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Thursday, January 27, 2011

CADUNGOG-NADELA RONDALLA


CADUNGOG-NADELA RONDALLA ORCHESTRA 
HALAD SA SANTO NIÑO 2011

Harmonico Tayong Cadungog  -  banjo, saxophone
Quintiliano Cadungog Nadela banduria, trumpet
Fortunato Cadungog Nadela trumpet
Alfonso Nadela Ondoy, Jr Saxophone
Aurelio Ferolin Cadungog  -   tenor
Victoriano Nadela banduria
Marcelino Apag  -  Guitar
Boy Omotong  -  Bass 

Monday, January 10, 2011

LANGUAGE 102 : VERBS and COUNTING

According to the English grammar, any kind of verb has its own conjugation in the sentence such as :     
  go        goes      went        gone          going 
(root)   (present) (past) (past particle) (present participle)


  iku – to go  
 
  PRESENT TENSE      PRESENT TENSE    PRESENT TENSE  
   Affirmative        Negative       Interrogative    
    ikimasu       ikimasen        ikimasu ka            


  PAST TENSE      PAST TENSE       PAST TENSE      
 Affirmative       Negative       Interrogative
  ikimashita   ikimasen deshita   ikimashita ka

   mashoo – (lets)         nasai – (order form) 

   In Nihongo, all verbs have their own conjugations, too. Most of the
common verbs have the letter “u” at the end of the word, such as “iku”.
When the verb is used in the sentence, the letter “u” can get each
conjugation depending on the use in the sentence. English and Japanese
grammars are similar to each other when it comes to adjectives and
adverbs. Adjectives modifying nouns become complements in the sentence
and adverbs modify adjectives, verbs and other adverbs.
    
Kare wa Midsayap e (ni) ikimasu.  He goes to Midsayap.   
 S                        V       S   V      adverb

In the example, “to Midsayap” is written after the verb “goes” but
“Midsayap e” is written before the verb  “ikimasu”. Japanese adverbs
should be placed before the modified words.

1. Tanaka-san wa Cotabato e ikimashita.   Mr. Tanaka went to Cotabato.   
2. Kare wa gakkoo e ikimasen deshita.       He did not go to school.
3. Anata wa sakki doko e ikimashita ka.   Where did you go a while ago?
4. Watashi wa doko emo ikimasen deshita.  I didn’t  go anywhere.
5. Anata wa doko e ikimasu ka.           Where will you go? (do you go)
6. Utto-san wa Davao e ikimasen.      Mr. Utto will not go to Davao.    
7. Watashi wa Baguer e  ikimasu.      I go to Baguer.


   In Japanese as in other languages, there are regular and irregular
verbs. However, irregular verbs are very few in number – only two
basic verbs in all. All the rest of the verbs are regular with
exception of a few which are slightly irregular in some respects. 
   The Japanese verbs are of two kinds – “Godan” verbs and “ichidan”
verbs. The “godan” verbs, literally five row verbs, are so called
because they have different bases of conjugation. They are sometimes
called strong verbs or consonant verbs.  The “ichidan” verbs,
literally one row verbs, are so called because they have been thought
to have only one base of conjugation. They are sometimes called weak
verbs  or vowel verbs.
   The dictionary form of all “ichidan” verbs ends either in “iru” or
“eru”. All other regular verbs belong to the “godan” verbs with few
exceptions. The following verbs ends in “eru” or “iru” but they are
“godan” verbs ; hairu; kaeru; mairu; iru; kiru and shiru. 


Some common verbs

Group 1 (Godan verbs) Ends in u,ku,gu,su,tsu,nu,bu,mu and ru.
        -– replace final u with imasu

 1. nomu – to drink                  
 2. yomu – to read                  
 3. kau – to buy                          
 4. narau – to learn                      
 5. kaku – to write              
 6. isogu – to hurry              
 7. morau – to receive; be given 
 8. aruku – to walk                 
 9. hairu – to enter, get in       
10. okuru – to send                    
11. kiru – to cut ; to wear      
12. hanasu – to speak            
13. iku – to go                  
14. oyogu – to swim                 
15. wakaru – to understand; to be comprehensible  
16. kaesu – to return something; to give back       
17. toru – to take    
18. kaeru – to return
19. mairu – to come, go   
20. matsu – to wait         
21. shinu – to die    
22. aru – to be ;  exist    
23. asobu – to play         
24. machigau – to make a mistake 
25. kasu – to lend
26. tsukau – to use      
27. ageru – to give, to offer
28. taberu – to eat
29. iru – to need; to require   


Group 2  (Ichidan verbs) replace the final ru with masu.

