Experiment testing Rotational Kinetic Energy with a hollow cylinder

[aniion]

1. I'm doing an experiment where I have a brass hollow cylinder, and I roll it down an inclined plane, and after each trial, I add fish weights to the inside of it to increase the mass.

I'm testing the proportionality between mass and angular velocity, with my hypothesis being an inverse proportionality. What I was wondering was that in my calculations, would I need to calculate just E_r=1/2(Iω²)? Or also normal kinetic energy?

The Attempt at a Solution

What I plan to do is first roll the naked ring down, measure the distance, time, radius and mass of said ring, then repeat with 3 added fish weights to the inside of the ring. I would do this a total of 5 times. Then I would calculate the angular velocity of each trial, and make a graph of the relationship with mass. My hypothesis would be that it would be a curve. Then, I would take the inverse of the angular velocity, and the graph would then become a straight line. Bear in mind, this is for a senior level course in high school, end-of-year cumulative assignment.

Although the aforementioned is my main query, if there's any other mistakes or things that I have not taken into account, I would greatly appreciate being aware of them. Thank you for your time.

 

[Delphi51]

You do need to include .5*m*v². The rolling cylinder has both translational and rotational kinetic energy. I'm wondering how you are adding the fish weights. They will change the moment of inertia. I don't see a nice straight line relationship coming out of this for mass and angular velocity.

I'm impressed that you are doing moment of inertia and rotational energy in high school!
Are you in an advanced class or an advanced country?

The only high school experiment with rotational energy that I'm familiar with is rolling marbles down a ramp so they then cross a short horizontal distance and finally fly in a parabolic path down to the floor where they hit a carbon paper and mark the horizontal flight distance. From this distance and the known heights, you can calculate the velocity at the bottom of the ramp. There is a nice relationship between the ramp height and the horizontal flight distance that you can find graphically and derive theoretically. Students not knowing about rotational energy get a mysterious loss of energy to think about. Changing to cylinders, rings, etc. results in different portions lost. Perhaps you could use it to deduce formulas for the moment of inertia.

 

[aniion]

Thanks for the reply!

Do I really need to include translational energy if all I'm testing is the relationship between angular velocity and mass? Will it change the angular velocity?

I am adding the 3/4 ounce fishweights on the inside of the ring. I am aware that it will change the radius, but that's the only way I could think of to noticeably change the mass, yet keep the radius as constant as posssible. I would include this in the possible sources of error too.

Why may I ask do you not think a straight line would be the result after doing the necessary transformation to angular velocity? I did a similar experiment a couple months back with .5*m*v² with a relationship between mass and velocity. After I took the inverse of the velocity, the graph then became a straight line.

Haha, no, not an advanced class or country, just a regular senior class in Canada. No, I haven't done any experiment regarging marbles, or any such thing, and it's a little late for me to change my topic haha. Thanks for the quick reply though!

 

[Delphi51]

My thinking is that the potential energy at the top of the ramp is converted into rotational and translational kinetic energy like this:
mgh = ½mv² + ½Iω² Using v = r*ω to eliminate the v:
2mgh = mr²ω² + Iω² Approximating I as mr²:
2mgh = mr²ω² + mr²ω² Solve for 1/ω:
2gh = 2r²ω²
1/ω² = r²/gh
It appears that 1/ω is a constant that does not vary with mass.
This reflects the fact that the final kinetic energy is proportional to ω² and the initial potential energy is proportional to the mass.

 

[aniion]

Yeah, I just realized while reading what you typed that it didn't make much sense. I have now decided to make my hypothesis mass is inversely proportional to angular acceleration. Would that produce a straight line? I'd assume so, if it's anything like translational kinetic energy.

Would I need to include translational energy at all if I'm using this proportionality? I was thinking about just including the conservation of energy as something on the side, just as some extra calculations, and to check if energy was conserved relatively well, but I wasn't thinking that it would have any effect on my calculations to prove this proportionality.

 

[Delphi51]

Could you work out the theory for your new idea and find out how angular acceleration depends on mass? That would be an important part of your investigation.

 

[aniion]

Yeah, I wrote that moment of inertia can be thought of as the resistance an object has to angular acceleration, so naturally, as the moment of inertia increases, the subsequent angular acceleration of the object would decrease, proportional to the moment of inertia.

Now this may be kind of simple, but I can't wrap my head around how I am to get the theoretical value for torque, to calculate the experimental error? I would get my experimental value for torque once I graph the relationship for moment of inertia, and inverse angular acceleration. I would then draw the line of best fit, and get the slope of the line, thus giving me the experimental value for torque. But how would I get the theoretical value for torque? I'd assume that I would just calculate it normally, T = I*[itex]\alpha[/itex] . But what values would I put in for moment of inertia and angular acceleration?

Since I'm doing 5 different trials, with 5 different moment of inertias and angular acceleration, would I have to average the total moment of inertias, and angular acceleration, and use those values to find the average torque, and compare that to the experimental value?

 

[Delphi51]

Okay, so T = I*α suggests α = T/I and angular acceleration proportional to 1/I.
This does assume that Torque is constant, doesn't change when you add masses to the ring. Is that a reasonable assumption? What causes the Torque anyway?

There is a terrific analogy between linear and rotational motion, summary here:
http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html#rlin
Earlier you were using this idea by looking at the linear analog for your initial proposed experiment. Maybe a good idea to do that again with this one. Consider first a very simple case of an object falling. F = ma, a = F/m and perhaps acceleration is proportional to 1/m. Is it really? Does a heavier mass fall with smaller acceleration? If you look into this, you'll soon find it was Galileo who figured out how mass affects the acceleration of falling objects and in his day things fell too fast to measure so he used the very idea you have, of rolling things down a long ramp.

