GERAK TRANSLASI DAN ROTASI

Gerak Translasi dan Rotasi

A. Momen Gaya

Momen gaya merupakan salah satu bentuk usaha dengan salah satu titik sebagai titik acuan. Misalnya anak yang bermain jungkat-jungkit, dengan titik acuan adalah poros jungkat-jungkit. Pada katrol yang berputar karena bergesekan dengan tali yang ditarik dan dihubungkan dengan beban.

Momen gaya adalah hasil kali gaya dan jarak terpendek arah garis kerja terhadap titik tumpu. Momen gaya sering disebut dengan momen putar atau torsi, diberi lambang t (baca: tau).

Gambar:

Menarik beban menggunakan katrol

t = F . d

Satuan dari momen gaya atau torsi ini adalah N.m yang setara dengan joule.

Momen gaya yang menyebabkan putaran benda searah putaran jarum jam disebut momen gaya positif. Sedangkan yang menyebabkan putaran benda berlawanan arah putaran jarum jam disebut momen gaya negatif.

Gambar:

Skema permainan jungkat jungkit

Titik 0 sebagai titik poros atau titik acuan.

Momen gaya oleh F₁ adalah t₁ = + F₁ . d₁

Momen gaya oleh F₂ adalah t₂ = – F₂ . d₂

Pada sistem keseimbangan rotasi benda berlaku resultan momen gaya selalu bernilai nol, sehingga dirumuskan:

∑ t = 0

Pada permainan jungkat-jungkit dapat diterapkan resultan momen gaya = nol.

∑ t = 0

- F₂ . d₂ + F₁ . d₁ = 0

F₁ . d₁ = F₂ . d₂

Pada sistem keseimbangan translasi benda berlaku resultan gaya selalu bernilai nol, sehingga dirumuskan:

∑ F = 0

Pada mekanika dinamika untuk translasi dan rotasi banyak kesamaan-kesamaan besaran yang dapat dibandingkan simbol besarannya.

Perbandingan dinamika translasi dan rotasi

Translasi		Rotasi
Momentum linier	p = mv	Momentum sudut*	L = I
Gaya	F = dp/dt	Torsi	 = dL/dt
Benda massa Konstan	F = m(dv/dt)	Benda momen inersia konstan*	 = I (d/dt)
Gaya tegak lurus terhadap momentum	F =  x p	Torsi tegak lurus momentum sudut	 =   L
Energi kinetik	Ek = ½ mv2	Energi kinetik	Ek = ½ I2
Daya	P = F . v	Daya	P =  . 

Analogi antara besaran translasi dan besaran rotasi

Konsep	Translasi	Rotasi	Catatan
Perubahan sudut	s		s = r.
Kecepatan	v = ds/dt	 = d/dt	v = r.
Percepatan	a = dv/dt	 = d/dt	a = r.
Gaya resultan, momen	F		 = F.r
Keseimbangan	F = 0	 = 0
Percepatan konstan	v = v0 + at	 = 0 + t
	s = v0t = ½ at2	 = 0t + ½t2
	v2 = + 2as	2 = + 2
Massa, momen kelembaman	m	I	I = miri2
Hukum kedua Newton	F = ma	 = I
Usaha	W =  F ds	W =   d
Daya	P = F.v	P = I 
Energi potensial	Ep = mgy
Energi kinetik	Ek = ½ mv2	Ek = ½ I2
Impuls	 F dt	  dt
Momentum	P = mv	L = I

^Contoh

_F2

₃₀^o

_O A

_B 37^o

^F₁

Dari gambar di atas, tentukan momen total terhadap poros O. Jarak

OA = 4m dan OB = 8 m, gaya F₁ = 10 N, dan F₂ = 6 N.

