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Proposal of an Experimental Apparatus for Learning about Torque

Donggeul Hyun

Department of Science Education, Teachers College, Jeju National University, Jeju 63294, Korea

Hunkoog Jho

Department of General Education, Dankook University, Cheonan 31116, Korea (Received 17 September 2015 : revised 21 October 2015 : accepted 31 October 2015)

From primary to tertiary education, the concept of torque is learned as one of key concepts in physics education. The concept of torque can be understood properly on the basis of the concept of the vector product. Also, the concept of torque is utilized commonly as an important means for understanding the abstract concept of the vector product because the concept of torque is physical.

In this study, the problems occurring in learning related to torque were analyzed, and an apparatus for torque experiments was developed for solving those problems. The apparatus provides students with useful ways to learn the concepts of torque the vector product in a visible way. This study also suggests some experiments that can be done with the apparatus.

PACS numbers: 01.40.Fk, 07.10.Pz, 01.50.My

Keywords: Torque, Experimental apparatus, Vector, Vector product

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PACS numbers: 01.40.Fk, 07.10.Pz, 01.50.My Keywords: [taË>j˵, z´+«> ©œu, 7˜', 7˜'YL

∗E-mail: [email protected]

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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?/Hú¨î ×æd”‚õ s\ f”“§ 



Húf” ×æd‚”s Õª#Q4R e”. [taË>j˵›'aº z´+«>\"fH Ã

º¨î ×æd”‚õ úf” ×æd”‚_ “§ &hs r„óøÍ_ r„

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¼l\¦
?/H Â҂ÃÌ÷&#Q e”. ttóøÍõ r„óøÍ



s\H ü<39 +þAI_ Z…#QaA`¦ z0> ¿º óøÍ s_ 

¹1

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Ž

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 Œ•6 x~½Ó†¾Óܼ–Ð yŒ••¸–Ð
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\

 sØԍH ‚ìr_ ß¼lü< ~½Ó†¾Ó`¦ 0Au 7˜' ~r`¦
·p



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¦ (d)\¦ 6 x# Œ j˵`¦Œ•6 xr~´ M: j˵_ Œ•6 x&h_ 0A u

 7˜'ü< j˵ 7˜' s_ yŒ••¸\¦ 8£¤&ñ½+É Ãº e”. j˵_



Œ

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Q4R r„óøÍ\ Â҂Ã̝)a_ èHFKõ yŒ••¸óÍ_ø yŒ• èHFK`¦

~ 1

> “d”½+É Ãº e”•¸2Ÿ¤÷&#Q e”. Õªo“¦ j˵_ Œ•6 x@/_ 8

£¤€\H ^¦Ô–àН)a “¦&ñûZ (bolt for fixing force action slider) (g) Â҂ÃÌ÷&#Q e”#Q j˵_ Œ•6 x@/\¦r„óøÍ_ "é¶

H0Au\ “¦&ñr~´ ú e”. ¢¸ôÇ tt@/\Hr„ ó

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•

¸2Ÿ¤>h`‚d” r„ $tûZ (rotation stop bar) (h) ú¨î

»

¡¤“ x»¡¤_¢,aAᤁ©œo_ 0Aü< A\ Â҂ÃÌ÷&#Q e”#Q



r„ $tûZ\¦ ?/§4`¦ M: r„ H r„óøÍs $tûZ\ ]X

8ú¤ %i`¦M: r„`¦Æ3"Ò>)a. s z´+«> ©œu\"f r

„

 $tûZHr„óøÍs r„ $ tûZ\ Œ•6x  H j˵\ @/ ô

Ç ìǿŒ•6 x§4\ _ôÇ [ta>jË˵\ _ #Œ r„óøÍ_ &ñt©œ I

“ ¨î+þAœI©\ e”H `כ¦ ˜Ð#ŒÅҍH כ s m s z

´+«> ©œu_ 3q&lh ©œ éߖíH yr„óøÍs r„ H~½Ó†¾Óõ [

taË>j˵_ ß¼l\¦ Ö¼z> l 0AôÇ כ sÙ¼–Ð, r„óø͓Ér Å

Ò#Q” [taË>j˵\ _ #Œ >5Åq r„`¦ “¦ e”Hכ ܼ–Ð ç

ߖÅÒK ôÇ. r„ $tûZ 0A\H r„óøÍ_ r„~½Ó†¾Ó s

 o¶ú˜³ð–Ð ³ðr÷&#Q e”. r„óøÍs r>~½Ó†¾Óܼ–Ð r

„

½+É M: [taË>j˵_ ~½Ó†¾Ó“Érš¸Ér’<HZOgË:\  x − y¨î€ _

 Ï?@~½Ó†¾Ó“ −z» ¡¤ ~½Ó†¾Ós 9, r„óøÍ`¦Írøì>~½Ó†¾Óܼ

–

Ð r„½+É M: [taË>j˵_ ~½Ó†Ó“¾Ér x − y¨î€_ 0A~½Ó†¾Ó“ z»¡¤

~

½Ó†¾Ós.

