PENJUMLAHAN GAYA
TUJUAN PEMBELAJARAN:
• Mahasiswa dapat menentukan besar
dan arah resultan dari beberapa gaya
dengan metode analitis.
• Mahasiswa dapat menentukan besar
dan arah resultan dari beberapa gaya
dengan metode grafis.
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Review
PRINSIP DASAR
Dalam Mekanika
1. Hukum PARALELOGRAM dalam
2.
3.
4.
5.
6.
penjumlahan gaya
Prinsip TRANSMISIBILITAS
Hukum NEWTON 1
Hukum NEWTON 2
Hukum NEWTON 3
Hukum GRAFITASI NEWTON
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Sistem Satuan
• Four fundamental physical quantities. Length, Time, Mass, Force.
• We will work with two unit systems in static’s: SI & US Customary.
Bagaimana konversi dari SI ke US atau sebaliknya ?
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GAYA
Apakah gaya itu ?
4
Menyusun atau menjumlahkan gaya
dimaksudkan untuk menentukan resultante (R),
dengan kata lain dua buah gaya atau lebih
dapat digabung menjadi satu gaya pengganti
yang disebut resultante (R).
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Dapat dilakukan dengan 2 cara
Cara lukisan
Cara hitungan
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APPLICATION OF VECTOR
ADDITION
There are four
concurrent cable forces
acting on the bracket.
How do you determine
the resultant force acting
on the bracket ?
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Addition of Vectors
• Trapezoid rule for vector addition
• Triangle rule for vector addition
C
B
C
• Law of cosines,
R 2  P 2  Q 2  2 PQ cos B
  
R  PQ
• Law of sines,
B
sin A sin B sin C


Q
R
P
• Vector addition is commutative,
   
PQ  Q P
• Vector subtraction
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Sample Problem
SOLUTION:
• Trigonometric solution - use the triangle
rule for vector addition in conjunction
with the law of cosines and law of sines
to find the resultant.
The two forces act on a bolt at A.
Determine their resultant.
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Sample Problem (Lanjutan)
• Trigonometric solution - Apply the triangle rule.
From the Law of Cosines,
R 2  P 2  Q 2  2 PQ cos B
 40N 2  60N 2  240N 60N  cos155
R  97.73N
From the Law of Sines,
sin A sin B

Q
R
sin A  sin B
Q
R
 sin 155
A  15.04
  20  A
  35.04
60N
97.73N
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ADDITION OF SEVERAL VECTORS
• Step 1 is to resolve each force
into its components
• Step 2 is to add all the x
components together and add all
the y components together. These
two totals become the resultant
vector.
• Step 3 is to find the magnitude
and angle of the resultant vector.
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Example of this
process,
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You can also represent a 2-D vector with a
magnitude and angle.
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EXAMPLE
Given: Three concurrent forces
acting on a bracket.
Find: The magnitude and
angle of the resultant
force.
Plan:
a) Resolve the forces in their x-y components.
b) Add the respective components to get the resultant vector.
c) Find magnitude and angle from the resultant components.
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EXAMPLE (continued)
F1 = { 15 sin 40° i + 15 cos 40° j } kN
= { 9.642 i + 11.49 j } kN
F2 = { -(12/13)26 i + (5/13)26 j } kN
= { -24 i + 10 j } kN
F3 = { 36 cos 30° i – 36 sin 30° j } kN
= { 31.18 i – 18 j } kN
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EXAMPLE (continued)
Summing up all the i and j components respectively, we get,
FR = { (9.642 – 24 + 31.18) i + (11.49 + 10 – 18) j } kN
= { 16.82 i + 3.49 j } kN
y
FR
FR = ((16.82)2 + (3.49)2)1/2 = 17.2 kN
 = tan-1(3.49/16.82) = 11.7°

x
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Sample Problem
SOLUTION:
• Resolve each force into rectangular
components.
• Determine the components of the
resultant by adding the corresponding
force components.
Four forces act on bolt A as shown.
Determine the resultant of the force
on the bolt.
• Calculate the magnitude and direction
of the resultant.
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Sample Problem (cont’)
SOLUTION:
• Resolve each force into rectangular components.
force mag
x  comp
y  comp

 129.9
 75.0
F1 150

 27.4
 75.2
F2
80

 110.0
F3 110
0

 96.6
 25.9
F4 100
Rx  199.1 R y  14.3
• Determine the components of the resultant by
adding the corresponding force components.
• Calculate the magnitude and direction.
Ry 14.3 N
tan  

  4.1   4.1
Rx 199.1 N
R
14.3 N
 199.6 N
sin 4.1
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READING QUIZ
1. The subject of mechanics deals with what happens to a body
when ______ is / are applied to it.
A) magnetic field
B) heat
D) neutrons
E) lasers
C) forces
2. ________________ still remains the basis of most of today’s
engineering sciences.
A) Newtonian Mechanics
B) Relativistic Mechanics
C) Euclidean Mechanics
C) Greek Mechanics
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READING QUIZ
3. Which one of the following is a scalar quantity?
A) Force B) Position C) Mass D) Velocity
4. For vector addition you have to use ______ law.
A) Newton’s Second
B) the arithmetic
C) Pascal’s
D) the parallelogram
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CONCEPT QUIZ
5. Can you resolve a 2-D vector along two directions, which
are not at 90° to each other?
A) Yes, but not uniquely.
B) No.
C) Yes, uniquely.
6. Can you resolve a 2-D vector along three directions (say
at 0, 60, and 120°)?
A) Yes, but not uniquely.
B) No.
C) Yes, uniquely.
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ATTENTION QUIZ
7. Resolve F along x and y axes and write it in
vector form. F = { ___________ } N
y
A) 80 cos (30°) i - 80 sin (30°) j
x
B) 80 sin (30°) i + 80 cos (30°) j
C) 80 sin (30°) i - 80 cos (30°) j
30°
F = 80 N
D) 80 cos (30°) i + 80 sin (30°) j
8. Determine the magnitude of the resultant (F1 + F2)
force in N when F1 = { 10 i + 20 j } N and F2 =
{ 20 i + 20 j } N .
A) 30 N
B) 40 N
D) 60 N
E) 70 N
C) 50 N
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