How to analyze the compression of a clay ball

[dannyboy2233]

Hello all! This is my first post on the forum. As some background, this issue pertains to the IB Physics Internal Assessment that I am in the process of writing. Thanks in advance for your help!

Homework Statement

My research question is: How does the drop height of a consistent-size clay ball affect the extent to which it is compressed? My independent variable was the drop height of the clay ball, and my two dependent variables were the vertical compression of the clay ball and its horizontal expansion. I have already taken data for the experiment, but I'm not sure I need to post it here; if anyone would like me to, please let me know. I also graphed my data, and the best curve fit was a natural exponent function. There was a moderately small RMSE.

Homework Equations

Not entirely sure; that's what I'm wondering.

The Attempt at a Solution

So far, my "solution" is very qualitative and lacks real mathematics. I essentially just explain that the graph's leveling-off makes sense, as a ball can only compress or expand to a certain degree within the restraints of real-world physics. What I'm having trouble with, however, is determining a). whether or not this curve fit is correct, and b). how to mathematically explain what I have observed. Thanks!
Danny

 

[CWatters]

The old fashioned way of checking for curve fit was to plot the graph using modified axis to see if you get a straight line. If it's straight over a range but not outside that range then you have limits for the range over which your original function holds.

Perhaps you could measure the compression another way? One that allows you to measure the energy used? Then relate that to the height?

 

[Chestermiller]

Is this a college project or a high school project?

If you are trying to develop a mathematical model of the clay deformation, then you need to know the basic mechanical properties of the clay. Clay is not an elastic solid or a Newtonian fluid, and it would more appropriately be modeled as a Bingham plastic. Even having the properties of the clay, this is still not a simple system to analyze, and would require setting up and solving a set of partial differential equations for the deformation as a function of time and position in the clay. A better approach might be to use dimensional analysis to get a handle on this. I would try using the Buckingham pi theorem. It won't give you the exact solution, but it would give you the key dimensionless groups to use in correlating your data. This would alleviate the need to vary each parameter individually.

 

[dannyboy2233]

The old fashioned way of checking for curve fit was to plot the graph using modified axis to see if you get a straight line. If it's straight over a range but not outside that range then you have limits for the range over which your original function holds.

Perhaps you could measure the compression another way? One that allows you to measure the energy used? Then relate that to the height?

Thank you for your help! Unfortunately, we were only given two days in class to collect data, and we aren't allowed to continue outside of class, so I'm stuck with what I have already. I will try using your suggestion, and see if it helps my situation. Thanks!
Danny

 

[dannyboy2233]

Is this a college project or a high school project?

If you are trying to develop a mathematical model of the clay deformation, then you need to know the basic mechanical properties of the clay. Clay is not an elastic solid or a Newtonian fluid, and it would more appropriately be modeled as a Bingham plastic. Even having the properties of the clay, this is still not a simple system to analyze, and would require setting up and solving a set of partial differential equations for the deformation as a function of time and position in the clay. A better approach might be to use dimensional analysis to get a handle on this. I would try using the Buckingham pi theorem. It won't give you the exact solution, but it would give you the key dimensionless groups to use in correlating your data. This would alleviate the need to vary each parameter individually.

Thank you for your response; this is a high school project for second-year IB Physics. I have pretty extensive experience with calculus, but have never used a differential equation in the context of physics. Would the Buckingham pi theorem be beyond my skill set? Thanks!
Danny

 

[Chestermiller]

Thank you for your response; this is a high school project for second-year IB Physics. I have pretty extensive experience with calculus, but have never used a differential equation in the context of physics. Would the Buckingham pi theorem be beyond my skill set? Thanks!
Danny

The B pi theorem is not beyond your skill set.

 

[dannyboy2233]

The B pi theorem is not beyond your skill set.

Ok. How would I apply this theorem to the work that I'm doing? I tried looking it up, and it is very general. Thank you again for your help!

 

[Chestermiller]

Are you familiar with the physical properties: viscosity (of clay) and density? What other parameters do you think might be important in this physical system?

 

[dannyboy2233]

Are you familiar with the physical properties: viscosity (of clay) and density? What other parameters do you think might be important in this physical system?

I don't know that information off the top of my head, but I'm sure I could find it. My guess is that mass, volume, and projected area (surface are?) may be other relevant parameters when considering this physical system. Do I also need information about atmospheric density/viscosity? Thanks!

 

[Chestermiller]

I don't know that information off the top of my head, but I'm sure I could find it. My guess is that mass, volume, and projected area (surface are?) may be other relevant parameters when considering this physical system. Do I also need information about atmospheric density/viscosity? Thanks!

I'm not asking you the values of the viscosity and the density of clay. I'm just asking if you know what these properties mean. Knowing the diameter of the clay sphere is the same as knowing its volume and projected area. Knowing the diameter and density of the clay sphere are the same as knowing its mass. So, so far, the parameters we feel would be involved would be diameter D, density ##\rho##, and viscosity ##\eta##. Any others you can think of that might determine the decrease in height of the sphere as a result of the collision?

 

[Chestermiller]

Here's a hint: If the clay were traveling 100 m/s when it hit the ground, would the clay compress as much if it were traveling 1 m/s?

 

[dannyboy2233]

Here's a hint: If the clay were traveling 100 m/s when it hit the ground, would the clay compress as much if it were traveling 1 m/s?

It would compress more! I assume velocity is also a relevant parameter here.

 

[Chestermiller]

OK. Se we have identified all the parameters that seem to be important in this:

Clay viscosity ##\eta##
Sphere diameter D
Clay density ##\rho##
Impact Velocity v (later, we'll express the velocity v in terms of the height from which the clay was dropped)
Squished thickness h

We would now like to apply the Buckingham Pi theorem. Do you have any idea how to do this so that we can identify the two dimensionless groups that are involved here?

 

[Chestermiller]

$$\frac{h}{D}=f\left(\frac{\rho v D}{\eta}\right)=f\left(\frac{\rho \sqrt{2gl} D}{\eta}\right)$$where l is the height from which the clay ball is dropped. So, if the viscosity and density of the clay is fixed, you make a graph of d/D versus ##\sqrt{l}D## (or ##lD^2##). All the data should fall on a single curve.