Jawab

Pada sistem keseimbangan translasi benda berlaku resultan gaya selalu bernilai nol,

Untuk gaya F₁

r₁ = OB = 8 m

^{Besar momen gaya} t₁ = ^F₁ ^sin ₁. r₁

= 10 . sin 37. 8

= 10 . 0,6 . 8

= 48 N.m

Arah momen gaya t₁ searah perputaran jarum jam

Untuk gaya F₂

r₂ = OA = 4 m

^{Besar momen gaya} t₂ = ^F₂ ^sin ₂. r₂

= 6 . sin 30. 4

= 6 . 0,5 . 4

= 12 N.m

Arah momen gaya t_{2 berlawanan arah} perputaran jarum jam

Momen gaya total adalah

t = t₂ + t₂

= 48 + 12

= 60 Nm

_Momen Kopel

Kopel adalah pasangan dua buah gaya yang sejajar, sama besar dan berlawanan arah. Kopel yang bekerja pada sebuah benda akan menghasilkan momen kopel yang mengakibatkan benda berotasi. Momen kopel disimbolkan M

_F F F ^-

₊

MFd

_d d d

^F F F

_{(a) (b) (c)}

Gambar (a) menunjukkan sebuah kopel bekerja pada suatu benda.Untuk

gambar (b) menunjukkan bahwa kopel bertanda positif jika putarannyasearah

dengan perputaran jarum jam, tetapi jika perputaran kopelberlawanan dengan arah perputaran jarum jam, maka kopel bertandanegatif seperti gambar (c).

Jika pada benda bekerja beberapa kopel maka resultan momen kopel total benda tersebut adalah

M = M₁ + M₂ + M₃ + … + M_n

Contoh

_F4

^F₁

^P _1m 2m 1m

Q

F₃

F₂

_Jawab:

Batang PQ panjangnya 4m. Pada batang tersebut bekerja empat buah gaya F₁ = F₃ = 5 N, dan F₂ = F₄ = 8 N, seperti tampak pada gambar di samping. Tentukan besar dan arah momen kopel pada batang PQ tersebut.

Gaya F₁ dan F₃ yang berjarak d = 3m membentuk kopel yang arahnya searah perputaran jarum jam (+) dan besarnya:

^M ₁ = ^{F x d} = 5 x 3 = 15 N m

Gaya F₂ dan F₄ yang berjarak d = 3 m membentuk kopel yang arahnya berlawanan arah perputaran jarum jam (-) dan _besarnya:

^M ₂ = ^{F x d} = 8 x 3 =  24 N m

_{Resultan momen kopel adalah:}

M = M₁ + M₂

= 15 + (  24)

=  9 N m

Tanda negatif (-), menunjukkan bahwa momen kopel resultan

arahnya berlawanan dengan arah perputaran jarum jam.

_{Koordinat Titik Tangkap Gaya Resultan}

Jika terdapat beberapa gaya yang bekerja pada bidang XY, maka setiap gaya tersebut dapat diuraikan atas komponen-komponennya pada sumbu-X dan sumbu-Y. Misalkan, komponen-komponen gaya pada sumbu-X adalah ^F_1x^, F_2x^, F_3x^,…,F_nx^{, yang jaraknya masing-masing terhadap} sumbu-X adalah ^y₁^{, y}₂^, y₃^,…,y_n ^.

^{Sedangkan komponen-}komponen gaya pada sumbu-Y adalah ^F_1 y ^, F_2y ^, F _3y ^, …,F_ny ^, yang jaraknya masing-masing terhadap sumbu-Y adalah x₁, x₂, x₃,…,x_n . Semua komponen gaya pada sumbu-X dapat digantikan oleh sebuah gaya resultan F _x yang jaraknya y_o dari sumbu-X, demikian juga semua komponen gaya pada sumbu-Y dapat digantikan oleh sebuah gaya resultan F _y yang jaraknya x_o dari sumbu-Y.

^{Koordinat titik tangkap dapat ditentukan dengan persamaan sebagai berikut.}

x_o = =

y_o = =

Jadi koornitat titik tangkap (x_o,y_o)

_Contoh

Y

F₂=5N

F₃=7N

_X

Dari gambar di samping, tentukan besar, arah, dan letak titik tangkap resultan.

_-3 -1 0 2 ³

F₁=-3N

^F₄^=-2N

Jawab

Semua gaya sejajar sumbu-Y, gaya ke atas positif dan ke bawah negatif, resultan gaya adalah:

F_y = F₁ + F₂ + F₃ + F₄

= -3 + 5 + 7 – 2 = 7 N (arah ke atas)

_{Letak titik tangkap gaya resultan adalah:}

x_o =

x_o =

x_o =

1. Momen Inersia Benda Tegar

Benda tegar adalah benda padat yang tidak berubah bentuk apabila dikenai gaya luar. Dalam dinamika, bila suatu benda tegar berotasi, maka semua partikel di dalam benda tegar tersebut memiliki percepatan sudut yang sama. Momen gaya atau gaya resultan gerak rotasi  didefinisikan sebagai berikut.