#

ŒQ >h_ s1lxd” j˵_ Œ•6 x@/\¦y©œ^‰ %i½É`+¦ Hr

„

óøÍ\ “¦&ñr& #ŒQ >h_ j˵s Óüt^‰\ ½+É M:, s[þt _

 ½+Ë$í [ta>jË˵õ ¨î+þAs ÷&•¸2Ÿ¤¢¸ Ér j˵`¦ #Œ r

„

»¡¤ O\ @/ôÇ ½ +Ë$í [taË>jµ`˦ ½¨½+É Ãº e•”¸2Ÿ¤ ÷&#Q e”



. Fig. 4\"f ˜Ð#ŒÅҍHâĺH x − y¨î€ ?/_ [j >h_ jË

µ ~F1, ~F2, ~F3 r„óøÍ_ #ŒQ 0Au\  tH âĺ_ s

 [j j˵\ _ôÇ ½+Ë$í [taË>jµ ~τË r`¦½¨ Hâĺs. [j

>

h_ j˵ ~F₁, ~F₂, ~F₃\ _ôÇ r„»¡¤ O\ @/ôÇ ½Ë$+í [taË>j˵

~

τr`¦©œWr~´ ú e”H¸ ¢
_ r„»¡¤ O\ @/ôÇ [taË>

jË

µ ~τR, 7£¤ [ta>jË˵\ _ôÇ Óüt^‰_ ¨î+þA›¸|  ~τr+ ~τR= 0\ s

ÀÒ> H ¢¸ 
_ j˵ ~F_R¦`r„óøÍ_ ú¨î ×æd”‚



©œ_ ôÇ 0Au 7˜' ~rR\ úfs” ÷&•¸2Ÿ¤ôÇ. j˵ ~F_R\ _

ôÇ [taË>j˵ ~τ_Rr ~ɓτR= rRFRzˆ_ éߖíHÇ +ôþAI–Ð
è­q Ã

º e”ܼ 9, r„»¡¤ O\ @/ôÇ [j j˵_ ½+Ë$í [taË>j˵ ~τrH Ó

ü

t^‰_ ¨î+þA›¸|  ~τ_r+ ~τ_R = 0`¦ H–Ð [taË>j˵ ~τ_R_ ß¼ l

ü< °ú “¦ Õª ~½Ó†¾Ós ìøÍ@/“, 7 £¤ ~τr= −~τR= −rRFRzˆ_ é

ߖíHôÇ +AIþ–Ð ½¨½+É Ãº e”. 7£¤r„»¡¤ O\ @/ôÇ [j j˵ _

 ½+Ë$í [taË>j˵ ~τr_ ß¼lH τr= rRFRs 9, Õª ~½Ó†¾Ó“Ér jË

µ ~F_R\¦]j 
 Õª ß¼l\¦yŒ™™èr& q¨î+þA©œI\ s

ØÔ!3`¦ M: r„óøÍ_ r„~½Ó†¾Ó`¦ ›'a¹1Ï #Œ &ñ½+É Ãº e”

. s z´+«> ©œu\"f r^Rü< F^RHpt_ [taË>j˵`¦½¨

l 0AôÇ >íߖ'ܑ¼–Ð •¸{9)a.

{9

ìøÍ&hܼ–Ð yœ^©‰“ Óüt^‰\ •6Œ x H #ŒQ j˵[þts ôÇ

¨î

€“ x − y¨î€\ e”€, y• jŒ ˵\ _ôÇ [taË>j˵“Ér 0A u

 7˜'_ ß¼l, j˵_ ß¼l, Õªo“¦ 0Au 7˜'–ÐÂÒ' j˵ 7˜' ~½Ó†¾Ó_ e± yŒ•\ _ #Œ &ñ)a. 7£¤,yŒ• j˵\ _ ô

Ç r„»¡¤ O\ @/ôÇ [taË>j˵_ ~½Ó†¾Ó“Ér š¸f z»” ¡¤ ~½Ó†¾Ós



. 7£¤ y©œ^‰_ ôÇ 0Au 7˜' ~rⁱ\ Œ•6 x H j˵ ~Fi\ _ ô

Ç r„»¡¤ O\ @/ôÇ [taË>jµ ~τË iH ~τi = f (ri, Fi, θi) ˆZ–Ð

è­q ú e”. #Œl\"f l ri, F_i, θ_iH yŒ•yŒ• 0Au 7˜ '

 ~ri, j˵ ~Fi, 0Au 7˜' ~ri ~_ ½Ó†¾Óܼ–ÐÂÒ' j˵ ~Fi_ ~½Ó

Fig. 4. (Color online) The torque ~τ caused by a force ~F acting at the position vector ~r with a clockwise angle θ on the rotary plate (a) and the equilibrium state of the rotary plate by another torque ~τ_R caused by a force ~F_Racting at the position vector ~rR with a counter-clockwise right angle θR for measuring the torque ~τ (b).

