AC/DC ELECTRICAL ENGINEERING REVIEW CENTER Contact #: 09236884939

CENTROID 3. Find the ordinate of the centroid of the solid t = time WORK DONE IN STRETCHING A SPRING
generated by revolving the area bounded by y = 4x –
The centroid, or center of gravity, of any object is the x^2 and y = 0 about x = 0. 6. A particle moves along a straight line with velocity v
A. 1.4 C. 1.6 F=0
point within that object from which the force of = 4t^3 + 3t^2 + 5. The distance the body covers from t
B. 2.0 D. 1.8 = 0 to t = 2 equals ___.
gravity appears to act.
A. 55 C. 34
PAPPUS PROPOSITION B. 24 D. 16 F = kx
Note: In solving for the coordinates of the centroid of
x
plane area, use if possible a vertical strip when solving  Surface of revolution (1st proposition) – if an arc is
for the x – coordinate and horizontal strip when revolved about a coplanar axis not crossing the arc, WORK DONE IN WINDING CABLES / ROPES
solving the y - coordinate. Hooke’s law:
the area of the surface generated is equal to the
Work in lowering the cable
product of the arc length and the circumference
 Centroid of a plane area axis F = kx
described by the centroid of the arc.

y - axis h Work done

ω x k
A = S • 2πr̅ h Work = ∫ (W + ωx)dx
x W = (x2 2 − x1 2 )
x2 r 0 2
x y A • x̅ = ∫ dA • x W
x1 A = ∫ dS • 2πr̅ k = spring constant
x - axis
x1 = initial elongation
Work in winding up the cable x2 = final elongation
y - axis 4. Find the area of the surface generated by revolving
x y2 about the x-axis the arc 3y=4x from x=0 to x=3. 9. A spring of natural length 10 in is that a force of 6
A • y̅ = ∫ dA • y A. 18π C. 20π lbs will stretch it 2 inches. Find the work done in
y y y1 B. 19π D. 21π stretching it from a length of 14 inches to a length of
h
x - axis ω h-x Work = ∫ (W + ω(h − x))dx 18 inches.
 Volume of revolution (2nd proposition) - if a plane h 0 A. 72 in –lb C. 70 in –lb
is revolved about a coplanar axis not crossing the area, B. 80 in –lb D. 74 in –lb
1. Determine the abscissa of the centroid of the area W
the volume of the solid generated is equal to the x
bounded by the parabola y = x^2 and the line y = 2x + axis
product of the plane area and the circumference WORK DONE IN PUMPING LIQUIDS OUT OF THE
3.
A. 1.0 C. 1.2 described by the centroid of the plane area. TANK
B. 1.6 D. 1.4 ω = weight per unit length of the cable
W = weight attached to the cable or rope Point of discharge
2. Find the ordinate of the centroid of the area under h = length of cable or rope being pulled
Ap V = AP • 2πr̅ Work = ρVd
one arch of the sine curve, y = sin x.
A. π/2 C. π/5 d
r
B. π/6 D. π/7 Work = ρ ∫ dV • d
7. A 5-lb monkey is to climb a 20 ft hanging rope that c.g.
5. Find the volume generated by revolving a rectangle weighs 0.3 lb/ft. If the monkey climbs up to the top at
 Centroid of a volume of revolution the same time carrying the rope, how much work has
of sides a and b about side a.
A. πab^2 C. πba^2 it done?
Abscissa of the centroid B. 2πab^2 D. 2πba^2 A. 200 lb-ft C. 180 lb-ft d = distance of the centroid of the liquid volume to the
B. 150 lb-ft D. 160 lb-ft discharge point
x2
V • x̅ = ∫ dV • x VELOCITY AND ACCELERATION PROBLEMS
x1 8. A bucket weighing 10 N when empty is loaded with Common container shapes with their centroids:
ds dv 90 N of sand and then lifted to a height of 10 m at
Ordinate of the centroid v= → s = ∫ vdt a= → v = ∫ adt constant speed. Sand leaks out of a hole in a bucket at
dt dt a uniform rate. One third of the sand is lost by the end
y2 of the lifting. Find the work done.