1.  okiru – to get up ; to wake up          
2.  oshieru – to teach           
3.  miseru – to show , to let someone see
4.  hajimeru – to begin           
5.  akeru – to open            
6.  neru – to sleep ; to go to sleep
7.  wasureru – to forget
8.  miru – to see ;  to watch ; to look at
9.  shimeru – to shut


Group 3  (Irregular verbs)

1. kuru – to come            
2. suru – to do             
3. itte kuru – to go and come back    
4. katte kuru – to go and buy        
5. kiite kuru – to go and inquire    
6. totte kuru – to fetch          
7. motte kuru – to bring
8. mite kuru – to go and see   
9. karite kuru – to go and borrow
10. chumon suru – to order; to give an order
11. ryori suru – to cook; to prepare food
12. sentaku suru – to wash ( clothing )
13. benkyo suru – to study; to work 
                      
Dictionary form   Present tense    Past tense   
    nomu         nomimasu       nomimashita      
    kasu        kashimasu      kashimashita      
    aruku        arukimasu      arukimashita      


Present negative    Past negative     Meaning 
nomimasen        nomimasen deshita     drink
kashimasen       kashimasen deshita    lend
arukimasen       arukimasen deshita    walk 


 New words

 1. riyu – a reason         
 2. imi – meaning        
 3. koppu – a glass; tumbler   
 4. kotoba –a word;a language  
 5. machi – a town;a street    
 6. yane – roof        
 7. tarai – basin        
 8. ame – rain            
 9. shiyakusho –City hall  
10. chawan – cup          
11. saji – a spoon         
12. banjo – toilet         
13. dempo - telegram       
14. kagi – key             
15. hari – needles        
16. tamago – eggs          
17. denwa – telephone      
18. tama – ball         
19. ana – hole               
20. fujin – lady         
21. gohan – meal          
22. bara – rose          
23. nezumi – mouse         
24. koori – ice          
25. sakki – a while ago    
26. doko emo – anywhere    
27. doko nimo – nowhere
28. depaato – department
29. yuubinkyoku– post office
30. shokudoo – dining room
31. kado – corner
32. kusuriya – drugstore
33. Rizal kooen – Rizal Park
34. Santiago Yoosai–Fort Santiago
35. ginkoo – bank
36. shootengai–shopping district
37. taiikukan – gymnasium
38. keibajoo – horse racing track
39. teiryuujoo–stopping place,bus stop
40. Kokunai kuukoo – Domestic Airport
41. shoogakkoo – elementary school
42. taishikan – embassy
43. hakubutsukan – museum
44. Monbushoo – Ministry of Education
45. kootoogakkoo – senior high school
46. toshokan – library
47. chuugakkoo – junior high school
48. kissaten – coffee shop
49. eigakan – movie theater
50. asoko – over there
51. soko – there
52. koko – here
53. resutoran – restaurant    
54. toire – comfort room         
55. juutakuchi – residential area
56. rajio kasetto – radio cassette
57. suupaamaaketto – supermarket 
58. eki – station
59. yomikata– reading;how to read
60. ichiba – market ( place )
61. Bunka Kaikan – Cultural Center   
62. Kokusai kuukoo – International Airport
63. nyuugaku shaken – entrance examination
64. Nihon taishikan – Embassy of Japan
65. Gaimushoo – Ministry of Foreign Affairs
66. daigakusei – university student
67. gimukyooiku – compulsory education
68. daigaku – university / college
69. Kokusai kaigijoo – International Convention Center
                                             


   Nani ka        Nani ga        Nani mo    

  Nani ga is use as a subject, nani mo is use with negative
       expressions.
  Ya – ya and to are different in what they imply; i. e.
             Anata to watashi  == you and I

  Anata ya watashi  ==  you and I and other people like us.
  Nado  –  and so forth; and the like; and so on. 
           Nado is used as a final element in enumerating.

 Ex. Hon ya jibiki ya empitsu nado ga arimasu
       ==> There are books, dictionaries pencils and the like.


 Expression of Existence
   a) Ga arimasu – is used inanimate things
        (things which cannot move by itself)
   b) Ga imasu – is used for animate things.

1. Nani ka arimasu ka.   
       -- Is there anything?(inanimate object)
2. Nani ga arimasu ka.   
       -- What is there?(inanimate object)
3. Nani mo arimasen.     
       -- There is nothing. 
4. Nani ka imasu ka.     
       -- Is there one?(living object)
5. Dare ka imasu ka.      
       -- Is there anybody?
6. Dare ga imasu ka.      
       -- Who is there? (people only)
7. Iie, nani mo arimasen.
       -- No, there is nothing.(inanimate object)
8. Iie, dare mo imasen   
       -- No, there is nobody. (people only)
9. Nani ga imasu ka.     
       -- What is there? (animate object)