†

¾Ó_ e± yŒ•s. ëߖ{9 n>h_ jµsË ôÇ Óüt^‰\ Œ•6 xôÇ

€

 s j˵[þt _\ ôÇ r„»¡¤ O @\ /ôÇ ½+Ë$í [taË>j˵ ~τ_rH

~ τ_r=Pn

i=1 f (r_i, F_i, θ_i)ˆz–Ð"f, ½+Ë$ [í taË>j˵ ~τ_r %ir Óüt^‰

\

 Œ•6 x H j˵[þt_ 0Au 7˜'_ ß¼l, j˵_ ß¼l, Õªo

“

¦ 0Au 7˜'–ÐÂÒ' j˵ 7˜'_ ~ӆ½¾Ó_ e± y•\Œ _”>rôÇ



Hכ `¦_pôÇ. sכ “Ér Óüt^‰\ Œ•6 x H j˵[þt_ Œ•6 x

&

h_ 0Au 7˜'_ ~½Ó†Ós¾ ØÔ8•¸ Fig. 3\"f ˜Ð#ŒÅÒ



Hü< °ú s 
_ 0Au 7˜'\¦
?/H f”‚ 0A\ Óüt

^

‰\ Œ•6 x H j˵[þt`¦
è­q ú e”ܼ 9, s[þt j˵\ _ ô

Ç ½+Ë$í [taË>j˵`¦s ƒ½¨\"f >hµ1ϝ)az´+> «©œu\¦6 x

#Œ 8£¤&ñ½+É Ãº e”Hכ `¦_pôÇ.

s

 ƒ½¨\"f >hµ1Ïa [)taË>j˵ z´+«> ©œuHœí1px†<Ɠ§ ú ï

r\"fH œ¼ª€1Ï $Ö¦_ "¶éo\¦ sK l 0A #Œ, ×æ†<Ɠ§ Ã

ºïr\"fH tYU_ "é¶o\¦ sK l 0A #Œ 6 x½+É Ãº e”

. “¦1px†<Ɠ§ úïr\"fH [taË>j˵_ >h¥Æ`¦sK “¦, [

taË>j˵`¦ 8¤&£ñ  9, #ŒQ j˵[tþ\ _ôÇ [taË>j˵[þt_ ¨î+þA



©œI\¦sK l 0AôÇ z´+«> ©œu–Ёָ6 xÉ Ã½+º e”. Õªo

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jË

µ[þt\ _ôÇ Ótü^‰_ ¨î+þA ›¸| ÷rëߖ m 7˜'YL_ >h

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IV. ô v Ú k ÈÌ   ö8 ý ] K ¡ ©ù p ø p §  ü” X ¢ ÷ m Ç] M ö8 ý < gŽ ˜ m

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&

ñ†<Êܼ–Ð+‹ [taË>j˵\ _ôÇ Ótü^‰_ ¨î+þA›¸| `¦sK½+É Ãº e”

H z´+«>[þt`¦]jîߖôÇ.



™

‹“@ 1: žâ·ÍÏÏט+ ¥´Ça ™‹“@ ô

Ç y©œ^‰“ Óüt^‰_ r„»¡¤ O–ÐÂÒ' 0Au 7˜' ~r“ &h\ 0

Au 7˜' ~r_ ~½Ó†¾Óõ θ_ yŒ•Ü¼–Ð j˵ ~F Œ•6 x½+É M: Óüt^‰

\

 [taË>j˵ ~τ l)a. sM: Ótü^‰\ Œ•6 x H [taË>j˵

~

τ_ ~½Ó†¾Ó“Ér 0Au 7˜' ~rõ jµ ~ËF sÀҍH 2" é¶ ¨î€\ Ã

ºf”ôÇ ~½Ó†¾Ós 9 Õª ß¼lH r F sin θ –Ð &ñ_÷&H Óüto|¾Ó Ü

¼–Ð 0Au 7˜' ~rõ j˵ ~F_ ß¼lü< s[þt_ e± yŒ• θ\ _

#Œ &ñ)a. sQôÇ [taË>j˵ ~τH 0Au 7˜' ~rõ j˵ 7˜ '

 ~F_ 7˜'YL“ ~τ = ~r × ~F–Ð ú†<Æ&h“ ~½ÓZO“ ¿º 7˜ _

 7˜'YLܼ–Ð ³ð‰&³½+É Ãº e”.

