If ##r_0## is at its maximum possible value, compared to its minimum possible value, what does that do to the calculated velocity?User849483 said:
the velocity decreased as the inner radius increasedPeroK said:
Can you calculate the max and min values for ##v## based on the uncertainties in the other measurements?User849483 said:
i do have the v values yesPeroK said:
yes i meant the uncertainty in the radii and formulakuruman said:
That doesn't help much. Can you tell us what task you were given as was given to you? It would also help if you briefly describe the experiment that you did and how you did it. Experimental uncertainties depend on what you did and how.User849483 said:
I had to find whether the inner radius of a cylinder affects its velocity when it rolls down an inclined plane. I timed each trial and I found there was a negative correlation. The formula I provided is a derived equation. I found the velocity for each inner radii value based on this formula, and now I have to find the uncertaintykuruman said:
I dont believe that the formula allows for there to be only 1 instance of each variableharuspex said:
Can you arrange ##\frac{x+y}x## so that there is one instance of each?User849483 said:
i cant seem to figure it out, i've been trying but im not sure how to.haruspex said:
So you have pairs of velocities and inner radii at fixed ##h## and ##r_o##? If so, did you make a plot of velocity vs. inner radius? You must have otherwise you wouldn't know that there is "a negative correlation." For each of the measurements of ##v## you can use the formula to calculate a maximum and minimum value given your estimated uncertainty in ##r_i## and draw error bars on the graph. Then you can add a continuous theoretical solid line using the formula and see how close it comes to your measured values.User849483 said:
Not sure what you mean by "only 1 variable each". What you want is that each variable occurs only once in the expression. For ##\frac{x+y}x## that really is trivial.User849483 said:
what do you mean by a "continuous theoretical solid line using the formula"? As in a line of best fit?kuruman said:
This is what I mean.User849483 said:
Did you mean values of ##r_i##?kuruman said:
Ok, I misunderstood your need. In post #1 I thought you wanted to find how uncertainties in the radius values combine to form uncertainty in v.User849483 said:
Can you tell us how you obtained these ##v## values? Cylinders usually are not equipped with speedometersUser849483 said:
From your relevant equation (was it given or did you deduce it? -- how?) I gather you mean the instantaneous velocty at the bottom end of the ramp.User849483 said:
)To calculate the uncertainty for a simple equation, you typically use the propagation of uncertainty formulas. For a function \( f(x, y, \ldots) \), the uncertainty \(\sigma_f\) can be approximated using partial derivatives: \(\sigma_f = \sqrt{\left(\frac{\partial f}{\partial x}\sigma_x\right)^2 + \left(\frac{\partial f}{\partial y}\sigma_y\right)^2 + \ldots}\), where \(\sigma_x\), \(\sigma_y\), etc., are the uncertainties in the variables \(x\), \(y\), etc.
When adding or subtracting quantities, the uncertainties add in quadrature. For \( z = x + y \) or \( z = x - y \), the uncertainty \(\sigma_z\) is given by \(\sigma_z = \sqrt{\sigma_x^2 + \sigma_y^2}\), where \(\sigma_x\) and \(\sigma_y\) are the uncertainties in \(x\) and \(y\), respectively.
For multiplication and division, the relative uncertainties add. For \( z = x \cdot y \) or \( z = \frac{x}{y} \), the relative uncertainty \(\frac{\sigma_z}{z}\) is given by \(\frac{\sigma_z}{z} = \sqrt{\left(\frac{\sigma_x}{x}\right)^2 + \left(\frac{\sigma_y}{y}\right)^2}\). The absolute uncertainty \(\sigma_z\) can then be found by multiplying the relative uncertainty by \(z\).
For a function of the form \( z = x^a \), the relative uncertainty is given by \(\frac{\sigma_z}{z} = |a| \cdot \frac{\sigma_x}{x}\). Therefore, the absolute uncertainty \(\sigma_z\) is \(\sigma_z = |a| \cdot x^{a-1} \cdot \sigma_x\).
For more complex functions, you can use the general formula for uncertainty propagation involving partial derivatives: \(\sigma_f = \sqrt{\left(\frac{\partial f}{\partial x}\sigma_x\right)^2 + \left(\frac{\partial f}{\partial y}\sigma_y\right)^2 + \ldots}\). This involves taking the partial derivative of the function with respect to each variable