  Sentences :  
   1. Anatagata no gakkoo wa doko ni arimasu ka.        
        -- Where is your school ?
   2. Watashi no gakkoo wa Nihon Taishikan no soba ni arimasu. 
        -- Our school is beside the Japnese embassy.
   3. Rustan depaato wa doko ni arimasu ka.   
        -- Where is Rustan department store ?
   4. Intercontinental Hoteru no mae ni arimasu.              
        -- It is in front of Intercontinental  hotel.
   5. Taishikan no soba ni nani ga arimasu ka.               
        -- What is beside the embassy ?
   6. Ginkko ya depaato ya suupaamaketto nado ga arimasu.      
 -- There are banks, department store,supermarket and others.
   7. Kusuriya mo arimasu ka.                   
        -- Is there a drugstore too ?
   8. Iie, kusuriya wa arimasen.                      
        -- No, there is no drugstore.
   9. Anata no uchi wa doko ni arimasu ka.             
        -- Where is your house ?
 10. Watashi no uchi wa Tokyoo ni arimasu.            
        -- My house in in Tokyo.
 11. Tokyoo no doko ni arimasu.                     
        -- Where is it in Tokyo ?
 12. Tokyoo no Meguro ni arimasu.                
        -- It is in Meguro, Tokyo.
 13. Meguro yuubinkyoku no chikaku desu.            
        -- It is near the Meguro post office.
 14. Sato-san wa doko ni imasu ka.                 
        -- Where is Mr. Sato ?
 15. Toire wa asoko ni arimasu.                   
        -- The comfort room is over there.


Numerals and Counting

Rudiments of Japanese counting : Native Japanese counting system
  Hitotsu   –  one  
   Futatsu   –  two   
   Mittsu    –  Three  
   Yottsu    –  four      
   Itsutsu   –  five       
   muttsu    –  six 
   nanatsu   –  seven 
   yattsu    –  eight 
   kokonotsu –  nine 
   too       –  ten 

  Chinese Counting system  used by the Japanese
   rei            –  zero     ( 0 )
   Ichi        –   one   ( 1 )
   Ni          –   two      ( 2 )
   San         –   three ( 3 )
   Shi/yon     –   four     ( 4 )
   Go          –   five  ( 5 )     
   roku        –   six      ( 6 )
   shichi/nana –   seven ( 7 )
   hachi       –   eight ( 8 )
   kyuu        –   nine  ( 9 )
   juu            –   ten   ( 10 )
                                      
   juuichi      –  eleven             
   juuni        –  twelve             
   juusan       –  thirteen         
   juushi,etc,  –  fourteen , etc.     
   nijuu        –  twenty                 
   nijuuichi    –  twenty one         
   nijuuni      –  twenty two           
   nijuusan     –  twenty three     
   nijuusan     –  twenty three      
   nijuushi,etc.–  twenty four, etc.     
   sanju     –  thirty  
   sanjuu-ichi    –  thirty one 
   yonjuu     –  forty 
   yonjuu-ichi     –  forty one    
   guu/ gojuu    –  fifty     
   gojuu-ichi      –  fifty one 
   hyaku     –  one hundred        
   nihyaku     –  two hundred
   sen         –  one thousand 
   juuman     –  one hundred thousand    


  Numeral Classifiers or Counters

 1. nin – for persons               
 2. wa – for birds              
 3. hiki – for animals and fish        
 4. hon – for long or cylindrical objects 
 5. hai – for cupfuls, containerful         
 6. mai – for flat things           
 7. tsu – for pairs               
 8. en – suffix for Japanese money
 9. dai – for vehicles
10. ken – for houses and buildings
11. choo – for things with handles
12. soku – for things worn on feet,
           pairs of footwears
13. soo – for ships or boats
14. kai – for number of times,
          classifier for floors or storeys 
15. ko – for bundles, parcels


 Days of the week :

  1. Nichiyoobi – Sunday     
  2. Getsuyoobi – Monday     
  3. Kayoobi – Tuesday          
  4. Suiyoobi – Wednesday
  5. Mokuyoobi – Thursday
  6. Kinyoobi – Friday
  7. Doyoobi – Saturday


 Months of the Year :

  1. ichigatsu – January           
  2. nigatsu – February            
  3. sangatsu – March        
  4. shigatsu – April                   
  5. gogatsu – May       
  6.  rokugatsu – June      
  7. shichigatsu – July  
  8. hachigatsu – August
  9. kugatsu – September   
 10. juugatsu – October  
 11. juuicjigatsu – November
 12. juunigatsu – December


  Counting People

 1. hitori – one            
 2. futari – two    
 3. sannin – three      
 4. yunin – four    
 5. gonin – five
 6. rokunin – six
 7. sichinin – seven   
 8. hachinin – eight
 9. kyuunin – nine        
 10. juunin – ten

     

Counting in multiples of ten  

   10 – juu                
   20 – nijuu                  
   30 – sanjuu   
   40 – yonjuu  
   50 – gojuu    
   60 – rokujuu
   70 - nanajuu
   80 - hachijuu
   90 – kyuujuu    
  100 – hyaku    



  Counting in multiples of one hundred

   100 – hyaku                 
   200 – nihyaku     
   300 – sanhyaku           
   400 – yonhyaku      
   500 – gohyaku
   600 – roppyhaku     
   700 – nanahyaku    
   800 – happyaku
   900 – kyuuhyaku   
  1000 – sen     
   