[

taË>j˵ ~τ\¦hµ>1ϝ)az´+> «©œu\¦6 x #Œ 8£¤&ñ l 0A ô

Ç z´+«> ]X ü< ~½ÓZO“Ér6§£õ °ú . r„óøÍ_ ú¨î ×æd”

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 Œ•6 x@/_ ×æd”s 0Au •¸2¤ jŸ ˵_ Œ•6 x/\@¦ r„óøÍ

\

 “¦&ñûZ\¦ 6 x #Œ “¦&ñôÇ. j˵_ Œ•6 x@/_ ×æd”}Œ•

@

/\ ƒ“¦o\¦ :ŸxK 6 xú^o= $Ö¦`¦ 6x  #Œ j˵ ~F \¦ 0

Au 7˜' ~rü< °ú “Ér ¨î€“ x − y¨î€ ?/\"f 0Au 7˜'

~

rü< yŒ••¸ θ_ ~½Ó†¾Óܼ–Ð ôÇ. sM: r„óøÍ\ Œ•6 x H jË

µ ~F\ @/ #Œ j˵_ Œ•6 x&h_ 0Au 7˜' ~rü< yŒ••¸ θ\ 



 r„óøÍs #QÖ¼ ~½Ó†¾Óܼ–Ð r„ HtSX‰½“+É Ãº e”.

Õ

ªo“¦ j˵`¦>5Åq °ú “ÉrœI©–Ð  €"f tt@/_ r„

$

tûZ\¦6 x #Œ r„`¦tñ&r(`¦M: [taË>j˵ ~τ_ ~½Ó

†

¾Ó`¦ Fig. 4_ (a)\"fü< °ú “Ér©œI\¦SX‰“½+É Ãº e”. 7£¤



r„óøÍs r>~½Ó†¾Óܼ–Ð r„ #Œ Fig. 4_ (a)\"fü< °ú  s

 r„óøÍ_¢,aAᤠ=åQÂÒìrs 0AAá¤_ r„ $tûZ\ ]X 

>

|¨cM: [taË>j˵ ~τ_ ~½Ó†Ó“¾Ér x − y¨î€_ AAá¤Ü¼–Ð †¾Ó

Fig. 5. (Color online) The torque ~τ_i caused by a force ~F_i acting at the position vector ~r_i with a clockwise angle θ_i = 90^◦ on the rotary plate (a) and the equilibrium state of the rotary plate by another torque ~τR caused by a force ~FR

acting at the position vector ~rR with a counter-clockwise right angle θR for measuring the torque ~τi (b).

H ~½Ó†¾Ó“ −ˆ z~½Ó†¾Ós. ëߖ{9 r„óøÍs ìøÍr>~½Ó†¾Óܼ

–

Ð r„ #Œ r„óøÍ_¢,aAᤠ=åQÂÒìrs AAá¤_ r„ $ t

ûZ\ ]X >)a€, [taË>j˵ ~τ_ ~½Ó†¾Ó“Ér x − y¨î€_ 0

AAá¤Ü¼–Ð †¾Ó H ½Ó†~¾Ó“ +ˆzӆ~½¾Ós )a. s H ¢¸ôÇ š¸

 É

r’<HZOgË:`¦6 x #ŒSX‰“½+É Ãº e”.

[

taË>j˵ ~τ r„óøÍ\ Œ•6 x H j˵ ~F_ Œ•6 x&h_ 0Au 7˜' ~rü< j˵_ Œ•6 x~½Ó†Óõ¾_ yŒ••¸ θ\  #QbG>  



Ht &ñ|¾Ó&Üh¼–Ð 8£¤&ñ 9€  6£§õ °ú “rÉ ]X ü< ~½ÓZO s

 €9כ¹ . Fig. 4_ (b)ü< °ú s r„óøÍ_ r„»¡¤ O–Ð Â

Ò'¢,aAá¤ÒìÂr_ e”__ 0Au 7˜' ~r^R\ s1lxd” j˵_ Œ• 6

x@/\¦“¦&rñv“¦ Õª ×æd”}Œ•@/\ j˵_ Œ•6 x~½Ó†¾Ó θ^R 0

Au 7˜' ~rR\ úf”s ÷&•¸2Ÿ¤, 7£¤ θR = 90^◦“ yŒ••¸–Ð jË

µ ~F_R`¦•6Œ x #Œ r„óÍ_ø ×æd” ú¨î‚s ttóøÍ_ ú

¨î

»¡¤“ x»¡¤õ {9uôÇ ©œI\"f ¨î+þA©œI ÷&•¸2Ÿ¤ ôÇ



. sM: [taË>j˵ ~τ₁_ ß¼lH¨î+þA›¸| “ τ = rF sin θ = r_RF_R_ a'›>d”`¦ s6x  #Œ ½¨½+É Ãº e”. Õªo“¦ [taË>

