Homework 4
Mechanical Engineering Lab I - MECE E3018 - Fall 2020 - Professor Yevgeniy Yesilevskiy
Due: October 15, 2020 at 11:59pm
1. You are given the task to measure the number of moles of a gas. To do so you fill a cylindrical
pressure vessel with a gas and wait for the system to come to thermal equilibrium with the
room before measuring the temperature and pressure of the system. The Ideal Gas Law, 𝑃𝑉 =
𝑛𝑅𝑇, can be used to describe the system, where P is absolute pressure, V is volume, n is number
of moles, R is the universal gas constant, and T is the absolute temperature. What follows are
some questions which will guide you through the process to measure the number of moles and
also to estimate the uncertainty in the measurement.
a. You need to calculate the best estimate of the volume and the uncertainty in the
volume. The cylindrical pressure vessel has internal diameter, d, and internal length, L,
so the internal volume of the cylinder is 𝑉 = (𝜋4) 𝑑2𝐿. To calculate the best estimate
and the uncertainty in volume, you must therefore first calculate the best estimates and
the uncertainties of d and L.
i. The file `diameter.txt' contains measurements (units of mm) of the inside
diameter of the cylinder. Calculate the sample mean, 𝑑̅, which is the best
estimate for the diameter.
ii. Calculate the precision (or random) absolute uncertainly in the measurements
as:
and also the precision (or random) relative uncertainty as
where the d subscripts indicate diameter.
iii. The specification sheet for the caliper used to measure the inside diameter
states that the accuracy of the instrument is ±0.01 mm, which is determined by
calibration against a standard from the National Institute of Standards and
Technology (NIST). The accuracy quantifies the bias (or systematic) absolute
uncertainty that we denote as 𝐵𝑑. Calculate the bias relative uncertainty
defined as:
where again the d subscripts indicate diameter.
iv. The total absolute uncertainty, 𝑈𝑑,
defines the total relative uncertainty. Calculate both 𝑈𝑑 and 𝑒𝑈𝑑
and show that
𝑒𝑈𝑑
can also be calculated directly from 𝑒𝑃𝑑
and 𝑒𝐵𝑑
v. The file ‘length.txt' contains measurements (units of mm) of the inside length of
the cylinder. A different caliper with accuracy 0.02mm was used to make the
measurements. Repeat steps (i) through (iv) to calculate the best estimate, 𝐿̅,
and the absolute and relative total uncertainties denoted as 𝑈𝐿and 𝑒𝑈𝐿
,
respectively.
vi. Calculate the best estimate of the volume denoted as 𝑉̅.
vii. As we discussed in class, the uncertainty in a result (in this case in volume) can
be calculated in terms of how the standard deviations associated with the
measurements (in this case of 𝜎𝑑 and 𝜎𝐿
) combine to determine the standard
deviation of the result (in this case 𝜎𝑉). Specifically
for the case when the measurements of d and L are uncorrelated. Upon
normalizing this expression with the equation for the result (or volume) one
obtains
Show that for this case, and determine the total
uncertainty in volume as 𝑒𝑈𝑉
by calculating 𝜎𝑉𝑉.
b. You need to calculate the best estimate of the temperature and the uncertainty in the
temperature. The temperature measurements (with units of Kelvin) contained in the le
`temperature.txt' were made with a Hart Scientic 5665 thermistor.
i. Calculate the best estimate of the temperature denoted as 𝑇̅.
ii. Calculate the absolute and relative precision uncertainties denoted as 𝑃𝑇 and
𝑒𝑃𝑇
for the temperature.
iii. Refer to the specification sheet (DataSheets2.pdf) to determine the absolute
bias uncertainty for the temperature measurement denoted as 𝐵𝑇. Calculate
the relative bias uncertainty denoted 𝑒𝐵𝑇
.
iv. Calculate the absolute and relative total uncertainties denoted as 𝑈𝑇 and 𝑒𝑈𝑇
for the temperature.
c. You need to calculate the best estimate of the pressure and the uncertainty in the
pressure. The pressure in the cylinder is measured with a Honeywell PPTR pressure
transducer with a full-scale (FS) of 500 psi in analog mode with a maximum voltage of
5V. The output voltage of the pressure transducer is measured by a data acquisition
card (DAQ) and the measured voltage is then numerically converted to a pressure with
the calibration constant of 100 psi/V after which it is converted to MPa. The file
‘pressure.txt' contains the results of the absolute pressure in units of MPa.
