Unformatted text preview: ME 365 Homework Set 4 Out September 15, 2011 Due September 22, 2011 Problem No. 1 (a) To compute the standard deviation σ x of N measurements x1, …, xN the sum ∑ (x − x )2 need to be computed. Show by manipulation that this term can be rewritten as: i ∑ (x i − x )2 = ⎡ ∑ ( xi )2 ⎤ − N x 2 . ⎣
⎦ (b) Suppose two variables x and y are known to satisfy the relationship y(x) = Bx; i.e. they lie on a straight line that is know to pass through the origin. Suppose further that there are N measurements (xi, yi), in which the uncertainties is xi are negligible and those in yi are know to be of the same population. Show that the least
squares best estimate for B is: B* = ∑ x y . ∑x
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i Problem No. 2 The following data were taken from a small water
turbine experiment: rpm Torque (N
m) 100 4.89 201 4.77 298 3.79 402 3.76 500 2.84 601 4.00 699 2.05 799 1.61 (a) Fit a least
squares straight line to these data. (b) Montgomery, et al. (Engineering Statistics, John Wiley & Son, New York, 1998) suggests that a method to identify outliers in x
y data sets is to compute residuals (i.e. the difference between the measured y values and the computed y values) and then to plot the residuals divided by the standard error in the fit vs. the x values. Using this method determine whether any of the torque values appear to be outliers at the 95% confidence level (greater than ~±2 on this plot). ME 365 Homework Set 4 Out September 15, 2011 Due September 22, 2011 Problem No. 3 In testing the power output of a large number of a certain brand of alternating
current motors, the average output power is measured to be 3.75 kW. This result is obtained by measuring the rotational speed of the motor and its output torque. Power is computed as P = τω, where P is power in kW, τ is torque in (Newton – meters) and ω is angular speed in (radian per second). The angular speed is related 2π N
. Here N is the number of to the measured rpm by the equation: ω =
60
revolutions per minute. The following information is available regarding this measurement of torque and speed based on the instrument’s characteristics: Parameter Mean Value Systematic Uncertainty N (rpm) 1760 3.0 20.3 0.3 τ (Newton
meter) Calculate the total uncertainty in the measurement of the output power of the motors. (Assume a 95% confidence level for the given systematic uncertainties.) ...
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