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Energy Lost when Steel Sphere collides with a soft plate of material

  • #1
Hi everyone. This isn't a homework question; just something I was thinking about and would like some advice on how I should tackle it.

Homework Statement



How can I approximate the Kinetic Energy Lost of a Steel Sphere when it collides with a soft sheet of material.
4597494387_fd3f8a579e_m.jpg


A solid steel sphere of radius R impacts a homogeneous material (e.g clay) of depth D at a velocity v. I want to find the energy imparted to the material when one side of the ball just touches the other side of the material. In other words, I can also find the final velocity of the ball and calculate the energy lost to the material.

Homework Equations



I first write:

[tex]l=\sqrt{R^2-\left(R-x\right)^2}[/tex]
where x is the depth the ball has penetrated
and note that l refers to half of the line indicated in the diagram.

[tex]dl=\frac{l-x}{\sqrt{R^2-\left(R-x\right)^2}}dx[/tex]

I assume the steel ball and soft material have uniform density and Young's Modulus.
Where,

[tex]F=\frac{E A \Delta L}{L}[/tex]

The Attempt at a Solution



I consider the force acted by the material with Young's Modulus E on the sphere as it burrows through facing increasing surface area of the material, given by [tex]\pil^2[/tex], where l depends on x. Then I integrate over the thickness of the material, D.

Using Hooke's Law, I write

[tex]dF=\frac{ExdA}{D}=\frac{Ex\pi\left(dl\right)^2}{D}[/tex]

Which equates the force with respect to x the amount the ball has traveled into the material. Then I integrate it from 0 to D with respect to x, and hence find the work done by the material. Is this a sound approach or did I get something really wrong?

Thanks.
 

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