356 Higher Engineering Mathematics
Using equation (3), the approximate change int,
δt≈
∂t
∂l
δl+
∂t
∂g
δg
Sincet= 2 π
√
l
g
,
∂t
∂l
=
π
√
lg
and
∂t
∂g
=−π
√
l
g^3
(from Problem 6, Chapter 34)
δl=
0. 2
100
l= 0. 002 landδg=− 0. 001 g
henceδt≈
π
√
lg
( 0. 002 l)+−π
√
l
g^3
(− 0. 001 g)
≈ 0. 002 π
√
l
g
+ 0. 001 π
√
l
g
≈( 0. 001 )
[
2 π
√
l
g
]
+ 0. 0005
[
2 π
√
l
g
]
≈ 0. 0015 t≈
0. 15
100
t
Hence the approximate error intis a 0.15% increase.
Now try the following exercise
Exercise 142 Further problems on small
changes
- ThepowerPconsumed in aresistor is given by
P=V^2 /Rwatts. Determine the approximate
change in power whenVincreases by 5% and
Rdecreases by 0.5% if the original values ofV
andRare 50 volts and 12.5ohms respectively.
[+21 watts] - An equation for heat generatedHisH=i^2 Rt.
Determine the error in the calculated value of
Hif the error in measuring currentiis+2%,
the error in measuring resistanceRis−3%
and the error in measuring timetis+1%.
[+2%]
- fr=
1
2 π
√
LC
represents the resonant
frequency of a series connected circuit
containing inductanceLand capacitanceC.
Determine the approximate percentage
change infrwhenLis decreased by 3% and
Cis increased by 5%. [−1%]
- The second moment of area of a rectangle
about its centroid parallel to sidebis given by
I=bd^3 /12. Ifbanddare measured as 15cm
and 6cm respectively and the measurement
errors are+12mm inband−1.5mm ind,
find the error in the calculated value ofI.
[+1.35cm^4 ] - The sidebof a triangle is calculated using
b^2 =a^2 +c^2 − 2 accosB.Ifa,c andB are
measured as 3cm, 4cm and π/4radi-
ans respectively and the measurement errors
which occur are+0.8cm,−0.5cm and+π/ 90
radians respectively, determine the error in the
calculated value ofb.
[−0.179cm] - Qfactor in a resonant electrical circuit is given
by:Q=
1
R
√
L
C
. Findthe percentage change in
QwhenLincreases by 4%,Rdecreases by 3%
andCdecreases by 2%.
[+6%]
7. The rate of flow of gas in a pipe is given by:
v=
C
√
d
√ (^6) T 5 ,whereCisaconstant,disthediam-
eter of the pipe andTis the thermodynamic
temperature of the gas. When determining the
rate of flow experimentally,dis measured and
subsequently found to be in error by+1.4%,
andThas an error of−1.8%. Determine the
percentage error in the rate of flow based on
the measured values ofdandT.[+2.2%]