There is a description of some of Galileo's work here: http://galileo.phys.virginia.edu/classes/109N/lectures/gal_accn96.htm
And a pertinent excerpt from his play "Two New Sciences" here: http://galileo.phys.virginia.edu/classes/109N/tns61.htm
Galileo wrote the play so he could have mere characters speak his arguments and theory and therefore avoid blaspheming himself.

I can't wrap my head around how I am to get the theoretical value for torque

Make a sketch of your ring on a ramp. It will touch the ramp at one point only, so consider that the pivot point. A force acts on the center of mass of the ring, which is not directly above the pivot point, so it will cause a torque. You can calculate the torque from the definition of torque.

 

[Delphi51]

Don't despair; this is looking like a terrific experiment and analysis (as well as possibly making you the high school student best ever prepared for rotational motion at university).

I'm just leading you away from dead ends while being careful not to actually tell you very much so you'll figure everything out and be able to claim the whole thing as your own work. I trust others posting here will do so as well. You should, of course, include a link to this discussion in your paper so everything is above board.

You are leading us into a rather different approach to rotational motion than you see in a typical textbook, so it is all most interesting to me and surely to your teacher. I like your

Do I really need to include translational energy if all I'm testing is the relationship between angular velocity and mass? Will it change the angular velocity?

as a topic.

Don't wait for the theory to sing and dance before running your experiment. There is nothing as good as real data to test a theory! How will you measure the velocity at a known distance down the ramp? Angular velocity can easily be calculated from linear velocity. Measurement of the time to roll down the ramp would do, but nicer to have the velocity directly. Maybe you have one of those TI calculator attachments? Or use my trajectory and carbon paper idea? Whatever you do, try to get an estimate of the accuracy of the measurements as you do them, perhaps by repeating them.

 

[aniion]

Wow, I totally passed over that the torque should be remaining constant, if my hypothesis is correct, and moment of inertia is inversely proportional to angular acceleration. It look like I would just be able to calculate the torque using any of the trials, and use that as my theoretical value, to calculate experimental error. Thanks!

I'm planning on starting my experiment in the next couple of days, I just like to visualize everything in my head, and write the lab report up beforehand, just to make sure I have everything understood.

I have settled on increasing the mass of the ring, while also decreasing the radius of the ring, which would then in turn increase the moment of inertia of the ring for each trial. Then, after gathering all my results, I would calculate the angular velocity of each trial, then divide it by t to calculate the angular acceleration. Then I would graph the relationship with I being on the y-axis, and α being on the x-axis. I hypothesize that this would be a curved line. Then I would calculate the inverse of α, and put that on the y-axis, which would then give me a straight line.

Then I would calculate the slope, and calculate my experimental error. My last question is that I feel I should do a precision calculation, but I am unsure on what exactly to test the precision of. Any suggestions?

 

[Delphi51]

What will you measure? How will you calculate the angular velocity from the measurement?
The method for finding angular acceleration looks good. Be sure to record the ring mass and radius in each case. Estimate the other radius, out to the center of mass of a ring segment. Record the initial height (above the point where you measure the speed or time) in case you want to compare initial and final energies to find out "if the translational energy must be included."

It might help to have an objective in mind. It should not be just to make a straight line graph, but rather something like "to find a way to predict the final velocity of a ring rolled down a ramp."

What do you mean by "experimental error"? Some people think it is how close they get to the "right" answer. It actually means estimating the error in each of your measurements and following those errors through to calculated values so you can put error bars on your graphs. Then you can see if a straight line fits to within the experimental error. This approach enables you to tell if a formula you find fits the data to within the accuracy of the measurements. It is meaningful when you don't know the "right" answer. The whole point of doing experiments is to find things you don't already know.

 

[aniion]

I've conducted the experiment, and all seems well, but then I encountered a problem with the conservation of energy part.

Gravitational potential energy = Linear Kinetic Energy + Rotational kinetic energy
mgh = 1/2mv² + 1/2Iω²

m= 0.598kg h=0.98m v=1.35m/s I=2.15E-3 and ω=8.10E-2

I get 5.749J for Eg but my result for El and Er total comes out to be 0.54493. I thought there may be something wrong with my decimals, since it looks like it's one decimal off, but no, those are the correct measurements. Could there be that much loss of energy?

 

[Delphi51]

It would be best if you gave raw measurement numbers, including the radius of the ring.
Is h the height of the ramp? Did you measure the time or the velocity? Make it clear which values were measured and which were calculated.
Using v = r*ω -> r = v/ω = 1.35/.081 = 16.7 m, it appears you have a ring much larger than your ramp. If so, then I = .5*m*r² = 83. Something wrong with those numbers, all right.

 

[aniion]

Wow, this book that I'm using, The Laws of Physics by Milton A. Rothman told me that to calculate angular velocity, you multiply v by r. Now I have to redo my analysis and graphs. Thanks a lot man, you really saved me.

 

[aniion]

The radius is 0.060m, height is 0.98 m, displacement of the ring is 1.16 m, and the time is 0.860.

 

[Delphi51]

Hey, those numbers look pretty good!
Perhaps even better, if you can estimate the radius out to the center of the ring to use in calculating the moment of inertia m*R².
Any idea how accurate you measured the time? A 1% error in t is a 1% error in v, which is a 2% error in t² and in the energy.
Also, can you get r a little more accurate? r = .06 suggests it was measured to the nearest cm and is only accurate to .5/6*100 = 17%.

 

[aniion]

I took three times for each trial and averaged them out. For the radius I measured to a tenth of a centimeter, so its 6.0E-2 cm.

 

[Delphi51]

It would be nice to have all the data on a spreadsheet. Measured values in the first few columns, calculated values in the next ones. An estimate of the % error for each measured quantity at the bottom of its column.