   Counting in multiples of  one thousand

  1,000 – sen                                       
  2,000 – nisen                               
  3,000 – sanzen                    
  4,000 – yonsen                   
  5,000 – gosen    
  6,000 – rokusen 
  7,000 – nanasen
  8,000 – hassen
  9,000 – kyuusen
 10,000 – ichiman
100,000 – juuman
1,000,000  – hyakuman      
10,000,000 – senman 
100,000,000– ichioku



  How to tell the time?
   1. ichiji – 1 o’clock      
   2. niji   – 2 o’clock      
   3. sanji  – 3 o’clock          
   4. yoji   – 4 o’clock   
   5. goji   – 5 o’clock   
   6. rokuji – 6 o’clock   
   7. shichiji  – 7 o’clock       
   8. hachiji   – 8 o’clock       
   9. kuji      – 9 o’clock         
  10. juuji     – 10 o’clock
  11. juuichiji – 11 o’clock
  12. juuniji   – 12 o’clock


  “ji” – means o’clock  and  “nanji”  means  what time?   
 

   Counter for number of minutes:

   “fun” or “pun” is the numeral classifier for minute
    and “nanpun” means “how many minutes”

 1. ippun   – 1 minute            
 2. nifun   – 2 minutes                  
 3. sanpun  – 3 minutes            
 4. yonfun  – 4 minutes            
 5. gofun   – 5 minutes        
 6. roppun  – 6 minutes    
 7. nanafun – 7 minutes    
 8. happun  – 8 minutes    
 9. kyuufun – 9 minutes    
 10.juppun  – 10 minutes   
11. juugofun    – 15 minutes 
12. nijuppun    – 20 minutes 
13. nijuugofun  – 25 minutes     
14. sanjuppun   – 30 minutes
15. sanjuugofun – 35 minutes
16. yojuppun    – 40 minutes    
17. yonjuugofun – 45 minutes
18. gojuppun    – 50 minutes
19. gojuugofun  – 55 minutes
20. rokujuppun  – 60 minutes



 Examples :

  8:15  – hachiji juugofun       
  10:20 – juuji nijuppun       
  11:45 – juuichiji yonjuugofun    
  12:10 – juuniji juppun       
   9:10 – kuji juppun          
  3:35  – sanji sanjuugofun
  2:25 – niji nijuugofun 
  1:30 – ichiji han    
  5:30 – goji han 
  4:30 – yoji han




5 minutes before three  
            – sanji gofun mae   
10 minutes before 12    
            – juuniji juppun mae  
15 minutes before 10    
            – juuji juugofun mae
5 minutes past 9        
            – kuji gofun sungi.   
10 minutes past 12      
            – juuniji juppun sugi.   
It’s 5 minutes advanced 
             – gofun susundeimasu
It’s 10 minutes advanced
             – juppun susundeimasu
It’s 20 minutes advanced
              – Nijuppun susundeimasu
It’s 5 minutes late.    
          – gofun okureteimasu.
It’s 10 minutes late.   
             – juppun okureteimasu. 


NOTES:

  susudeimasu – means “It’s  advanced”.
  okurete – means “it’s late”    
 
  7:30 AM   –  gozen shichiji han 4:30 PM  –  gogo yoji han     
  11:30 AM – gozen juuichiji han  5:30 PM  –  gogo goji han 

  from 8:00 to 5:00 – hachiji kara goji made

  from 10:30 in the morning to 9:00 in the evening  
           – asa juuji han kara yoru kuji made


Dialogue I (taiwa)
 A: Ima nanji desu ka.             
        What time is it now?
 B: Rokuji desu.       
        It’s six o’clock.
 A: Nihon wa ima nanji desu ka. 
        What time is it now in Japan?
 B: Kuji desu.                
        It’s nine o’clock.
 A: Gogo kuji desu ka.           
        Is it 9 PM ?
 B: Hai, soo desu.       
        Yes it is. 



Dialogue II    
 A: Sumimasen, nanji desu ka ?    
        Excuse me, what time is it?
 B: Yoji nijuupun desu.    
        It’s  four twenty.
 A: Arigatoo gozaimasu.       
        Thank you.
 B: Doo itashimashite.       
        You’re welcome.


Days of the Month

 tsuitachi – 1st day        
 futsuka – 2nd day        
 mikka – 3rd day        
 yokka – 4th day            
 itsuka – 5th day            
 muika – 6th day        
 nanoka – 7th day        
 yooka – 8th day        
 kokonoka – 9th day         
 tooka – 10th day        
 juuichinichi – 11th day   
 juuninichi – 12th day        
 juusannichi – 13th day    
 juuyokka – 14th day        
 juugonichi – 15th day        
 juurokunichi – 16th day  
juushichinichi – 17th day    
juuhachinichi – 18th day    
juukunichi – 19th day    
hatsuka – 20th day    
nijuuichinichi – 21st day    
nijuuninichi – 22nd day    
nijuusannichi – 23rd day    
nijuuyokka – 24th day    
nijuugonichi – 25th day    
nijuurokunichi – 26th day
nijuushichinichi – 27th day
nijuuhachinichi – 28th day    
nijuukunichi – 29th day
sanjuunichi – 30th day
sanjuuichinichi – 31st day







