THE BINOMIAL SERIES 65

A

New values of b and l are (1+ 0 .035)b and
(1− 0 .025)lrespectively.

New second moment of area

=

1

12

[(1+ 0 .035)b][(1− 0 .025)l]^3

=

bl^3

12

(1+ 0 .035)(1− 0 .025)^3

≈

bl^3

12

(1+ 0 .035)(1− 0 .075), neglecting

powers of small terms

≈

bl^3

12

(1+ 0. 035 − 0 .075), neglecting

products of small terms

≈

bl^3

12

(1− 0 .040) or (0.96)

bl^3

12

, i.e. 96%

of the original second moment of area

Hence the second moment of area is reduced by
approximately 4%.

Problem 18. The resonant frequency of a

vibrating shaft is given by:f=

1

2 π

√

k

I

, where

kis the stiffness andI is the inertia of the

shaft. Use the binomial theorem to determine

the approximate percentage error in determin-

ing the frequency using the measured values of

kandI when the measured value ofkis 4%

too large and the measured value ofIis 2% too

small.

Letf,kandIbe the true values of frequency, stiffness
and inertia respectively. Since the measured value of
stiffness,k 1 , is 4% too large, then

k 1 =

104

100

k=(1+ 0 .04)k

The measured value of inertia,I 1 , is 2% too small,
hence

I 1 =

98

100

I=(1− 0 .02)I

The measured value of frequency,

f 1 =

1

2 π

√

k 1

I 1

=

1

2 π

k

1

2

1 I

−^12

1

=

1

2 π

[(1+ 0 .04)k]

1

(^2) [(1− 0 .02)I]−
1
2

1
2 π
(1+ 0 .04)
1
(^2) k
1
(^2) (1− 0 .02)−
1
(^2) I−
1
2

1
2 π
k
1
(^2) I−
1
(^2) (1+ 0 .04)
1
(^2) (1− 0 .02)−
1
2
i.e. f 1 =f(1+ 0 .04)
1
(^2) (1− 0 .02)−
1
2
≈f
[
1 +
(
1
2
)
(0.04)
][
1 +
(
−
1
2
)
(− 0. 02 )
]
≈f(1+ 0 .02)(1+ 0 .01)
Neglecting the products of small terms,
f 1 ≈(1+ 0. 02 + 0 .01)f≈ 1. 03 f
Thus the percentage error in f based on the
measured values of k and I is approximately
[(1.03)(100)−100], i.e.3% too large.
Now try the following exercise.
Exercise 35 Further practical problems
involving the binomial theorem

1. Pressurepand volumev are related by
   pv^3 =c, wherecis a constant. Determine the
   approximate percentage change incwhenp
   is increased by 3% andvdecreased by 1.2%.
   [0.6% decrease]


2. Kinetic energy is given by^12 mv^2. Deter-
   mine the approximate change in the kinetic
   energy when massmis increased by 2.5%
   and the velocityvis reduced by 3%.
   [3.5% decrease]


3. An error of+1.5% was made when meas-
   uring the radius of a sphere. Ignoring the
   products of small quantities determine the
   approximate error in calculating (a) the vol-
   ume, and (b) the surface area.
   [
   (a) 4.5% increase
   (b) 3.0% increase

]

4. The power developed by an engine is given
   byI=kPLAN, wherekis a constant. Deter-
   mine the approximate percentage change in
   the power whenPandAare each increased
   by 2.5% andLandNare each decreased by
   1.4%. [2.2% increase]