V • y̅ = ∫ dV • y where: v = velocity A. 850 J C. 890 J
y1
a = acceleration B. 990 J D. 925 J
s = distance
AC/DC ELECTRICAL ENGINEERING REVIEW CENTER Contact #: 09236884939

13. What is the moment of inertia of a circle of radius 9. Find the volume generated by revolving about x = 3
h/2 xo-axis of 5 m with respect to its tangent? the area bounded by y = 4x – x^2 and y = x.
c.g. 3/8 r A. 2454 C. 2254 A. 20π/5 C. 64π/3
c.g. B. 2050 D. 2540 B. 36π/7 D. 27π/2
Ix = Ixo + Ad2
d
Cylindrical Hemispherical PRACTICE PROBLEMS 10. Find the volume formed by revolving about y-axis
tank tank (full) the area in the first quadrant bounded by y = cosx and
x-axis the two axes.
1. Find the volume generated by rotating about the y-
A. 2 π – 2 C. π(π – 1)
axis the area in the first quadrant bounded by y = x^2
h/3 1/4 h B. π(π – 2) D. π – 1
d = distance between parallel axes and y = 4.
c.g. c.g. A. 8π C. 64π/3
11. The area bounded by the curve y = sin x from x = 0
B. 32π/3 D. 4π
Derived Formulas for Moment of Inertia on Common to x = π is revolved about the x-axis. What is the
Parabolic Conical Geometric Figures: volume generated?
2. Find the volume of the solid generated by revolving
tank tank A. 2.145 C. 3.452
about the x-axis the region bounded by the parabola y
B. 4.935 D. 5.214
= x^2 + 1 and the line y = x + 3.
10. A right circular cylindrical tank of radius 2 m and a bh3 bh3 A. 120π/3 C. 117π/5
Ix = Ixo = 12. Suppose that 2 J of work is needed to stretch a
height of 8 m is full of water. Find the work in h xo-axis B. 125π/4 D. 110π/7
3 12 spring from its natural length of 30 cm to a length of
pumping the water to the top of the tank.
x-axis 42 cm. How far beyond its natural length with a force
A. 3945 kN-m C. 3732 kN-m b 3. Find the volume of a solid generated by revolving
of 30 N keep the spring stretched?
B. 3932 kN-m D. 4032 kN-m about the y-axis the region bounded by the parabola, y
A. 11.5 cm C. 12.2 cm
= -x^2 + 6x – 8, and the x-axis.
B. 13.6 cm D. 10.8 cm
MOMENT OF INERTIA bh3 bh3 A. 8π C. 10π
Ix = Ixo = B. 9π D. 11π
12 36 h 13. The natural length of a spring is 10 cm. A force of
Moment of Inertia wrt the x – axis: xo-axis
50 N will stretch it to a total length of 15 cm. Find the
x-axis 4. Find the volume obtained if the region bounded by y
b work done in stretching it from its original length to a
y2 = ( x squared) and y = 2x is rotated about the x-axis.
total length of 20 cm.
Ix = ∫ y 2 dA A. 64π/15 C. 72π/3
A. 6 J C. 5 J
y1 B. 81π/10 D. 90π/7
B. 4.8 J D. 5.2 J
r
Moment of Inertia wrt the y – axis: πr4 xo-axis 5. The area bounded by the curve y^2 = 12x and the
Ixo = 14. A spring has a natural length of 8 inches. If a force
4 line x = 3 is revolved about the line x = 3. What is the
of 20 lbs stretches the spring ½ inch, find the work
x2 volume generated?
done in stretching the spring from 8 in to 11 in.
Iy = ∫ x 2 dA A. 196 C. 202
A. 100 in-lb C. 180 in-lb
x1 B. 181 D. 218
πr 4 B. 150 in-lb D. 110 in-lb
r
Ix = x-axis
12. Find the moment of inertia of the area bounded by 4 6. Find the volume generated by rotating the area
15. There is a required work of 124 ft-lb to compress
the curve x2 = 8y, the line x = 4 and the x – axis on the bounded by the curves, y = x^2, x = 1, x = 2 about x = 1.
aspiring to its free length x1 to x2 = 2.5 in. The constant
first quadrant with respect to x-axis. A. 17π/6 C. 24π/3
scale is 100 lbs/in. Find the free length.