 Vocabulary words:
 takusan –  many, a lot, plenty    
 ten–in    –  sales clerk      
 peso – peso            
 doru – dollar           
 sangurasu – sunglasses       
 koohii – coffee        
 ocha – green tea        
 koocha – black tea        
 miruku – milk            
 koora – coke            
 biiru – beer            
 pan – bread            
 gohan – rice ( cooked )    
 mizu – water            
 juusu – juice            
 choodo – exactly ; just    
 yoru – night; evening        
 kara – from            
 made – until ;  up to        
 susundeimasu – advanced/ be too fast    
 okureteimasu – late / be too slow      
 hiru – afternoon            
 shoogo – noon            
 kinoo – yesterday        
 ashita – tomorrow          
 konban – tonight           
 asatte – the day after tomorrow 
 tokidoki – sometimes          
 sengetsu – last month       
 konshuu – this week         
 raishuu – next week           
 kyonen – last year            
 nin – numeral classifier for persons. It is added to
       the Chinese-style numerals.  

 ikutsu – how many ( inquire about the number of things); 

 how old ( inquire about somebody’s age )

 mise – store; shop
 nannin – how many (persons)
 Ikura – how much
 kippu – ticket
 tamago – egg
 Nihon ryoori – Japanese food
 Chuuka ryoori – Chinese food
 Firipin ryoori – Filipino food
 Kankoku ryoori – Korean food
 Kaisen ryoori – seafoods
 ebi – fish
 sakana – fish
 yasai – vegetables
 niku – meat
 ima – now
 asa – morning
 mae – before
 sugi – past
 han – half past
 ji – o’clock
 pun/ fun – minute
 tokei – watch; clock    
 atteimasu – right; keep good time 
 kyoo – today
 yuube – last night
 ototoi – the day before yesterday
 mainichi – everyday
 kongetsu – this month
 raigetsu – next month
 senshuu – last week
 kotoshi – this month
 rainen – next year


 Sentences :

1. Anata wa nani o nomimasu ka.  
    -- What do you drink?
2. Watashi wa koocha o nomimasu. 
    -- I drink tea.
3. Anata wa biiru o nomimasu ka. 
    -- Do you drink beer?
4. Hai, biiru o nomimasu.      
   -- Yes, I drink beer.
5. Anata wa doo desu ka.
   -- How about you ?
6. Watashi mo biiru o nomimasu.  
   -- I drink beer, too.
7. I eat Japanese food.      
   -- Watashi wa Nihon ryoori o tabemasu.
8. She doesn’t eat Chinese food. 
   -- Kanojo wa Chuuka ryoori o tabemasen.




 AI  NI  TSUITE  (ABOUT LOVE)  

 1. ai suru – to love                 
 2. honto ni – truly                
 3. ai – love                   
 4. eien ni – forever                 
 5. watashi aishiteimasu ka – do you love me    
 6. deto – date                  
 7. koi ni chiru – fall in love        
 8. kudoku – woe                
 9. hatsukoi – first love            
 10. kokorogawari nai – will never change    
 11. Aishiteimasu – I love you           
 12. rabureta – love letter            
 13. konyaku – engagement            
 14. kisu – kiss                
 15. daku – hug                
 16. aibu – caress                 
 17. yakimochi – jealous             
 18. ki ga au – compatible            
 19. ki ga awa nai – incompatible        
 20. yakusoka – promise               
 21. aishiau – love each other            
 22. aishiaimasho – let us love each other    
 23. kekkon shimasho – let us get married     
 24. aishimasho – let us make love        
 25. shinkon ryoko - honeymoon       
 26. uwaki – unfaithful
 27. wakareru – break-up / separate
 28. rikon suru – divorce
 29. kinenbi – anniversary
 30. tanoshimimasu – enjoy
 31. funo – impotent
 32. dosei suru – live together out of wedlock
 33. ai no monogatari – love story
 34. konyakusha/ koibito – sweetheart 
 35. konyakuchu no otoko – fiance ( nobyo )
 36. konyakuchu no onna – fiancée ( nobya )
 37. Shinumade hanasanai – Till Death do us part
 38. renai kankei – love affair
 39. joji – romance
 40. koi wazurai – love-sick
 41. iro jikake – fake love
 42. renai kekkon – love marriage
 43. miai kekkon – arranged marriage
 44. au basho – meeting place
 45. kekkon mae no sekkus – premarital sex
 46. ninshin – pregnant
 47. ryuzan – abortion
 48. akachan o orosu – miscarriage
 49. aibiki – secret rendezvous

Saturday, January 8, 2011

ANALYTIC GEOMETRY : LINES



LINEAR EQUATIONS


The equation of a line can be written in the form

Ax + By + C = 0

where A, B and C are constants. This is called the general equation of a line. If A = 0 and B and C are not zero, the line is horizontal. If B = 0 and A and C are not zero, the line is vertical. If C = 0 and A and B are not zero, the line passes through the origin or the point (0, 0).