jË

µ ~τ_ ~½Ó†¾Ó“Ér j˵ ~FR\¦]j 
 Œ•> %i`¦M: r„ ó

øÍ_ r„ H ӆ½~¾Ó`¦ z´+«>&hܼ–Ð ›'a¹1Ï #Œ SX‰“½+É Ãº e”

. s z´+«> ©œu\"f r^RFRH pt_ [taË>j˵_ ß¼l

“

 τ = rF sin θ\¦½¨ l 0AôÇ >í–'ߑܼ–Ёָ6 x)a.



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 j˵_ Œ•6 x&h_ 0Au 7˜' ~rõ Œ•6 x H j˵ ~F_ ß¼lü<

~

½Ó†¾Óõ_ ›'a>\¦ ·ú˜?/l 0AôÇ z´+«>_ ] Xü< ~½ÓZO“Ér



6£§õ °ú . ĺ‚ j˵_ Œ•6 x&h_ 0Au 7˜' ~r_ ß¼l

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taË>j˵ ~τ\ puH%ò†¾Ó`¦ ·ú˜l 0A #Œ, r„óøÍ_ r

„

»¡¤ O–ÐÂÒ' š¸ÉrAá¤Ü¼–Ð j˵_ Œ•6 x&h_ 0Au 7˜' ~r_ ß

¼l\¦ {9&ñôÇ ß¼l–Ð 7£xrv€"f yŒ• âĺ_ 0Au 7˜ '

 ~rⁱ\ j˵_ •6Œ x@/\¦r„óøÍ\ “¦&ñr†. Õªo“¦ yŒ•

â

ĺ\_ j˵_ Œ•6 x&h_ 0Au 7˜' ~rⁱ\"f {9&ñôÇ yŒ••¸ θ_i(θ_i = constant)ܼ–Ð {9&ôñÇ j˵ ~F_i (F_i = constant)\¦ jË

µ_ Œ•6 x@/_ ×æd”}•@Œ/\ Œ•6 xrv€"f Fig. 5_ (a)\

"

fü< °ú s r„óÍ_ø r„~½Ó†¾Ó`¦ ›'a¹1Ï #Œ s ©œS!\"f _

 [taË>j˵ ~τi ~_ ½Ó†¾Ó`¦‰SX“ôÇ. Fig. 5_ (a)“ âĺH {9

&ñôÇ j˵ ~Fi 0Au 7˜' ~rⁱ–ÐÂÒ' r>~½Ó†¾Óܼ–Ð θⁱ = 90^◦_ {9&ñôÇ yŒ••¸\¦Ä»tôÇ âĺs.

{9

&ñôÇ j˵ ~F_i 0Au 7˜' ~ri–ÐÂÒ' {9&ñôÇ yŒ••¸ θi–Ð



Œ

•6 x Hâĺ j˵_ Œ•6x& h_ 0Au 7˜' ~rⁱ_ ß¼l_ o

\

 Ér [taË>j˵ ~τi_ ß¼l\¦F9€ ĺ‚ Fig. 5_ (b)ü<

° ú

 s r„óøÍ_ r„»¡¤ O–ÐÂÒ'¢,aAᤠÂÒìr_ e”__ 0A u

 7˜' ~rR\ j˵_ Œ•6 x@/\¦ “¦&ñrv“¦ Õª ×æd”}Œ•@/\ jË

µ_ Œ•6 x~½Ó†¾Ó θR 0Au 7˜' ~rR\ úf”s ÷&•¸2Ÿ¤, 7£¤ θR = 90^◦“ y••Œ¸–Ð 6 xú^o= $Ö¦–Ð j˵ ~FR¦` #Œ r„ ó

øÍ_ ×æd” ú¨î‚s tt@/_ ú¨î»¡¤“ x»¡¤õ {9uôÇ ©œ I

\"f ¨î+þA©œI ÷&•¸2Ÿ¤Çô. sM: yŒ• Äâº_ [taË>j˵

~

τ_i_ ß¼lH+î¨þA›¸| “ τ = r_RF_R_›'a>d`”¦s6 x #Œ

½

¨½+É Ãº e”. ¢¸ôÇ [taË>j˵ ~τ_i ~_½Ó†¾Ó“Ér j˵ ~F_R\¦]j 


 Œ•> %i`¦ M: r„óøÍ_ r„ H~½Ó†Ó`¾¦ z´+«>&h Ü

¼–Ð ›'a¹1Ï #Œ SX‰“½+É Ãº e”. r^RFRH ¡_ú· z´+«>\"f ü

< °ú s pt_ [taË>j˵_ ß¼l“ τi\¦½¨ l 0AôÇ >íߖ '