”Apabila sebuah benda tegar diputar terhadap suatu sumbu tetap, maka resultan gaya putar (torque, baca torsi) luar terhadap sumbu itu sama dengan hasil kali momen inersia benda itu terhadap sumbu dengan percepatan sudut”.

Dirumuskan sebagai berikut.

 =  F_i R_i Sin _i atau  = (  m_i R² _i ) ^. 

m_i R_i² disebut momen inersia atau momen kelembaman benda terhadap sumbu putar, yaitu penjumlahan hasil kali massa tiap partikel dalam suatu benda tegar dengan kuadrat jaraknya dari sumbu.

Dirumuskan:

I =  m_i . R_i²

Definisi lain dari momen inersia adalah perbandingan gaya resultan (momen) terhadap percepatan sudut.

Dirumuskan:

I =

maka  = I . 

 = I

Karena  = F . R dan  = I . 

maka  F . R = I . 

Percepatan tangensial adalah juga percepatan linier a, yaitu percepatan singgung tepi roda.

a =  . R

 =

persamaan menjadi :

 F . R = I .

Momen inersia harus dinyatakan sebagai hasil kali satuan massa dan kuadrat satuan jarak. Untuk menghitungnya harus diperhatikan bentuk geometri dari benda tegar homogen.

Tabel berikut menunjukkan momen inersia beberapa benda homogen.

THE LEGENT OF SITU BAGENDIT

THE LEGEND OF SITU BAGENDIT

Far away in an isolated village there was a young rich woman. The house that she had been living in was very big. Her wealth was plentiful. The young woman lived by herself. She didn’t have any friend at all.

“Wouw, I am very rich! Ha…ha…ha, I am the richest woman in this village!” said the young woman while she was looking at her gold and jewelries. It was so pity, that the young woman was very miserly. Her plentiful wealth never been used to help others.

“All of the wealth is mine, isn’t it? So what am I give it all to other for?” The young woman thought. However, many of villagers were poor. They lived in less condition. Sometimes some villagers were hunger, and didn’t get any food for days.

Because of the young woman miserly, the villagers called her Bagenda Endit. Bagendit meant the miserly rich person. “Bagendit, have mercy on me! My child has not eaten for few days”, said an old woman sadly.

“Hi, you crazy old woman! Get away from me!” yelled Bagendit threw the old woman away. Because the old woman didn’t want to go, Bagendit splashed her with water. Splash!, and all over the old woman body and her baby became wet.

Bagendit was a feeling less woman. She didn’t even have a little bit mercy to the old woman and her baby. She even got more angry. After that, she asked the old woman to get out of her house yard. And then, she was dragging her out of the yard.

Although Bagendit was very miserly, the village people kept coming in. The came for the water wheel. “No, I won’t let you to take away the water from my wheel! The water in the wheel is mine!” Bagendit yelled angrily.

“Ha…ha…ha…you’re all stupid! You think you just can take the water from my wheel!” Bagendit said while she was watching the thirst villagers outside the fence. Suddenly, a decrepit man was standing in Bagendit house yard. He was walking tottery to the wheel while holding his stick.

When the old man was trying to take the water, Bagendit saw it. Then, she hit the old man with a founder. “Have mercy on me Bagenda Endit! I want to take the water just for a drink”, said the old man when he was trying to get up.

Bagenda Endit kept beating the old man. And then, an astonishing thing happened. Suddenly the old man got up with a healthy body. He walked closer to Bagendit. He pointed his stick at the cruel rich woman’s nose.

“Hi, Bagendit, take the punishment from me!” said the old man loudly. Then he pointed at the wheel with his stick. Wus…byuur, the wheel was sprinkling the water swiftly. Not long enough, the water was flooding up. Bagenda Endit couldn’t save herself. She drawn with all of her wealth.

