ACE+ REVIEW CENTER

APRIL 2023 REVIEW PROGRAM

CIVIL ENGINEERING LICENSURE EXAM

APPLIED MATH, SURVEYING, TRANSPORTATION & HIGHWAY ENG’G, CONST. MGT.

Thursday, December 08, 2022 Module 09

KINEMATICS

Motion of Particles

I. Translation

The motion of a rigid body in which a straight line passing through any two of its particle always remain
to be parallel to its initial position.

II. Rotation

The motion of a rigid body in which the particles move in circular paths with their centers on a fixed
straight line called the axis of rotation

III. Plane Motion

The motion of a rigid body in which all particles in the body remain at a constant distance from a fixed
reference plane

Notations:
s = distance t = time
r = horizontal displacement y = vertical displacement
vo = initial velocity (velocity at time = 0) v = velocity at any time (final velocity)
a = acceleration g = acceleration due to gravity (9.81 m/s2, 32.2 ft/s2)

TRANSLATION

Rectilinear Translation - Motion along a straight line

 Uniform Motion (constant velocity)

s = vt

 Variable Acceleration
dv ds
a= v=
dt dt
 Constant Acceleration
1
v = vo + at s = vo t + at 2 v2 = vo 2 + 2as
2
 Free-falling body (vo = 0, s = h)
1 2
v = gt h= gt v 2 = 2gh
2

Note:
a is positive (+) if v is increasing (accelerate) g is positive (+) if the particle is moving downward
a is negative (-) if v is decreasing (decelerate) g is negative (-) if the particle is moving upward

Curvilinear Translation (Projectile Motion)

 At any point B in the projectile

x = vox t vy = voy − gt

1
y = voy t − gt 2 vy 2 = voy2 − 2gy
2

gx 2
y = x tan θ −
2vo 2cos2 θ

 At the highest point A: (vy = 0)

voy2 voy
H= t=
2g g

 Maximum horizontal range through the plane where it was fired, at point C: (𝑦 = 0)

vo 2 sin 2θ 2 voy 2 vo sin θ

R= t= =
g g g

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APRIL 2023 REVIEW PROGRAM

ROTATION

 Uniform Motion
Where: θ = angular displacement, radians
θ = ωt ω = angular speed, rad/sec
α = angular acceleration, rad/sec 2
 Uniform Acceleration t = time

ω = ωo + αt
1
θ = ωo t + αt 2
2
ω2 = ωo 2 + 2αθ

Relationship between Translation and Rotation

S=rθ
a= rα
v=rω

KINETICS

Newton’s Laws of Motion

1. A body at rest will remain to be at rest or in motion will remain in motion along a straight path unless
acted upon by an unbalanced force.

2. A particle acted upon by an unbalanced force system has an acceleration in line with and directly
proportional to the resultant of the force system and inversely proportional to its mass.

𝐤𝐅
𝐚= 𝐨𝐫 𝐅 = 𝐦𝐚 ; 𝐤 = 𝟏
𝐦

3. In every action there is always an equal and opposite reaction.

Reversed Effective Force, REF

W
REF = ma = a
g

If F < μ N, no motion
If F = μ N, impending motion
If F > μ N, motion starts (use kinetic friction)

Centrifugal Force (Reversed Normal Effective Force)

W 2 Wv 2
CF = Man = Mω2 r = ω r=
g gr

Centripetal Force (Reversed Tangential Effective Force)

W
T = Mat = rα
g

Pendulum

W 2
CF ω r ω2 r v2
g
tan θ = = = =
W W g gr

g
cos θ = for ω > √g/L
ω2 L

T = W sec θ
r
sin θ =
L

h L
Time to complete one revolution, t = 2π√ g Max. time for a revolution, t max = 2π√g

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FRICTION

Friction is a contact resistance by one body when the second body moves or tends to move past the first body.
The friction force always acts opposite to the motion or to the tendency to move.

Where: N = total normal reaction. The sum of all forces perpendicular to the surface
f = friction force = μ N
μ = coefficient of friction = tan ϕ
R = total surface reaction. The resultant of f and N
ϕ = angle of friction = arctan μ
W = weight of the body

Maximum angle that a plane may be inclined without causing the body to slide down, θ = ϕ = arctan μ.

BELT FRICTION

T1
= efβ
T2
Where: f = coefficient of friction
β = angle of contact in radians
e = 2.71828
T1 = tension in the tight side
T2 = tension in the slack side

BANKING OF CURVES

The maximum speed that a car can round a highway curve without skidding:
v2
tan (θ + ϕ) =
gR

If the car is on the point of slipping down the plane because of insufficient speed:
v2
tan (θ − ϕ) =
gR
Ideal Angle of Banking

The ideal angle of banking is the angle θ such that the car has no tendency to slide up or down the road. With
this angle, there will be no shearing stress (or friction does not work) at the tires of the car, hence ϕ = 0.
v2
tan θ =
gR
v2
The ratio is also known as the impact factor of centrifugal ratio.
gR

Horizontal Rotating Platform

The maximum speed that the platform may be rotated so that the block will not slide:

ω2 R v 2
tan ϕ = μ = =
g gR

WORK AND ENERGY

 Work
Work = force x distance

 Work on Spring
1 2
W= kx
2
 Kinetic Energy
1
KE = mv 2
2
 Potential Energy
PE = Wh = mgh

 Work-Energy Equation

Work is (+) if it helps the motion and (-) if it is against the motion.
KE1 ± Work ± PE = KE2
PE is (+) if the body goes down, and (-) if it goes up.

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IMPULSE AND MOMENTUM

Impulse = Force x time

W
Momentum = mv = v
g
W (+) Impulse = impulse in the same direction as the motion
∑(+)Impulse − ∑(−)Impulse = (v − vo )
g f (-) Impulse = impulse in the opposite direction as the motion

LAW OF CONSERVATION OF MOMENTUM

m1 v1 + m2 v2 + ⋯ = m1 v ′1 + m2 v ′2 + ⋯

COEFFICIENT OF RESTITUTION

The coefficient of restitution is defined as the ratio of the relative velocities of colliding bodies after
impact to their relative velocities before impact. e is always positive.

Relative velocity after impact

Coefficient of restitution, e =
Relative velocity after impact

v2 ′ → v1 ′ vseparation
e= =
v1 → v2 vapproach

Note: → denotes vectorial subtraction.

Where:
0 > e < 1 for elastic or inelastic collision
e = 0 for perfectly inelastic collision
e = 1 for perfectly elastic collision

UNIVERSAL LAW OF GRAVITATION

m1 m2
F=G
R2

ve = √2gR

Where:
Gravitational constant, G = 6.67 x 10-11 kg-1m3s-2
Mass of Earth, me = 5.972 x 1024 kg
Radius of Earth, Re = 6.38 x 106 m
R = center to center distance
ve = velocity of escape

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