chet

Gold Member
So i was thinking about it, and everyones right, you do need another force, but realistically, you have one: rolling friction. Since it acts between the 2 surfaces (your cylinder and your ramp) you could account for that in your sum of torques/forces equations. You need the coefficient of rolling friction, however. You could possibly determine that on a flat surface. The only concern i have is the accuracy of your hand with a stop watch. If the ramp is really long, you could calculate a reasonably sized delta within a reasonably small sigma value. But its all slightly conditional. Are you allowed to alter the cylinder and determine the moment of inertia of the altered cylinder? *Cough* sandpaper *\cough*

Homework Helper
MHB
The coefficient of friction won't help.
To force of friction is, just like everything else, proportional to the mass.

dean barry
Here maybe a clue:
Calculate the final velocity (v1) of a theoretical block of the same mass (m) as the cylinder, sliding down the same incline (without friction).
Calculate the final velocity (v2) of the cylinder over the same distance.
(use the elapsed time and distance)
Calculate the final linear KE of both the theoretical block ( ½ * m * v1² ) and the cylinder ( ½ * m * v2² )
Subtract the cylinder linear KE from the block linear KE, the remainder is the cylinder rotational KE at v2
This might help.

• Alettix
Gold Member
The coefficient of friction won't help.
To force of friction is, just like everything else, proportional to the mass.
True that ^
=[

Alettix
As I do neighter know the density of the cyliner, nor have access to any force independent of gravity, I'm now convinced that only the ratio I/m can be determined.
Thank you for the help everybody! :)

• I like Serena
dean barry
You dont need the density,
A lot of assumptions are made:
Acceleration down the incline is constant.
Gravitational acceleration (g) is deemed to be fixed at 9.81 (m/s)/s
(this should have been stated)
Rolling resistance is negligable and i presume is ignored.
This all serves to simplify the problem, and assumes that the result wont be to far from the theoretical.

However:
Measure the mass of the cylinder and the ouside diameter (d) (which gives you the radius ( r = d / 2) ):
Log the distance (s) down the incline, the incline angle (A) and the time it takes the cylinder to roll down the incline.

Calculate the KE of a (theoretical) block of the same mass as the cylinder sliding (without friction) the same distance (s) down the ramp, using the steps:
1) calculate the acceleration rate (a) of the block down the incline from: a = g * Sine A
2) calculate the final velocity from: v = sqrt ( 2 * a * s )
3) calculate the final KE (in Joules) from: KE = ½ * mass * v²
(call this result 1)

1) calculate the acceleration from: a = s / ( ½ * t ² )
2) calculate the final velocity from: v = sqrt ( 2 * a * s )
3) calculate the final linear KE of the cylinder from: KE = ½ * mass * v²
(call this result 2)

The difference between the two results tells you the KE that has been translated into rotational KE, with this (and the final rotation rate (ω) of the cylinder, which you get from the velocity and radius) you can find the moment of inertia by using the equation:
I = KE / ( ½ * ω ² )

dean barry
ω = v / r