The village was disappeared. The thing that left was a wide and deep lake. The lake was named Situ Bagendit. Situ means a wide lake. It was named Situ Bagendit, because the wide lake came from a wheel that belongs to Bagendit.

STORY OF BANYUWANGI

BANYUWANGI

   Banyuwangi is a name of a district in the province of east Java, Indonesia. It is located in the earternmost part of the island of Java, next to Bali. Bali strait is between Banyuwangi and the island of Bali. The culture of Banyuwangi is unique because it is a blend of Javanese, Balinese and Madurese influence. Here is a well known legend about Banyuwangi.

Long time ago Banyuwangi was called Blambangan. It was a kingdom under a wise king who had a handsome and smart son. Raden Banterang was his name. He liked hunting very much. He often went to forest around Blambangan to hunt for animals. One day when he was in a forest he saw a deer. He chased it and the deer ran deeper into the forest. His horse was so good and strong that he left his guards behind. Unfortunately he lost the deer. As he took a rest under a big banyan tree suddenly a lovely lady appeared in front of him. Raden Banterang was very surprised to see a beautiful girl alone in the forest. He was suspicious that she was not a human being. So he asked her.

'Excuse me lovely lady, do you live around here?'

 'No, I don't. I'm from Klungkung, Bali. My name is Surati. I'm a princess, the daughter of the king of Klungkung. I need your help'

 ‘I will gladly help you, but please tell me what your problem is'

 'I'm in danger. There was a rebellion in Klungkung. The rebel killed my father but I could escape. My guards took me here but I lose them. Now I'm alone. I don't know where to go. I have no relative here. Please help me'

 'You are coming to the right person. I'm prince Banterang from the kingdom of Blambangan. I will protect you. Please come with me.'

 Then Raden Banterang took Surati home. He fell in love with her and then several months later he married her. One day when Surati was in the street he met a man. The man called him.

 'Surati, Surati'

 She was surprised to realize that the man was her brother Rupaksa. Rupaksa told her that it was Raden Banterang who killed their father. He came to Blambangan to take revenge and asked surati to join him. Surati was shocked but she refused to join.

 'I'm really shocked to hear the news. But I'm not sure. Raden Banterang is now my husband. He's very kind to me. He never hurts me. He's protecting me. As a good wife I will never betray him. It is my duty to serve him.'

'But he killed our father'. 'It is hard for me to believe it. When I met him he was here, not in Klungkung'

 Rupaksa was disappointed with her sister. He was also very angry to her.

 'OK then. I have to go now. But please keep my head dress. Put it under your pillow'

Rupaksa gave his head dress to his sister Surati. To respect her older brother Surati put it under her pillow. Several days later Raden Banterang was hunting in a forest when he met a man that looked like a priest. The man greeted him politely. Then he said something.

 'Your life is in danger. Someone has an evil intention to you'

 'Who is he?'

 "Your wife Surati'

 "Surati? How do you know?'

 'I am a priest. I have clear spiritual vision. I just want to save you. Search her room. If you find a head dress under her pillow then my words are correct. It is from a man who will help her kill you'

 'Thank you your Holiness'

 Raden Banterang was shocked. He was very angry to his wife then he immediately went home. When he got to the palace he immediately searched Surati's bed room. As he found the head dress under her pillow he was sure that the priest was right.

 'You are unfaithful wife. I know that you want me dead. This is the evidence. This is from a man who will help you kill me. Tell me who he is'

Surati was shocked and cried.

 'It is my brother's head dress. I met him several days ago when you went hunting. He gave me his head dress and told me to put it under my pillow. So I put it there to respect him. It is him who want to kill you, not me'

But Raden Banterang did not trust her. He gave her a death sentence. He took his wife to a river bank as he would stab his wife and throw her body into the river.

 'Before I die, let me say a few words'

 'Please do' 'After I die, just throw my body into the river. If the water become dirty and smelly, it means that I am guilty. But if the water become clear and fragrance come out of it, it means that I am innocence'.

 Then as Raden Banterang would stab her wife with a kris Surati threw herself into the river. Amazingly the water became clear and fragrance came out of it. Surati was innocent! Raden Banterang regretted his emotional behavior. Since then on he changed the name of his kingdom into Banyuwangi. Banyu means water and Wangi menas fragrance.