‘ܼ–Ð Ö¸6 x)a. sM: r„óøÍ_ r„»¡¤Ü¼–ÐÂÒ' 0Au 7˜' ~ri_ ß¼l\  [ta>jË˵ ~τ_i_ ß¼l qYV #Œ 7£x

†<Ê`¦ z´+«>&hܼ–ÐSX‰“½+É Ãº e”.

Fig. 6. (Color online) The resultant torque ~τ_rcaused by the two forces ~F₁and ~F₂: ~F₁ acting at the position vector ~r₁ with a counter-clockwise angle θ1 on the rotary plate and ~F2 acting at the position vector ~r2 with a clockwise angle θ2 on the rotary plate (a), and the equilibrium state of the rotary plate by the torque ~τR caused by a force ~FR acting at the position vector ~rR with a counter-clockwise right angle θR for measuring the torque ~τr(b).



r„óøÍ\ Œ•6 x H jµ ~Ë F_ ß¼l [taË>j˵ ~τ\ pu



H %ò†¾Ó`¦ ·ú˜l 0A #Œ, r„óøÍ_ r„»¡¤ O–ÐÂÒ' {9

&

ñôÇ 0Au 7˜' ~rⁱ (ri = constant)\"f {9&ñôÇ ~½Ó†¾Ó θi

(θi = constant)ܼ–Ð j˵_ Œ•6 x@/_ ×æd”}Œ•@/\ Œ•6 x 



H j˵ ~Fi¦ {\9&ñôÇ ß¼l–Ð 7£xrv€"f j˵`¦ôÇ. s M

:, Fig. 5_ (a)_ ]X ü< ~½ÓZOܼ–Ð r„óøÍ_ r„~½Ó†¾Ó

`

¦›'a¹1Ï #Œ yŒ• âĺ_ [taË>j˵ ~τi_ ~½Ó†¾Ó`¦‰SX“ôÇ. Õª o

“¦ [taË>j˵ ~τi_ ß¼lHyŒ• âĺ Fig. 5_ (b)ü< °ú s



r„óøÍ_ r„»¡¤ O–ÐÂÒ'aA,¢á¤ÂÒìr e_”__ 0Au 7˜'

~

r_R\ j˵_ •6Œ x@/\¦“¦&rñv“¦ Õª ×æd”}Œ•@/\ j˵_ Œ• 6

x~½Ó†¾Ó θR 0Au 7˜' ~rR Ã\ºf”s ÷&•¸2Ÿ¤ôÇ. 7£¤ θR

= 90^◦“ yŒ••¸–Ð 6 xú^o= $Ö¦–Ð j˵ ~F_R¦` #Œ r„óøÍ _

 ×æd” ú¨î‚s tt@/_ ú¨»î¡¤“ x»¡¤ {õ9uôÇ ©œI

\

"f ¨î+þA©œI ÷&•¸2Ÿ¤ #Œ ¨î+þA›¸| “ τ_i = rRFR_

›'

a>d”`¦6s x #Œ ½¨½+É Ãº e”. sM: r„óøÍ\ Œ•6 x 



H j˵ ~Fi_ ß¼l\  [taË>j˵ ~τi_ ß¼l qYV #Œ 7£x

†<Ê`¦ z´+«>&hܼ–ÐSX‰“½+É Ãº e”.

Õ

ªo“¦ j˵_ Œ•6 x&h_ 0Au 7˜' ~rü< Œ•6 x H j˵ ~F



e± yŒ• θ [taË>j˵ ~τ\ puH %ò†¾Ó`¦ ·ú˜l 0A #Œ



r„óøÍ_ r„»¡¤ O–ÐÂÒ' {9&ñôÇ 0Au 7˜' ~rⁱ (ri = constant)\"f {9&ñôÇ ß¼l_ j˵ ~F_i (F_i = constant)\¦

