1-12-2014

Jurusan Teknik Mesin dan Industri, Fakultas Teknik, Universitas Gadjah Mada

Dinamika TKM 2302 / 3 SKS

Dr. Indraswari Kusumaningtyas

2 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Kinetika Benda Tegar 6 Impuls dan Momentum

This method is well suited to the solution of problems

involving force, velocity and time, particularly if the applied

forces were expressible as functions of the time.

It is the only practicable method for problems involving

impulsive motion and impact, i.e. when interactions between

rigid bodies occur during short periods of time.

3 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Principle of Impulse & Momentum

Consider a rigid body as made of a large number of particles Pi .

4 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

The momenta of the particles of a system can be reduced to a

vector attached to the mass center G, equal to their sum:

and a couple equal to the sum of their moments about G,

Principle of Impulse & Momentum

The system of momenta viΔmi

is equivalent to the linear

momentum vector attached

at G and to the angular

momentum couple .

5 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Principle of Impulse & Momentum

Based on the principle of impulse and momentum for plane motion

of rigid body, we have three diagrams.

6 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Principle of Impulse & Momentum

Three equation of motions obtained by:

Summing and equating the momenta and impulses in the x and y

directions.

Summing and equating the moments of these vectors with respect

to any given point.

Can be computed with respect to any points other than G, but consistent.

7 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Principle of Impulse & Momentum Translation

Rectilinear or Curvilinear,

Angular velocity ω = 0

Mass center velocity vG = v

If angular momentum is computed from

some other point A , the moment of the

linear momentum L must be calculated

about that point, so

8 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Principle of Impulse & Momentum Rotation

Centroidal Rotation

Mass center velocity vG = 0

Angular velocity ω

Non-Centroidal Rotation about O

Mass center velocity vG

Angular velocity ω

Computing about O,

9 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

System of Rigid Bodies

The principle of impulse and momentum can be applied to an entire

system of connected bodies rather than to each body separately.

This eliminates the need to include interaction impulses which occur

at the connections, since they are internal to the system.

Note that the system's

angular momentum and

angular impulse must be

computed with respect to

the same reference point O

for all the bodies of the

system.

10 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Conservation of Momentum

When no external force acts on a rigid body or a system of rigid

bodies, the system of the momenta at time t1 is equipollent to

the system of the momenta at time t2.

The total linear momentum of the system is conserved in any

direction and its total angular momentum is conserved about

any point.

Conservation of linear momentum

Conservation of angular momentum

11 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Conservation of Momentum

There are cases in which

the linear momentum is

not conserved yet the

angular momentum of

the system is conserved.

This occurs when the

external forces creating

the linear impulse pass

through either the

center of mass of the

body or a fixed axis of

rotation, so the angular

impulse is zero.

12 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Conservation of Momentum

Another classic example

of conservation of angular

momentum. When the

spinning wheel is moved

so its axis of rotation

becomes vertical, the

frictionless turn-table

spins in a direction

opposite to the wheel

(i.e. conserving angular

momentum about the

vertical axis).

13 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 1

14 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 1

15 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 1

16 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 1

17 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Eccentric Impact

Eccentric impact occurs when the line connecting the mass centers

of the two bodies does not coincide with the line of impact.

18 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Eccentric Impact

Assuming that the bodies are frictionless, we find that the forces

they exert on each other are directed along the line of impact.

Besides the principle of impulse and momentum, we use the

coefficient of restitution:

Study the text book to understand

the method for deriving the

coefficient of restitution.

19 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 2

It is normally assumed that the impact creates forces

which are much larger than the non-impulsive weights of

the bodies.

20 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 2

21 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 2

22 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 2

23 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 3

24 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 3

25 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 3

26 Dinamika – Kinetika Benda Tegar – Impuls & Momentum Jurusan Teknik Mesin dan Industri FT UGM

Contoh 3