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ME 3603 –
Spring 2024 Practice Problem Set 03 Work the following problems in your textbook: 4.30 (also calculate the 95 % confidence intervals of the slope and y-intercept) 4.32 4.33(a),(b) 4.34 4.39 5.4 5.6 5.9 5.19 …and the following problems:
SP 4.6)
A manufacturer of general aircraft dry vacuum pumps wishes to estimate the mean failure time of its product at 95% confidence. Initially, six pumps are tested to failure with these results (in hours of operation): 1272, 1384, 1543, 1465, 1250, 1319. Estimate the sample mean and the 95 % confidence interval of the true mean. How many more data points would be needed to improve the confidence interval to be within ±50 hours? SP 4.7)
The force on a spring was measured as the spring was elongated. The data are shown. 𝒙 (𝒊𝒏)
7 18 27 32 38 48 57 61 𝑭 (𝒍𝒃
𝒇
)
69.9 133.6 145.1 168.9 204.5 236.2 317.4 319.1 Calculate the spring constant 𝑘
and its 95 % confidence interval based on these measurements. (Continued on next page)
Challenge: SP 4.8)
Using an ammeter and a wattmeter, you measure the power usage of a device at different operating conditions (data shown below). Theoretically, you expect these values to follow a power law, 𝑃𝑜𝑤𝑒𝑟 = ?𝐼
𝑏
, where 𝐼
is the current in amps and ?
and ?
are coefficients. Current (𝑨)
Power (𝑾)
0.10 53 0.20 205 0.30 448 0.40 810 0.50 1263 0.60 1799 0.70 2465 0.80 3210 0.90 4076 a)
Find the coefficients ?
and ?
by first linearizing the power law equation, then performing a linear regression. b)
Find the coefficients ?
and ?
by plotting the data in Excel and adding a Power law trendline. Compare these to your results from part (a). Bonus Brainteaser: B 4.1)
Using a well-calibrated digital scale with a resolution of 0.1 𝑘𝑔
, you weigh an object three times. Each time, it displays 10.3 𝑘𝑔
. What is the 99 % confidence interval of the true weight based on these three measurements (assume a normal distribution)?
SP 5.1)
You find an old digital scale in the lab, and want to know if is still accurate. You run a calibration test by weighing a set of precision standardized weights you have in the lab. You decide to measure each weight multiple times (results below) to test the repeatability of the instrument. The scale indicates an operating range of up to 5 lb, and displays weights with a resolution of 0.01 lb. Calibrated weight Scale readings (lb) 0 lb (empty) 1 0.00 2 0.01 3 0.00 4 0.00 0.1000 lb ± 0.0005 lb 1 0.08 2 0.09 3 0.07 4 0.11 5 0.10 6 0.10 1.0000 lb ± 0.0025 lb 1 0.91 2 0.98 3 0.89 4 0.91 5 0.90 6 1.00 5.0000 lb ± 0.0050 lb 1 4.81 2 4.90 3 4.72 4 4.84 5 4.81 6 4.82 7 5.00 8 4.88 For each of the four nominal weights, estimate: a)
The random uncertainty (use 95 % confidence) b)
The systematic uncertainty/error* c)
The design-stage uncertainty d)
Based on your answers to the above, what is the overall maximum expected uncertainty (with 95 % confidence)? What is this in terms of a percentage of the full scale output? *Hint: Treat the known weights as the exact values (can you justify this?). SP 5.2)
You and two other labs are measuring stock parts for comparison. You (Lab A) measure the diameter of the part at five locations using a pair of calipers with an analog readout (resolution 0.001”) and a stated accuracy of 0.002”. Your readings are:
Lab A readings (in.) 2.038 2.031 2.023 2.041 2.030 Lab B and Lab C both use calipers with a stated accuracy of 0.01 cm (including resolution error). They measure the diameter at three and four locations, respectively: Lab B readings (cm) 5.15 5.14 5.13 - Lab C readings (cm) 5.11 5.12 5.10 5.11 a)
Using a 95 % confidence level, present the average diameter of all three parts. b)
Using a 95 % confidence level, answer the following questions: i.
Is Lab A’s part thicker than Lab B’s? …thinner than Lab B’s?
ii.
Is Lab A’s part thicker than Lab C’s? …thinner than Lab C’s?
iii.
