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intensity I o through the duct and measuring the light intensity transmitted to the other side, I. A
transmission factor is found by
T ¼ I=I o ¼ e
ÀKEW
0 T 1
Here W is the width of the duct, K is the solids density, and E is a factor taken as 2.0 Æ 0.04 (95%) for
spheroid particles. Determine how u K /K is related to the relative uncertainies of the other variables. If
the transmission factor and duct width can be measured to within Æ1%, can solids density be measured
to within 5%? 10%? Discuss your answer, remembering that the transmission factor varies from 0 to 1.
5.44 A step test is run to determine the time constant of a first-order instrument (see Chapter 3). If the error
fraction G t
ð Þ can be estimated to within Æ2% (95%) and time t can be estimated in seconds to within
Æ0.5% (95%), plot u t =t versus G t
ð Þ over its range, 0 G t
ð Þ 1.
5.45 The acceleration of a cart down a plane inclined at an angle a to horizontal can be determined by
measuring the change in speed of the cart at two points, separated by a distance s, along the inclined
plane. Suppose two photocells are fixed at the two points along the plane. Each photocell measures
the time for the cart, which has a length L, to pass it. If L ¼ 5 Æ 0.5 cm, s ¼ 100 Æ 0.2 cm,
t 1 ¼ 0.054 Æ 0.001 s, and t 2 ¼ 0.031 Æ 0.001 s, all (95%), estimate the uncertainty in acceleration:
a ¼
L
2
2s
1
t 2
2
À
1
t 2
1
Compare as a concomitant check a ¼ g sin a using values for a ¼ 30
and 90
. What uncertainty in a
is needed to make this the better method?
5.46 Golf balls are often tested using a mechanical player called an ‘‘Iron Byron’’ because the robotic
golfer’s swing was patterned after that of Byron Nelson, a famous golf professional. It is proposed to
use initial velocity and launch angle from such testing to predict carry distance (how far the ball
travels after impact). The following data represent test results for a particular golf ball when struck
with a particular driver (golf club):
Initial Velocity (mph)
Launch Angle (degrees)
Carry Distance (yd)
165.5
8
254.6
167.8
8
258.0
170.0
8
261.4
172.2
8
264.8
165.5
10
258.2
167.8
10
261.6
170.0
10
264.7
172.2
10
267.9
165.5
12
260.6
167.8
12
263.7
170.0
12
266.8
172.2
12
269.8
If the initial velocity can be measured to within a systematic uncertainty of 1 mph, and the launch
angle to within 0.1 degrees estimate the resulting uncertainty in the carry distance as a function of
initial velocity and launch angle. Over this range of initial velocity and launch angle, can a single
value of uncertainty be assigned?
206 Chapter 5 Uncertainty Analysis
14:36:33 Page 206
intensity I o through the duct and measuring the light intensity transmitted to the other side, I. A
transmission factor is found by
T ¼ I=I o ¼ e
ÀKEW
0 T 1
Here W is the width of the duct, K is the solids density, and E is a factor taken as 2.0 Æ 0.04 (95%) for
spheroid particles. Determine how u K /K is related to the relative uncertainies of the other variables. If
the transmission factor and duct width can be measured to within Æ1%, can solids density be measured
to within 5%? 10%? Discuss your answer, remembering that the transmission factor varies from 0 to 1.
5.44 A step test is run to determine the time constant of a first-order instrument (see Chapter 3). If the error
fraction G t
ð Þ can be estimated to within Æ2% (95%) and time t can be estimated in seconds to within
Æ0.5% (95%), plot u t =t versus G t
ð Þ over its range, 0 G t
ð Þ 1.
5.45 The acceleration of a cart down a plane inclined at an angle a to horizontal can be determined by
measuring the change in speed of the cart at two points, separated by a distance s, along the inclined
plane. Suppose two photocells are fixed at the two points along the plane. Each photocell measures
the time for the cart, which has a length L, to pass it. If L ¼ 5 Æ 0.5 cm, s ¼ 100 Æ 0.2 cm,
t 1 ¼ 0.054 Æ 0.001 s, and t 2 ¼ 0.031 Æ 0.001 s, all (95%), estimate the uncertainty in acceleration:
a ¼
L
2
2s
1
t 2
2
À
1
t 2
1
Compare as a concomitant check a ¼ g sin a using values for a ¼ 30
and 90
. What uncertainty in a
is needed to make this the better method?
5.46 Golf balls are often tested using a mechanical player called an ‘‘Iron Byron’’ because the robotic
golfer’s swing was patterned after that of Byron Nelson, a famous golf professional. It is proposed to
use initial velocity and launch angle from such testing to predict carry distance (how far the ball
travels after impact). The following data represent test results for a particular golf ball when struck
with a particular driver (golf club):
Initial Velocity (mph)
Launch Angle (degrees)
Carry Distance (yd)
165.5
8
254.6
167.8
8
258.0
170.0
8
261.4
172.2
8
264.8
165.5
10
258.2
167.8
10
261.6
170.0
10
264.7
172.2
10
267.9
165.5
12
260.6
167.8
12
263.7
170.0
12
266.8
172.2
12
269.8
If the initial velocity can be measured to within a systematic uncertainty of 1 mph, and the launch
angle to within 0.1 degrees estimate the resulting uncertainty in the carry distance as a function of
initial velocity and launch angle. Over this range of initial velocity and launch angle, can a single
value of uncertainty be assigned?
206 Chapter 5 Uncertainty Analysis