~

½Ó†¾Ó θi`¦r>~½Ó†¾Ós
 ìÍrø>~½Ó†¾Óܼ–Ð {9&ñôÇ yŒ•_ ß¼ l

–Ð 7£xrv€"f j˵_ Œ•6 x@/_ ×æd”}Œ•@/\ ƒ“¦o

\

¦ :ŸxK 6 xú^o= $Ö¦`¦6 x #Œ r„óøÍ\ ôÇ. sM: y

Œ

• âĺ_ [taË>j˵ ~τ_i_ ~½Ó†¾Óõ ß¼lH ·ú¡_ z´+«>[þt\"f ü

< °ú “Ér]X ü< ~½ÓZO\ _ #Œ ½¨½+É Ãº e”. sM: r„ ó

øÍ\ Œ•6 x H j˵ ~F_i_ ~½Ó†¾ÓyŒ• θi_ o\  [taË>j˵

~

τ_i_ ß¼lü< ~½Ó†¾Ós o H `כ¦ z´+«>&hܼ–Ð SX‰“½+É Ã

º e”.

s

QôÇ z´+«> õ\¦ :ŸxK [taË>j˵ ~τ_i_ 5Åq$`í¦ 7áx½+Ë&h Ü

¼–Ð sK > ½+É Ãº e”ܼ 9, ¢¸ôÇ Ã º†<Æ&h“ ³ð‰&³`¦ :Ÿx K

 7˜'ü< 7˜'ƒíߖ 1px_ ú†<Æ&h úéߖ`¦Ö¸6 xK [taË>j˵

`

¦
è­q ú e”•¸2Ÿ¤½+É Ãº e”.



™

‹“@3: ÆÁ 5˜+ ×ÏUc˜+ø5 TÒ¼Å]·ÍžâÏ×Ϙ+ ¥´Ça™‹“@ Fig. 6_ (a)\"f ˜Ð#ŒÅҍH ü< °ú s r„óøÍ_ r„

»

¡¤ O–ÐÂÒ' š¸ÉrAᤓ 0Au 7˜' ~r1\ ìøÍr>~½Ó†¾Óܼ–Ð θ₁_ yŒ••¸–Ð j˵ ~F₁`¦, r„»¡¤ O–ÐÂÒ' š¸ÉrAᤓ 0Au 7˜' ~r2\ j˵ ~F<sub>2</sub>`¦ 0Au ˜'7 ~r2\"f r>~½Ó†¾Óܼ–Ð yŒ••¸ θ₂_ ~½Ó†¾Óܼ–Ð ¿º j˵ ~F₁õ ~F₂¦ 1`lxr\ Œ•6 x½+É M:, r„ ó

øÍ\ Œ•6 x H s[þt¿º jµ ~Ë F₁õ ~F₂\ _ #Œ l)a r

„

»¡¤ O\ @/ôÇ ½+Ë$í [taË>j˵ ~τ_r\ _ #Œ r„óøÍs %ƒ6£§

~

½Ó†¾Ó“ ú¨»î¡¤“ x»¡¤õ {u9 H~½Ó†Ó\¾"f r„ $tûZ

\

 ]X H ӆ½~¾Óܼ–Ð r>~½Ó†¾Óܼ–Ð r„s {#9Qèߖ ©œS!

`

¦˜Ð#Œïr. sQôǛ'a¹1Ï\ _ #Œ ½+Ë$í [taË>j˵ ~τr_ ~½Ó

†

¾Ó“Ér−ˆz~½Ó†¾Óe”`¦SX‰½“+É Ãº e”“¦, Õª ß¼l\¦ ^‰+«>½+É Ãº e”

.

¿

º j˵ ~F1õ ~F2\ _ôÇ r„»¡¤ O\ @/ôÇ ½+Ë$í [taË>j˵

~

τr_ ß¼l\¦½¨ l 0A #Œ Fig. 6_ (b)ü< °ú s r„óøÍ_



r„»¡¤ O–ÐÂÒ' ¢,aAá¤ÂÒìr_ e”__ 0Au 7˜' ~r^R\ j˵ _

 Œ•6 x@/\¦“¦&ñrv“¦ Õª ×æd”}Œ•@/\ j˵_ Œ•6 x~½Ó†¾Ó θR 0Au 7˜' ~rR\ úf”s ÷&•¸2Ÿ¤, 7£¤ θR= 90^◦“ yŒ••¸

–

Ð j˵ ~FR`¦ #Œ r„óøÍ_ ×æd” ú¨î‚s ttóøÍ_ ú

¨î

»¡¤“ x»¡¤ {õ9uôÇ ©œI\"f ¨î+þA©œI ÷&•¸2Ÿ¤ôÇ.

s

M: ½+Ë$í [taË>j˵ ~τ_r_ ß¼lH¨î+þA›¸| “ τ_r= r_RF_R_

›'

a>d”`¦s6 x #Œ ½¨½+É Ãº e”ܼ 9, [taË>j˵[þt\ _ôÇ Óüt

^

‰_ ¨î+þA›¸| `¦sK½+É Ãº e”.

