Higher Engineering Mathematics, Sixth Edition

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

1. 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]


2. 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%]

3. 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%]

4. 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 ]


5. 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]


6. 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%]