Exercise A2 Mass calculations Modify the file R_GAUGE_BLOCK_A.m to evaluate the following exercise. The mass of a cylindrical artefact is given by M = xphr2, where p is the density at 20"C, r is the radius, h is the height. Suppose r and h are measured at temperature t and that the coefficient of thermal expansion is c. Express the uncertainty associated with M in terms of the uncertainties associated with p, r, h, t and C. Determine the 2 x2 variance matrix Vm for the masses Mi and Mz of two cylindrical artefacts made from the same material from the following data: ri = 25.005 mm is the radius measured at temperature to = 20.5 "C, h1 = 63.510 mm is the height measured at temperature t1. r2 = 25.020 mm is the radius measured at temperature tz = 20.8 "C, h2 = 63.515 mm is the height measured at temperature tz. p = 8000 kg mos; c = 10 X 105 m K-1; u(ri) = u(r2) = u(hi) = u(h2) = 0.001 mm, u(ti) = u(tz) = 0.1 "C; u(p) = 0.05 kg m 3; u(c) = 1 x 10-ki Use VM to evaluate the uncertainties associated with M1 : M2. Notes: The mass of a cylindrical artefact is given by M = nphr2, Taking into account temperature, we have: M = nphr=(1 + c(t - to)) Sensitivity coefficients: aM Ch = - oh = npr= (1 + c(t - to))3 aM C. = = 2nphr(1 + c(t - to)) aM Ct = at -= 3mphr=(1+ c(t - to)) c aM Co ac = 3mphr= (1+ c(t - to)) (t - to) aM Cp ap -= ihr= (1 + c(t - to))3 For two masses M1 and M2, we have: M1 = nphr (1 + c(t, - to)) My = nphyr?(1 + c(tz - to)) X = (h1,h2,r1,r2,t1,t2,c,rho)"; C1 = [ch O cp Oct 0 Cc Crho] C2 = [0 ch O cp 0 ct Cc "rho] C = 1 [M1] Y = M2] Vx = diag[up, up, up, up, up, up, up, uinho] Vy = V(Y) = CVXCT S = M1 + M2 = [1 1] [M;] u=(S) = V(S) ~ c'Vyc = [1 1]vy [;]