(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hi, I've a lab assingment, and the labwork must be planned beforehand, but I have some trouble figuring out some parts of my error analysis.

So, I'm supposed to measure (as in not use integrals to find out) the mass momentum of a ball. The plan is to place the ball on an inclined plane, and let it roll to the ground. I'm going to measure the mass, radius, the velocity just before the ball hits the ground, and the distance to ground from the beginning position.

This is done with 10 or so different starting distances (and consecutively with 10 different end velocities).

2. Relevant equations

First I derive the formula for my calculations from the principle of concervation of mechanical energy:

[tex]mgh &= \frac{1}{2}I_{cm} \omega _{cm}^2+\frac{1}{2}mv_{1}^2 \newline \ldots \Leftrightarrow mgh = \frac{I_{cm}+mr^2}{2r^2}v_{1}^2\\[/tex], where m is the mass, r radius, g is the acceleration due gravity, h is the initial position when the ball is at rest, v is the velocity just before the ground, and I_cm is the mass momentum of inertia.

Now, I have 10 different results, I use linear fit on a set of points [ m*g*h, v^2], so I get an equation like [tex]U(v^2) = kv^2+b[/tex]. Now, computer gives me k with it's error limit (and b, which is just the systematic error).

[tex]k=\frac{I_{cm}+mr^2}{2r^2} \Leftrightarrow 2kr^2 = I_{cm}+mr^2 \Leftrightarrow I_{cm} = 2kr^2-mr^2[/tex]

Now I have the equation for my mass moment of inertia.

3. The attempt at a solution

The problem is, the when using the propagation of uncertainty (http://en.wikipedia.org/wiki/Error_propagation): [Broken]

[tex]\displaystyle\delta I_{cm} &= \sqrt{\sum_{i}^n (\frac{\partial I_{cm}}{\partial x_i} \delta x_i)^2} =

\sqrt{(\frac{\partial I_{cm}}{\partial k} \delta k)^2+(\frac{\partial I_{cm}}{\partial r} \delta r)^2+

(\frac{\partial I_{cm}}{\partial m} \delta m)^2}

\\ &= \sqrt{4r^4\delta k^2+4r^2(2k-m)^2\delta r+r^4\delta m^2} = r\sqrt{4r^2\delta k^2+4(2k-m)^2\delta r+r^2\delta m^2 }.[/tex]

k is not an independent variable, as it depends on mass m and radius r.So the big question is, how am I supposed to change the last equation, when taking into account that k depends on m and r?

As for my bad english, I apologize, please ask, if I didn't make myself clear. Thank you.

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# Homework Help: Mass momentum of inertia in lab, error analysis

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