For your uncertainty propagation, estimate the uncertainty in the mass and radiu

Question

For your uncertainty propagation, estimate the uncertainty in the mass and radius measurements you took, as well as the uncertainty for the angular velocity recorded by the rotary motion sensor. How do i estimate uncertainty for the mass and the radius I measure of a disk? The following are the values I measured. Mass Radius Lower Disk 0.12056 0.048 Upper Disk 0.12033 0.048 Ring 0.46938 0.026 0.038 Also how do i get the uncertainty of the angular velocity I found on a program that recorded its velocity? Is it the standard deviations? The following are some of the angular velocities Two Discs ?i (deg/sec) 1452 1603.1 1961.7 1722 1822.8

Explanation / Answer

This is a complete method to solve this question. In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate to the combination of variables in the function. The uncertainty is usually defined by the absolute error ?x. Uncertainties can also be defined by the relative error (?x)/x, which is usually written as a percentage. f_k(x_1,x_2,dots,x_n) be a set of m functions which are linear combinations of n variables x_1,x_2,dots,x_n with combination coefficients A_{k1},A_{k2},dots,A_{kn}, (k=1dots m). f_k=sum_i^n A_{ki} x_i or mathbf{f}=mathbf{Ax}, and let the variance-covariance matrix on x be denoted by Sigma^x,. Sigma^x = egin{pmatrix} sigma^2_1 & ext{cov}_{12} & ext{cov}_{13} & cdots \ ext{cov}_{12} & sigma^2_2 & ext{cov}_{23} & cdots\ ext{cov}_{13} & ext{cov}_{23} & sigma^2_3 & cdots \ dots & dots & dots & ddots \ end{pmatrix} Then, the variance-covariance matrix Sigma^f,, of f is given by Sigma^f_{ij}= sum_k^n sum_ell^n A_{ik} Sigma^x_{kell} A_{jell}: Sigma^f=mathbf{A} Sigma^x mathbf{A}^ op. This is the most general expression for the propagation of error from one set of variables onto another. When the errors on x are un-correlated the general expression simplifies to Sigma^f_{ij}= sum_k^n A_{ik} left(sigma^2_k ight)^x A_{jk}. Note that even though the errors on x may be un-correlated, their errors on f are always correlated. The general expressions for a single function, f, are a little simpler. f=sum_i^n a_i x_i: f=mathbf {a x}, sigma^2_f= sum_i^n sum_j^n a_i Sigma^x_{ij} a_j= mathbf{a Sigma^x a^t} Each covariance term, M_{ij} can be expressed in terms of the correlation coefficient ho_{ij}, by M_{ij}= ho_{ij}sigma_isigma_j,, so that an alternative expression for the variance of f is sigma^2_f= sum_i^n a_i^2sigma^2_i+sum_i^n sum_{j (j e i)}^n a_i a_j ho_{ij} sigma_isigma_j. In the case that the variables x are uncorrelated this simplifies further to sigma^{2}_{f}= sum_i^n a_{i}^{2}sigma^{2}_{i}. f_k pprox f^0_k+ sum_i^n rac{partial f_k}{partial {x_i}} x_i where partial f_k/partial x_i denotes the partial derivative of fk with respect to the i-th variable. Or in matrix notation, mathrm{f} pprox mathrm{f}^0 + J mathrm{x}, where J is the Jacobian matrix. Since f0k is a constant it does not contribute to the error on f. Therefore, the propagation of error follows the linear case, above, but replacing the linear coefficients, Aik and Ajk by the partial derivatives, rac{partial f_k}{partial x_i} and rac{partial f_k}{partial x_j}. In matrix notation operatorname{cov}(mathrm{f}) = J operatorname{cov}(mathrm{x}) J^ op. That is, the Jacobian of the function is used to transform the rows and columns of the covariance of the argument. Nonetheless, the most common formula among engineers and experimental scientist to calculate error propagation for independent variables is the one proposed by the NIST: Any non-linear function, f(a,b), of two variables, a and b, can be expanded as fpprox f^0+rac{partial f}{partial a}a+rac{partial f}{partial b}b hence: sigma^2_fpproxleft| rac{partial f}{partial a} ight| ^2sigma^2_a+left| rac{partial f}{partial b} ight|^2sigma^2_b+2rac{partial f}{partial a}rac{partial f}{partial b} ext{cov}_{ab}. In the particular case that f=ab!, rac{partial f}{partial a}=b, rac{partial f}{partial b}=a. Then sigma^2_f pprox b^2sigma^2_a+a^2 sigma_b^2+2ab, ext{cov}_{ab} or left(rac{sigma_f}{f} ight)^2 pprox left(rac{sigma_a}{a} ight)^2+left(rac{sigma_b}{b} ight)^2+2left(rac{sigma_a}{a} ight)left(rac{sigma_b}{b} ight) ho_{ab}.

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