1 Sistem Gaya | Sine | Euclidean Vector

MEKANIKA TEKNIK JURUSAN TEKNIK INDUSTRI - FTI Universitas Islam Sultan Agung

Pengajar : A. Syakhroni, ST, M.Eng

Apa itu Mekanika? Cabang ilmu fisika yang berbicara tentang keadaan diam atau geraknya benda-benda yang mengalami kerja atau aksi gaya

Mechanics

Rigid Bodies Deformable Bodies (Things that do not change shape) (Things that do change shape)

Statics

Dynamics

Fluids

Incompressible

Compressible

Apa saja yang dipelajari? Sistem Gaya Momen dan Kopel Keseimbangan partikel Keseimbangan benda tegar Diagram gaya normal, diagram gaya geser, dan diagram momen • Konsep tegangan • Momen inersia dan momen polar • Teori kegagalan statis • • • • •

Review 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 ?

SISTEM GAYA

SISTEM GAYA SPACE (3D)

Fundamental Principles • The parallelogram law for the addition of forces: Two forces acting on a particle can be replaced by a single force, called resultant, obtained by drawing the diagonal of the parallelogram which has sides equal to the given forces

f1+f2

f2

f1

• Parallelogram Law

Fundamental Principles (cont’) • The principle of transmissibility: A force acting at a point of a rigid body can be replaced by a force of the the same magnitude and same direction, but acting on at a different point on the line of action

f2 f1 f1 and f2 are equivalent if their magnitudes are the same and the object is rigid.

• Principle of Transmissibility

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 ?

Addition of Vectors • Trapezoid rule for vector addition • Triangle rule for vector addition C B C

B

• Law of cosines, R 2  P 2  Q 2  2 PQ cos B    R  PQ • Law of sines, sin A sin B sin C   Q R A • Vector addition is commutative,     PQ  Q P • Vector subtraction

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.

• Trigonometric solution - Apply the triangle rule. From the Law of Cosines,

R 2 = P 2 + Q 2 − 2 PQ cos B = (40 N )2 + (60 N )2 − 2(40 N )(60 N ) cos155° R = 97.73N From the Law of Sines,

sin A sin B = Q R Q sin A = sin B R 60 N = sin 155° 97.73N A = 15.04° α = 20° + A

α = 35.04°

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.

Example of this process,

You can also represent a 2-D vector with a magnitude and angle.

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.

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

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

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.

Sample Problem (cont’) SOLUTION: • Resolve each force into rectangular components.

Sample Problem (cont’) force

mag

r F 1 r F 2 r F 3 r F 4

150 80

x − comp + 129

y − comp .9

+ 75 . 0

− 27 . 4

+ 75 . 2

110

0

100

+ 96 . 6

R x = +199.1

− 110

.0

− 25 . 9

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 α = = α = 4.1° Rx 199.1 N

14.3 N R= = 199.6 N sin

α = 4.1°

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

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

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.

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

TERIMA KASIH! 1