66 NUMBER AND ALGEBRA
- The radius of a cone is increased by 2.7%
and its height reduced by 0.9%. Determine
the approximate percentage change in its
volume, neglecting the products of small
terms. [4.5% increase] - The electric field strengthHdue to a magnet
of length 2land momentMat a point on its
axis distancexfrom the centre is given by
H=M
2 l{
1
(x−l)^2−1
(x+l)^2}Show that iflis very small compared withx, thenH≈2 M
x^3.- The shear stressτin a shaft of diameter
Dunder a torqueTis given by:τ=
kT
πD^3.
Determine the approximate percentage error
in calculatingτifTis measured 3% too
small andD1.5% too large.
[7.5% decrease]- The energyWstored in a flywheel is given
by:W=kr^5 N^2 , wherekis a constant,r
is the radius andNthe number of revolu-
tions. Determine the approximate percent-
age change inWwhenris increased by
1.3% andNis decreased by 2%.
[2.5% increase] - In a series electrical circuit containing
inductanceLand capacitanceCthe resonant
frequency is given by:fr=1
2 π√
LC. If the
values ofLandCused in the calculation are
2.6% too large and 0.8% too small respect-
ively, determine the approximate percentage
error in the frequency.
[0.9% too small]- The viscosityηof a liquid is given by:
η=kr^4
νl, wherekis a constant. If there is
an error inrof+2%, inνof+4% andlof
−3%, what is the resultant error inη?
[+7%]- A magnetic pole, distancexfrom the plane
of a coil of radiusr, and on the axis of the
coil, is subject to a forceFwhen a cur-
rent flows in the coil. The force is given
by:F=kx
√
(r^2 +x^2 )^5, wherekis a constant.Use the binomial theorem to show that when
xis small compared tor, thenF≈kx
r^5−5 kx^3
2 r^7- The flow of water through a pipe is given by:
G=√
(3d)^5 H
L.Ifddecreases by 2% andH
by 1%, use the binomial theorem to estimate
the decrease inG. [5.5%]