A. 1.43 C. 1.32 B. 32π/3 D. 12π/5
πr 4 A. 7.955 in C. 7.467 in
B. 1.78 D. 1.52 Ix = B. 8.125 in D. 8.228 in
8 r 7. The area under the portion of the curve y = cos x
MOMENT OF INERTIA for PARALLEL AXES x-axis from x = 0 to x = π/2 is revolved about the x-axis, Find
16. Find the force required to compress the spring of
the volume of the solid generated.
modulus 72,000 pounds per foot a distance of ½ inch.
y-axis A. 3.22 C. 2.47
A. 3200 lbs C. 4000 lbs
πab3 B. 4.10 D. 1.34
Ixo = B. 3000 lbs D. 3600 lbs
4 b
a x-axis 8. The area on the first quadrant bounded by the line x
17. A conical vessel is 12 m across the top and 15 m
+ y = 1 is rotated about the x-axis. Determine the
πa3 b deep. If it contains water (density = w) to a depth of 10
Iyo = volume generated.
4 m, find the work in pumping the liquid to a height 3 m
A. π/2 C. π/5
above the top of the vessel.
B. π/4 D. π/3
A. 500πw C. 490πw
B. 560πw D. 520πw
AC/DC ELECTRICAL ENGINEERING REVIEW CENTER Contact #: 09236884939

18. A cylindrical well is 2 m in diameter and 12 m 27. Find the volume of the solid generated by 36. Determine the moment of inertia about the x-axis, 45. The passing general weighted average rating shall be
deep. If there is 3 m of water in the bottom of the well, revolving about their common chord the region of the area bounded by the curve x^2 = 4y, the line x = 70% with no grade below ____% in any of the subjects in
determine the work done in pumping all this water to common to the circles: x^2 + y^2 = 16 and x^2 + y^2 = -4, and the x-axis. the REE licensure examination.
the surface. 8x. A. 9.85 C. 13.24 A. 50 C. 65
A. 900 kN-m C. 920 kN-m A. 62.238 C. 54.415 B. 10.17 D. 12.19 B. 60 D. 70
B. 990 kN-m D. 970 kN-m B. 50.635 D. 66.133
46. Under Rule 19, the Board upon the approval of the
37. Find the ordinate of the centroid of the solid
commission shall officially release the results of the
19. An aquarium 2 m long, 1 m wide and 1 m deep is 28. Find the volume generated by revolving a formed by revolving about the y-axis the first
examination not later than _____ days from the date of the
full of water. Find the work needed to pump half of the rectangle of sides a and b about side a. quadrant area bounded by the parabola y^2 = 24x and examination.
water out of the aquarium. A. πab^2 C. πba^2 the lines y = 0 and x = 6. A. 20 C. 30
A. 2450 J C. 2640 J B. 2πab^2 D. 2πba^2 A. 4 C. 4.5 B. 25 D. 50
B. 2590 J D.2850 J B. 5.5 D. 5
29. Find the volume of the solid generated by rotating 47. An Electrical Engineer shall be fair, impartial and
20. A right circular tank of depth 12 ft and radius 4 ft the curve x2 + y2 = 5 about the line 3x + 4y – 20 = 0. 38. Four particles having masses 2, 6, 4 and 1 slugs reasonable in rendering professional service to his clients,
is half full of oil weighing 60 pounds per cubic foot. A. 412.12 C. 394.78 located at the points (5, -2), (-2, 1), (0, 3) and (4, -1) employers and contractors regarding contracts or other
Find the work done in pumping the oil to a height 6 ft B. 421.27 D. 318.92 respectively. Determine the ordinate of their center of agreements.
above the tank. mass. A. True B. False
A. 133 ton-ft C. 140 ton-ft 30. A ship anchor weighs a ton and anchor chain A. 1 C. 1/2
B. 130 ton-ft D. 136 ton-ft weighs 50 lbs/linear ft. What is the work done in B. 3/4 D. 2/3 48. Article III Section 22 of RA 7920
pulling up the anchor if 100 ft of chain are out, A. Re-examination of Failed Subjects
21. A cylindrical tank with a base radius of 5 ft and a assuming that the lift is vertical? 39. Find the ordinate of the centroid of the solid B. Continuing Professional Education Program
height if 20 ft is filled with water. Find the work done A. 700,000 ft-lb C. 350,000 ft-lb generated by revolving about x = 0 the area bounded C. Report of Ratings
D. Oath
in pumping all the water out the top of the tank. B. 450,000 ft-lb D. 600,000 ft-lb by 2x + y = 2; x = 0 and y = 0.