The Standard Equation of Lines

1. The point-slope form

y – y1 = m ( x – x1)

where m is a slope and (x1, y1) is a given point

2. The slope-intercept form

y = m x + b

where m is a slope and b is the y – intercept

3. The two-point form

y – y1 = [(y2 – y1)/(x2 – x1)] (x – x1)

where (x1, y1) and (x2, y2) are given points.

4. The intercept form

x/a + y/b = 1

where a is an x – intercept and b is the y–intercept.

5. The Normal form

x cos a + y sin a = p

where a is the slope of the normal and p is the distance of the
line from the origin.



Exercises (Problems in red color are assignment)

I. Find the equation of the line given the conditions:
1. P( 3 , – 2) , m = ¾
2. P(0, 5 ), m = – 2/3
3. P1(-1, 2 ) , P2(3, 4 )
4. P1( 0, -3 ), P2( 4, 0 )
5. m = ½ , b = -2
6. m = 3, b = - 3/2
7. a = 60 degrees, p = 3
8. a = 45 degrees, p = 3sqrt(2)
9. a = - ½ , b = 2
10. a = 3 , b = –5

II. Find the intersection of the following lines.
1.   x + 2y = 3
      2x + 8 = 3y

2.   x – 2y = 2
      2x – 3y = 5

3.   3x – y = 2
       x + 2y = 3

4.   x – y = 2
      5x + 3y = 2

5.   3x - 2y = 4
      2x + y = 12

III. The vertices of the triangle MNO are M(-1, 1), N(6, 2 ) and O(2, 5 ).

1. Determine the equation of (a) the side MN, (b) the medial from M
to NO and ( c ) the altitude from M to NO.
Ans. 7y – x = 8, 2y – x = 3 , 3y – 4x = 7

2. Find the equation of ( a ) the line through N parallel to MO,
( b) the line through O parallel to MN and ( c) the point of
intersection of the lines in ( a) and ( b).
Ans. 4x – 3y = 18, 7y – x = 33 , ( 9 , 6 )

3. The angle between MN and NO and between MN and MO.

IV.   The equation of two sides of a parallelogram 2x – 3y + 7 = 0
and 4x + y = 21, and one of the vertex is (–1, –3 ).
Determine the three other vertices.
Ans. (5 , 1) , ( 4 , 5), (– 2, 1)


Tuesday, January 4, 2011

Gravity, Force and Newton's Laws

GRAVITY AND FALLING BODIES

Gravity is one of the most familiar forces in nature; its effect on motion has been a subject of discussion for centuries. If an object is dropped from a great height, it can be observed that it falls with ever increasing speed until air resistance balances the effect of gravity, at which time it is said to have reached its terminal velocity. The term free falling bodies is used for objects that are moving freely under the influence of gravity, whether they are moving upward or downward. Any object that has no forces other than gravity acting on it is said to be in free fall, whether it is moving upward, downward, or in any direction.
It is found that if air resistance can be made negligible, then falling bodies will accelerate toward the center of the earth at the same rate, regardless of their mass. The value for the acceleration of gravity, given the symbol g, has been measured on earth as g = 9.8 m/s2. Galileo was the first to demonstrate that all bodies fall at the same rate if air resistance is negligible. ( It is often said that he did this by dropping objects of various masses from the Leaning Tower of Pisa, although there is no historical evidence that he actually used the famed tower.) Galileo’s recorded experiments settled some very old controversies about falling bodies, proving less-popular ideas to be correct.
Even more important than his discoveries about falling objects was his breaking away from old methods of determining truth. Galileo is often credited with being the Father of Modern Science because of his forceful demonstration of the value of observation and the discoveries he made through his ingenious experiments.
The following is a data from one of Galileo’s earliest experiments of a ball rolling down an inclined plane. His data were recorded on his notes. Galileo held a ball at the top of an inclined, grooved board and marked its position. Releasing the ball, he marked its position at the end of equal intervals of time. This is much like dropping a ball from a height, except that the effect of gravity has been “reduced” by allowing the ball to roll slowly down the inclined board rather than falling straight down. The position as measured by Galileo are given in the following table :
Time t (equal intervals)  t2    Distance,S(points)     S/t2
 1                        1           33              33.0
 2                        4          130              32.5
 3                        9          298              33.1
 4                       16          526              32.9
 5                       25          824              33.0
 6                       36         1192              33.1
 7                       49         1620              33.1
 8                       64         2104              32.9
 
The observations show what was already known quantitatively to Galileo and others of his time – that a rolling (or falling) object picks up speed as it continues to roll (or fall). However, the debt we owe to Galileo is for his careful measurements and his quantitative (mathematical) interpretation of the data. His object was to find a general rule describing how distances increase with increasing time of fall. After some trial and error, and with considerable insight, Galileo realized that the distance traveled was proportional to the square of the elapsed time.