i. Calculate the best estimate of the pressure denoted as 𝑃̅.
ii. Calculate the absolute and relative precision uncertainties denoted as 𝑃𝑃 and
𝑒𝑃𝑃
for the pressure.
iii. Refer to the specification sheet (DataSheets2.pdf) to determine the absolute
bias uncertainty for the pressure measurement denoted as 𝐵𝑃 . Calculate the
relative bias uncertainty denoted 𝑒𝐵𝑃
.
iv. Calculate the absolute and relative total uncertainties denoted as 𝑈𝑃 and 𝑒𝑈𝑃
for the pressure.
d. You are now prepared to calculate the best estimate of the number of moles as well as
the related total uncertainty, based upon the best estimates and uncertainties in
pressure, temperature and volume.
i. From the Ideal Gas Law, calculate the best estimate, 𝑛̅ of the number of moles
of gas contained in the pressure vessel.
ii. Calculate the total absolute uncertainty of the measurement of moles, 𝑈𝑛 and
the total relative uncertainty of the measurement of moles, 𝑒𝑈𝑛
2. The shear modulus, G, of a metal can be determined by measuring the angular twist, 𝜃, resulting
from a torque applied to a cylindrical rod made of the metal. For a rod of radius, 𝑅, and a
torque, 𝑇, applied at a length 𝐿 from a fixed end, the modulus can be calculated to be
In this problem you will examine the effect of the relative uncertainty of each measured variable
on the shear modulus to assist with the design of an experimental setup. With the equipment
you have in the laboratory, you can measure 𝜃, 𝐿, and 𝑇 each with relative total uncertainty of
±1.0 %. However you need to purchase an instrument to measure 𝑅. The goal of your
measurements is to have a total relative uncertainty in 𝐺 of ∓2.0% or less. What is the
maximum relative uncertainty allowable in the measurement for R?
3. The tip deflection of a cantilever beam with rectangular cross-section subjected to a point load
at the tip is given by the formula.
where 𝑃 is the force, 𝐿 is the length of the beam, 𝐸 is the Young's modulus of the material, 𝑏 is
the width of the cross-section, and ℎ is the height of the cross-section. The instrument
uncertainties in 𝑃, 𝐿, 𝐸, 𝑏, ℎ are each 2 %.
a. Estimate the fractional (i.e. relative) uncertainty in 𝛿.
b. This beam is used in an experiment to determine the value of an unknown load, 𝑃𝑥, by
performing a series of four repeated measurements of 𝛿 at that load under the same
controlled conditions. The resulting sample standard deviation is 8 μm and the average
deflection is 20 μm. Determine the overall uncertainty in the deflection measurements
estimated at 90 % confidence assuming that the resolution of the instrument used to
measure 𝛿 is so small that it produces negligible uncertainty.
4. A hand-held velocimeter uses a heated wire and, when air blows over the wire, correlates the
change in temperature to the air speed. The reading on the velocimeter, 𝑣𝑠𝑡𝑑, is relative to
standard conditions defined as 𝑇𝑠𝑡𝑑 = 70 °F and 𝑝𝑠𝑡𝑑 = 14.7 psia. To determine the actual
velocity, 𝑣𝑎𝑐𝑡, in units of feet per minute, the equation
𝑣𝑎𝑐𝑡 = 𝑣𝑠𝑡𝑑 [460+𝑇460+𝑇𝑠𝑡𝑑][𝑝𝑠𝑡𝑑𝑝]
must be applied, where 𝑇 is in °F and 𝑝 is in psia. The accuracy of the reading on the velocimeter
is 5% or 5 ft/min, whichever is greater. The velocimeter also measures air temperature with an
accuracy of 1°F. During an experiment, the measured air velocity is 400 ft/min and the
temperature is 80°F. The air pressure can be assumed to be at standard conditions. Assuming
that the relative precision uncertainties are much smaller than the relative bias uncertainties,
determine
a. the best estimate of the actual air velocity
b. the absolute uncertainty in the actual air velocity
c. the percent uncertainty in the actual air velocity.
5. Seven identical model truss bridges are constructed. Each is loaded with increasing masses until
it collapses. The masses used are in 50g increments. The loads at failure are 1.40 kg, 1.40 kg,
1.45 kg, 1.50 kg, 1.55 kg, 1.60 kg, and 1.60 kg. If an additional truss is constructed, estimate with
95 % confidence the loaded mass at which the truss will fail.