Is Lab B’s part thicker than Lab C’s? ..thinner than Lab C’s?
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ME Undergraduate Lab General Guidelines assignment Watch or read the following sections of the MEULGG and complete the following quizzes: §§ 3.1, 3.2, 3.3 Quiz 10 § 4 Quiz 11
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3. Problem
Estimate the frictional resistance Rp for a container ship using the ITTC 1957 model-ship correlation line
Equation (2):
0.075
CF
[ log,,(Re) – 21
The ship has the following particulars:
Full scale ship data
length between perpendiculars Lep
length in waterline
length over wetted surface
195.40 m
Lwz
Los
For the wetted surface S you can use the following
formula by Kristensen and Lützen (2012) derived
for container ships.
200.35 m
205.65 m
breadth
B
29.80 m
draft
T
10.10 m
37085.01 m3
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displacement
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V
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Example: bottled water at Kroger (Homework)
=
Month
Actual
Forecasted
α = 0.3
Jan
1,325
1,370
Feb
1,353
1,361
Mar
1,305
1,359
Apr
1,275
1,349
May
1,210
1,334
Jun
?
1,309
solve the acutal sales in jun using
Exponential smoothing: the method
Assume that we are currently in period. We calculated the
forecast for the last period (F) and we know the actual demand
last period (4.) ...
F₁=F₁₁+a(A₁-F₁)
The smoothing constant a expresses how much our forecast will
react to observed differences...
If a is low: there is little reaction to differences.
If a is high: there is a lot of reaction to differences.
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Problem 1/MatlabGrader (20 points) (Core Course Outcome 4)
Develop a Matlab function myFitExam that finds the best fit of the function
p(t) = as
sin³ (t) + b sin² (t) + csin(t) + d
(1)
to a given set of data points (ti, pi) using regression. Here t is in radians.
The function input shall be
⚫t: column vector of data values t
⚫ p: column vector of data values p
The function output shall be
• a: scalar containing the best fit coefficient a
b: scalar containing the best fit coefficient b
c: scalar containing the best fit coefficient c
d: scalar containing the best fit coefficient d
In the function use only functions developed in this class in modules 1 - 4. You do not need to provide these functions in your
submission. They will be provided when assessing your function after the deadline.
Note: no assessments will be performed on your submitted function before the exam deadline. Scores on Canvas for this problem
before the exam has been fully graded are meaningless.
Required submission:
☐…
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791²
(1)
Calculate:
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analysis (the inputs are g and vo and the output is the time of landing. Think about
their units and how you might construct the output using the inputs, just by matching
units). Solve the problem exactly. Verify that the scaling analysis gives you (almost)
the correct answer.
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(d) The times at which the ball reaches the height v/g. What is the physical interpretation
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(e) Does your scaling analysis provide any insight into the answers for questions (a-e)?
Discuss. (Hint: Observe how your answers depend on g and vo).
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mu_01 = [153446.180; 41874155.872; 0; 3066.875; -11.374; 0] / 1000;
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0 0 1.205 0 0 -6.071e-5;
0.0160 -9.883e-4 0 4.437e-8 -1.212e-6 0;
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P_02 = [6494.224 -376.156 -4.492e-5 0.0160 -0.494 -5.902e-8;
-376.156 22.561 2.550e-6 -9.885e-3 0.0286 3.419e-9;
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0.0160 -9.883e-3 -1.180e-10 4.438e-8 -1.212e-6 -1.448e-13;
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QUESTION 3
Solve the following differential equation in Matlab using ode45.m. Use time interval of 0.1 second. (Don't use options in ode45 to change teh tolerance.)
df (t)
+ƒ (t) ×sin(t) =0
dt
f(0)=0.8.
What is f(t=3.7)?
0.120
0.126
0.491
0.847
0.635
○ 0.695
QUESTION 4
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T'sec
In helical spring experiment a
student plotted the graph of T2
versus the oscillating mass (M), and
Used the slope to find the spring
:constant, the value of k is
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
O.
O a. 13.3 N/m
O b. 6.8N/m
O c. 5 N/m
O d. 3.3 N/m
O e. 24 N/m
5
10 15 20 25 30 35 40 45 50 55 60 65
M (9)
70
75
80
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