Mechanical Measurements HW2 Solution

Mechanical Measurements HW2 Solution - PROBLEM 5.9 KNOWN:...

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PROBLEM 5.9 KNOWN: Four resistors are available: two rated at R = 500 ± 50 and two rated at R = 2000 ± 5% . FIND: Best design combination to form R T = 1000 SOLUTION We can combine the resistors in series or in parallel. Consider as Case 1, a series arrangement and as Case 2, a parallel arrangement. Case 1 R T = R 1 + R 2 If we use the two 500 resistors, then 1 R d u = ± 50 2 R d u = ± 50 The propagation of uncertainty through to R T is estimated by 12 2 2 1/2 ( ) [ ( ) ( )] TRR TT d Rdd RR u uu ∂∂ = ±+ = 22 [(1 * ) (1 *) dd = 71 ±Ω (95%) Case 2 T R = + If we use the two 2000 resistors, then 1 R d u = ± 100 2 R d u = ± 100 The propagation of uncertainty through to R T is estimated by 2 ( ) [ ( ) ( d u = = 2 2 1 2 1 1 2 1 2 [( { } ( { } * ( ) () R R R R R R R R ± - +- ++
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= 35 ±Ω (95%) Case 2 provides the smaller uncertainty at the design-stage. We should proceed using this design. COMMENT This is a classic illustration of using uncertainty analysis to determine the better of differing approaches. Although each individual resistor in Case 2 has a larger absolute uncertainty than those in Case 1, we find that the weighted combination of the two resistors in Case 2 yields a substantially lower uncertainty. This is not obvious by inspection alone. Our design results from a close analysis of the sensitivity of the resultant to each contributing uncertainty. The combination in Case 2 is just less sensitive to the individual uncertainties.
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PROBLEM 5.13 KNOWN: Heat transfer from a rod is to be determined. Nu = hD/k is the nondimensional heat transfer. u h = 150 ± 7% W/m 2 -K (95%) u d = 20 ± 0.5 mm (95% assumed) u k = 0.6 ± 2% W/m-K (95% assumed) FIND: u Nu SOLUTION At the known level of uncertainty provided: Nu = f(h,D,k) then, 1/2 2 22 N u hDk N u N u Nu u uuu hDk  ∂∂∂  = ± ++     = 2 22 N u h Dk 2 D h hD u u uu kkk - = = 2 Nu 2 0.0 2 15 0 150x0.02 u 10. 5 0.00 5 0.01 2 0.4 0. 6 0. 6 0.6 -
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This homework help was uploaded on 04/04/2008 for the course 650 350 taught by Professor Shan during the Spring '08 term at Rutgers.

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Mechanical Measurements HW2 Solution - PROBLEM 5.9 KNOWN:...

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