S => h = ½ at2 ,

for vertical motion, a ==> g ( acceleration due to gravity )

Problems
1. A ball is thrown vertically up with an initial velocity of 15 m/s. How high does the ball rise from its projection point ? How long does it take for this rock to reach the highest point. How high does it go in 2 seconds ? in 3 seconds ? What is the time required to travel a height of 9 m ? 5 m ?
Ans. ( 11.48 m , 1.53 s , 10.4 m , 0.9 m , 0.82 s , 2.24 s , 0.38 s , 2.68 s )
2. A rock is dropped from a bridge 60 m high relative to the water on a river below. How long will it take for the rock to reach the surface of the water ? Calculate the positions of the rock 1s, 2 s, 3s after its release (a) relative to the bridge and (b) relative to the water.
3. A metal sphere is dropped from a 55 m high tower. Determine the height traveled by the sphere in the time interval from 1.5 s to 2.5 s.


FORCE, THE CAUSE OF ACCELERATION; NEWTON’S LAWS OF MOTION

In 1642, several months after Galileo died, Isaac Newton was born. At age 23, Newton developed his famous laws of motion, which completed the overthrow of the Aristotelian idea that had dominated the thinking of the best minds for 2,000 years.
Every acceleration ( change in velocity ) is caused by forces acting on a body. Conversely, if a body does not accelerate, then the total force acting on it is zero even if several forces are present. The apparently simple idea of cause and effect, that forces cause acceleration, didn’t come easily. It was and still is tempting to think of common phenomena as having no cause and simple being “the nature of things”. For example, “Why does water flow downhill?” seems stupid. Yet such question have a serious answers; in this case, the force of gravity causes water to flow downhill. The genius of Newton and others was not only in providing answers to basic questions. But also in simply being curious enough to ask basic questions.
Force is defined intuitively as a push or a pull. If an applied force is the only one thing acting on a body, then the body will accelerate in the same direction as the force. The strength of the force determines the magnitude of the acceleration. If several forces act on a body, then its acceleration is in the same direction as the total force and has magnitude proportional to the total force.

NEWTON’S LAWS OF MOTION
Galileo had a major influence in the study of motion. What Newton did was to write down the relationships between the force and motion in a form that could be used to predict and describe motion. Those relationships were found to apply in every circumstance where an experiment could be performed to test them and came to be known as Newton’s laws of motion.

The First Law : Inertia (mass). A body rest remain at rest or in motion in a straight line with a constant velocity unless acted upon by an outside force. The property of a body that causes it to remain at rest or to maintain a constant velocity is called inertia. The law was a refinement of Galileo’s idea --- in the absence of force, a moving object will continue moving. Galileo considered the tendency of things to resist change in motion as inertia. Inertia is a measure of how difficult is it to set a body into motion, or if it is already moving, how difficult is it to stop.


The Second Law: The acceleration produced by forces acting on a body is directly proportional to and in the same direction as the net external force and inversely proportional to the mass of the body.
a = Fnet/m ==> Fnet= ma , m ==> mass and a ==> acceleration

Newton’s Second law gives a precise definition of force that is consistent with our intuitive notions of a force as a push or a pull. A large force produces a large acceleration, a large mass requires a large force to make it accelerate at the same rate as a small mass, and a body will accelerate in the same direction as the net force on it.

The Third Law : Action – Reaction. Whenever one body exerts a force on a second body, the second body exerts a force back on the first that is equal in magnitude and opposite in direction. This is paraphrase as, “For every action there is equal and opposite reaction”.
One force is called the action force and the other is the reaction force. In every interaction, the forces always occur in pairs. The action and the reaction pair of forces make up the interaction between two things. We know that forces can cancel when they are equal and act in the opposite direction on the same object. Even though action and reaction are equal and oppositely directed, they do not cancel each other for they are acting on different bodies.
An example is a swimmer that exerts a force on the side of the pool. By Newton’s third law, the side of the pool exerts a force back on the swimmer – an external force. If friction is negligible between the swimmer and the water, she will then move in a direction opposite to the force she exerted on the side of the pool with an acceleration proportional to the force she exerted.
Cars accelerate forward by exerting backward forces on the ground. The reaction force of the ground acts as an external force on the car in the forward direction.
UNITS OF FORCE
1. Newton – is the force required to give a mass of 1 kilogram an acceleration of 1 m/ s2.
1 newton ( N ) = 1 kg-m/s2
2. Dyne – is the force required to give a mass of 1 gram an acceleration of 1 cm/ s2.
1 dyne = 1 g-cm/ s2
3. Pound – is the force required to give a mass of 1 slug an acceleration of 1 ft/ s2.
1 lb = 1 slug-ft/ s2 = 4.448 N

WEIGHT, FRICTION, TENSION, AND OTHER CLASSES OF FORCES
The weight of an object is the gravitational force exerted on it by the earth. When an object is dropped near the earth’s surface, it is accelerated by the gravitational force with an acceleration g, thus by Newton’s second law, the weight w becomes
w = mg. ==> m = w/ g

We see in this equation the relation between mass and weight : Weight is a force proportional to the mass of a body and g is the constant of proportionality. Here, g is taken to be positive, since the direction of forces are indicated with plus or minus sign. Weight depends on the location of the object, since the acceleration of gravity varies with location. As you go higher, g decreases so that weight also is decreasing. On the moon g ==> 1/6 of the earth’s g.

