224 Higher Engineering Mathematics
- Two impedances, Z 1 =( 3 +j 6 ) and
Z 2 =( 4 −j 3 )are connected in series to
a supply voltage of 120V. Determine the
magnitude of the current and its phase angle
relative to the voltage.
[15.76A, 23. 20 ◦lagging] - If the two impedances in Problem 2 are con-
nected in parallel determine the current flow-
ing and its phase relative to the 120V supply
voltage. [27.25A, 3. 37 ◦lagging] - A series circuit consists of a 12resistor, a
coil of inductance 0.10H and a capacitance of
160 μF. Calculate the current flowing and its
phase relative to the supply voltage of 240V,
50Hz. Determine also the power factor of the
circuit. [14.42A, 43. 85 ◦lagging, 0.721] - For the circuit shown in Fig. 20.11, determine
the currentIflowing and its phase relative to
the applied voltage. [14.6A, 2. 51 ◦leading] - Determine, using complex numbers, the mag-
nitude and direction of the resultant of the
coplanar forces given below, which are act-
ingat a point.ForceA, 5N acting horizontally,
ForceB, 9N acting at an angle of135◦toforce
A,ForceC, 12N acting at an angle of 240◦to
forceA. [8.394N, 208.68◦from forceA]
I
R 1530 V
R 3525 V
V 5200 V
R 2540 V XL^550 V
XC 520 V
Figure 20.11
- A delta-connected impedance ZA is given
by:
ZA=
Z 1 Z 2 +Z 2 Z 3 +Z 3 Z 1
Z 2
Determine ZA in both Cartesian and polar
form givenZ 1 =( 10 +j 0 ),
Z 2 =( 0 −j 10 )andZ 3 =( 10 +j 10 ).
[( 10 +j 20 ), 22. 36 ∠ 63. 43 ◦]
- In the hydrogen atom, the angular momen-
tum, p, of the de Broglie wave is given
by: pψ=−
(
jh
2 π
)
(±jmψ). Determine an
expression forp.
[
±
mh
2 π
]
- An aircraftPflying at a constant height has
a velocity of( 400 +j 300 )km/h. Another air-
craftQat the same height has a velocity of
( 200 −j 600 )km/h. Determine (a) the veloc-
ity ofPrelative toQ, and (b) the velocity of
Qrelative toP. Express the answers in polar
form, correct to the nearest km/h.[
(a) 922km/h at 77. 47 ◦
(b) 922km/h at− 102. 53 ◦
]
- Three vectors are represented byP,2∠ 30 ◦,
Q,3∠ 90 ◦and R,4∠− 60 ◦. Determine in
polar form the vectors represented by (a)
P+Q+R,(b)P−Q−[R.
(a) 3. 770 ∠ 8. 17 ◦
(b) 1. 488 ∠ 100. 37 ◦
]
- In a Schering bridge circuit,
ZX=(RX−jXCX),Z 2 =−jXC 2 ,
Z 3 =
(R 3 )(−jXC 3 )
(R 3 −jXC 3 )
andZ 4 =R 4
whereXC=
1
2 πfC
At balance:(ZX)(Z 3 )=(Z 2 )(Z 4 ).
Show that at balance RX=
C 3 R 4
C 2
and
CX=
C 2 R 3
R 4