ME 309 Fall 2008
Section 2 (Merkle)
Homework # 2
Due Fri 29 Aug 2008
Problem 21.
Water running out of a faucet is to be measured by means of a digital stop
watch with 0.01
s
intervals and a glass container to catch the water.
The volume flow
rate of water in
m
3
/s
is to be determined by weighing the water on a scale with gradations
of 0.001 kg.
The glass container weighs 0.04 kg and contains a maximum of 0.4 kg of
water.
(Except for part e, ignore the effects of the mass of the container in your solution.)
a)
Find an algebraic expression for the volume flow rate.
The volume flow rate is the
result,
R
, in this problem;
Solution:
The volume flow rate of water, m
, is given by
t
M
V
Δ
=
ρ
/
,
where M is
the mass,
is the density, and
Δ
t is the increment of time.
b)
Using the expression from a), identify the measured variables,
x
i
, and estimate
their variations,
δ
x
i
; and relative uncertainty,
u
xi
; (Note you should be able to find a
numerical value for each quantity, but some of the
u
xi
cannot be computed until you
have the values in d)
Solution:
There are three variables in the problem.
Their variations, and relative
uncertainties are:
x
1
=density =1000kgm
3
δ
x
1
= 1kg/m
3
u
1
= 0.001
x
2
= measuring time, T
x
2
= 0.005 s
u
2
= 0.005/T
x
3
= mass of water, M
x
3
= .0005 kg
u
2
= 0.005/M
Comments:
The temperature of the water is not known, so the density must be estimated from
an (assumed) room temperature.
Looking at the variation in density with
temperature, a 1kg/m
3
estimate appears reasonable.
Something smaller might be
appropriate also.
The measuring time and mass varies as described in c)
c)
Find the variation in volume flow rate (the propagation of error),
( )
i
i
i
x
x
R
R
∂
∂
=
/
, for each variable;
Solution:
( )
i
i
i
x
x
R
R
∂
∂
=
/
, for each variable;
The propagation of error for the three variables is:
T
T
M
T
T
R
2

=
∂
∂
;
δρ
T
M
R
2

=
∂
∂
and
M
T
M
M
R
1
=
∂
∂
d)
Find the estimated uncertainty in the flow rate measurement if the container is
filled to the top, 1/10
th
full, and 1% full.
The container contains 0.4 kilogram (after
the container mass is subtracted out) when the container is full, and the fill time takes