Center of gravity. The force of gravity on solids can be considered to act on a single point, called center of gravity (c.g.).For symmetrical objects, c.g. is at its geometric center. For asymmetrical objects, the c.g. is closer to the more massive part of the body. A closer related concept is the center of mass (c.m.), is the point at which all of the mass in a body can be considered to be located.

Newton’s Universal Law of Gravitation. The law states that there a force of attraction between any two masses that is proportional to the product of the masses and inversely proportional to the square of the distance between their centers of mass.
F = G m1 m2/r2
where G ==> Newton’s Universal constant of gravitation
G = 6.67 x 10–11 N.m2/kg2 ,
m1, m2 ==> masses in kg and
r ==> distance between the centers of mass in meter.
FRICTION
Friction is any force that opposes every effort to start to slide or roll one body over another body. Frictional forces are specially important to us in our daily lives, for without them we could not walk or hold things with our hands; cars wouldn’t be able to start or stop; nails and screws would be useless. Frictional forces are not fundamental forces like gravity or electromagnetism, but arise as reaction to other applied forces. Friction is proportional to the force exerted by one substance on another perpendicular to the surface between them---that is, the normal force (perpendicular force). The mathematical expressions are :
1. f = ukFN , uk==> coefficient of kinetic friction , FN ==>Normal force
2. f = us FN , us ==>coefficient of static friction
Equation 1 is used for the friction between moving substance and equation 2 for stationary substances.
Coefficient of friction is the ratio of the force of friction f to the normal force, FN.

PRINCIPLES OF FRICTION
1. The force of friction always act in a direction opposite to the direction of motion, for objects in
relative notion --- that is, sliding or rolling.
2. The frictional force is proportional to the normal (perpendicular ) force between the two surfaces in
contact.
3. Frictional force is approximately independent of the area of contact between the surfaces.
4. The frictional force depends on the particular material that make up the surfaces.

* Synovial fluid – a fluid which looks like blood plasma which lubricates the joints and limbs of the body.
ADVANTAGES OF FRICTION
1. Walking would be impossible without friction.
2. Pulley driven machines depend on friction for their operation.
3. Friction prevents belts from slipping off their pulley.
4. Friction between the tires and the road prevents skidding of vehicles.
5. Clutch, bolts and nuts, nails, screws, matches, brakes, etc. depends on friction.

DISADVANTAGES OF FRICTION
1. Wearing out of parts of machines, thus causing extra expenses for maintenance.
2. It causes expansion on machine parts and heat loses thus reducing the efficiency of machines.



TENSION
A tension is any force carried by a flexible string, rope, cable, chain, etc. Because the medium carrying the force is flexible, it can only pull and can exert no force except along its length. Tension comes from a Latin word meaning “to stretch thin”. In muscle systems the fibrous cords that carry forces exerted by muscles to other parts of the body are called tendons. Tension is due to the cohesive atomic and molecular electromagnetic forces acting in a string.
For a body suspended on a string with zero or constant speed upward or downward, the tension is given by T = w = mg. If the body accelerates downward on a string, the tension is given by T + ma = mg and if the body has an upward acceleration on a string, the tension is given by T = mg + ma.

Problems :
1. Determine the weight of a 50 kg person on earth. On the moon if g is 1/6 of the earth’s g.
2. Determine the mass of a box if a force of 80 N is able to accelerate it at 1.25 m/ s2.
3. Calculate the mass of a flea in grams if its weight is 5 x 10–6 N.
4. Find the acceleration of a rocket with mass of 1.2 x 106 kg if its engine exerts a net force of 2 x 106 N.
5. A 70 kg gymnast climbs on a rope. Determine the tension in the rope if
(a) he climbs at constant speed,
(b) he has an upward acceleration of 0.5 m/s2 ; and
(c) he goes downward with a downward acceleration of 0.5 m/s2.
6. Determine the force of gravitation between the earth and the sun and between the earth and the
moon.
mE = 5.99 x 1024 kg , mS = 1.99 x 1030 kg , mM = 7.36 x 1022 kg
Searth–sun = 149.6 x 109 m , Searth–moon = 3.84 x 108 m , Radius of earth = 6.367 x 106 m
7. A man weighs a fish of mass m on a spring scale attached to the ceiling of a elevator. Show that if the elevator accelerates in either direction , the spring scale gives a reading different from the weight of the fish. What is the reading on the scale if the elevator moves up or down at constant speed?
8. How much torque do you exert if you push perpendicularly on a door with a force of 30 N at a distance of 0.85 m from its hinges?