A. 8.615 x 105 ft-lb C. 9.801 x 105 ft-lb A. 1/3 C. ¼
49. Applicants for admission to the REE examination must
B. 9.182 x 105 ft-lb D. 8.905 x 105 ft-lb 31. A 30-cm long cable weighing 15 N/m is to be B. ½ D. 1 be on or before date of examination at least twenty-one
wound about a windlass. Find the work done. years of age.
22. Find the area of the surface generated by revolving A. 6750 J C. 6507 J 40. Find the ordinate of the centroid of the solid A. SEC 17(a) C. SEC 17(c)
about the y-axis the arc y = ( x squared ) from x = 0 to B. 7650 J D. 5760 J generated by revolving about the y-axis the area in the B. SEC 17(b) D. SEC 17(d)
x = 6/5. first quadrant bounded by the curve y = 4 – x^2 and
A. 1036/375 π C. 1030/238 π 32. A cable that weighs 2 lb/ft is used to lift 800 lbs of the coordinate axes. 50. Any waterborne unit which is designed and built to
B. 1042/389 π D. 1052/320 π coal up a mine shaft 500 ft deep. Find the work done. A. 4/3 C. 3/4 have an electric plant
A. 540,000 ft-lb C. 650,000 ft-lb B. 5/4 D. 4/5 A. Electric locomotive C. Watercraft
23. Find the area of the surface formed by revolving B. 401,000 ft-lb D. 600,000 ft-lb B. Power barge D. Seacraft
the circle x^2 + y^2 = 9 about the line x = 5. 41. A notice issued by the System Operator when the
A. 592 C. 624 33. A 60 m cable that weighs 4 kg/m has a 500 kg Contingency Reserve is less than the capacity of the 51. When there is generation deficiency, the grid is to be
B. 502 D. 662 weight attached at the end. How much work (kg-m) is largest Synchronized Generating Unit or power import considered in the _______________ state.
done in winding up the last 20 m of the cable? from a single interconnection, whichever is higher. A. Alert C. Extreme
24. Find the total area generated by revolving a square A. 12,000 kg-m C. 10,000 kg-m A. Red Alert C. Blue Alert B. Emergency D. Restorative
with sides a about a line b ( b> a ) units from its center. B. 10,800 kg-m D. 11,200 kg-m B. Yellow Alert D. Green Alert
52. The reciprocal of resistance
A. 10πab C. 4πab
42. An alert issued by the System Operator when the Grid A. elastance C. conductance
B. 12πab D. 8πab 34. Find the moment of inertia with respect to the x- B. reluctance D. inductance
Contingency Reserve is zero.
axis of the area bounded by the parabola y2 = 4x and A. Red Alert C. Blue Alert
25. Find the area S of the surface of revolution the line x = 1. B. Yellow Alert D. Green Alert 53. Ohm’s law is not applicable in
generated by revolving about the y-axis the arc of x = A. 2.32 C. 2.13 A. vacuum tubes C. metallic resistors
y^3 from y = 0 to y = 1. B. 2.56 D. 1.42 43. A notice issued by the System Operator when a B. carbon resistors D. wire-wound resistors
A. 2.73 C. 3.27 tropical disturbance is expected to make a landfall within
B. 2.69 D. 3.56 35. Find the moment of inertia of the area bounded by 24 hours. 54. The relative permittivity of free space
the curve x2 = 8y, the line x = 4 and the x – axis on the A. Red Alert C. Blue Alert A. 1 C. 100
26. An ellipse whose major and minor semi-axes are a first quadrant with respect to x-axis. B. Yellow Alert D. Green Alert B. 10 D. 1000
and b respectively is revolved about a tangent which is A. 1.43 C. 1.32
parallel to the major axis. Find the volume generated. B. 1.78 D. 1.52 44. The professional license shall be renewed every << If there’s no hard work, there’s no prosperity.>>
A. 2π ab2 C. 2𝜋 2 a2b A. 1 year C. 3 years
B. 2𝜋 2 ab2 D. 2π a2b B. 2 years D. 4 years