¢

¸ôÇ ¿º j˵ ~F₁õ ~F₂ Z>•¸–Ð 0Au 7˜' ~r1õ ~r2\ yŒ• y

Œ

•_ 0Au 7˜'_ ~½Ó†¾Ó\"f ìÍrø>~½Ó†¾Óܼ–Ð θ¹_ yŒ••¸ü<

r

>~½Ó†¾Óܼ–Ð θ2_ yŒ••¸–Ð •6Œ xrv“¦ r„»¡¤ O\ @/ôÇ [

taË>j˵ ~τ₁ ~τõ 1_ ~½Ó†¾Óõ ß¼l\¦yŒ•yŒ• ½¨ôÇ. Õªo“¦ s [

þ

t [taË>j˵ ~τ1õ ~τ¹_ ½+Ës ¿º jµ ~ËF1õ ~F2¦ 1`lxr\ Œ•6 x

½

+É M:_ ½+Ë$í [ta>jË˵ ~τ_rõ q“§ôÇ.

V. ÷ m Ç] M ö + s ÇÊ Ý õ m Í À X Ø8 ý

s

 ƒ½¨\"f [taË>j˵_›'aº †<Æ_þv`¦0A #Œ >hµ1ÏôÇ z´ +

«> ©œu_ 7˜'›'aº †<Æ_vþ\"f_ †<Æ_þv&h“ _pü< Õª  {

©œ$í“Ér s ƒ½¨\"f ]jîߖ H z´+«>[þt_ õ&ñõ z´+«>  õ

\¦H–Ð 6£§õ °ú s 7H_|¨cú e”.

'Í

P: z´+«>“ Fig. 4_ î[ta>jË˵_ 8£¤&ñ z´+«>ï_ (a)_



©œS!\"f †<ÆÒqt[t“þÉrÄ»–ÐîrÐ҄Ðo`¦ :Ÿx #Œ #î”îr1lxõ



r„îr1lx\"f Óüt^‰\ Œ•6 x H j˵_ ´òõ\¦ SX‰“½+É Ãº e”

. Õªo“¦ Óüt^‰\¦ r„ > H´òõ–Ð"f_ [taË>j˵

~

τ \¦s:r&hܼ–Ðëߖ “d” Hכ s m z´|9&hܼ–Ð yŒ™ y

Œ

•`¦ :Ÿx #Œ Õª ß¼lü< ~½Ó†¾Ó`¦^‰+«>½+É Ãº e”. ¢¸ôÇ [t aË

>j˵ ~τH r„»¡¤`¦ ” Óüt‰“^ r„óøÍ\ Œ•6 x H j˵ F , j~ ˵ ~F_ Œ•6 x&h_ 0Au 7˜' ~r, Õªo“¦ j˵_ Œ•6 x&h_ 0

Au 7˜' ~rü< j˵ ~F_ e± yŒ• θõ›'aºH†d`¦ ·ú˜ ú e”.

†

<ÆÒqt[þts sQôÇ Óüto|¾Ó[þt s_ ›'a\>¦ sK l 0A K

"fH yŒ• Óüto|¾Ó_ ß¼lü< ~½Ó†¾Ó`¦ “¦9K ôÇ. s



H ú†<Æ&h“ •¸½¨“ 7˜'\¦ ¸6Ö x #Œ r„óøÍ\ Œ•6 x 



H j˵ ~F ,r„»¡¤ O–ÐÂÒ' jµ ~ËF_ Œ•6 x&ht_ 0Au 7˜ '

 ~r, ¿º 7˜'_ e± yŒ• θ`¦ :xŸK
è­q ú e”. sM: r

„

»¡¤ O\ @/ôÇ [taË>j˵ ~τ_ ß¼lü< ~½Ó†¾Ó`¦ 7˜'“ j˵ ~F , jË

µ ~F_ Œ•6 x&h_ 0Au ˜'7 ~r, Õªo“¦ s ¿º 7˜'_ e± y

Œ

• θçߖ_›'a>“ 7˜'d” f (~r, ~F , θ)ܼ–Ð
è­q ú e”ܼ 9 7˜'–Ð"f_ [taË>j˵ ~τ`¦ “d” > ½+É Ãº e”. ¢¸ôÇ [taË>

jË

µ ~τH0Au 7˜' ~rõ j˵ 7˜' ~F_ 7˜'YL“ ~τ = ~r × ~F–Ð

³

ð‰&³÷&HX<, {9ìøÍ&h“ ¿º 7˜' ~Aü< ~B_ 7˜'YL ~A × ~B“Ér îÕª ß¼lHº ¿7˜' sÀҍH'+þA_ €&hõ °ú “¦ Õ

ª ~½Ó†¾Ó“Ér ˜'7 ~Aü< 7˜' ~B /BNÄ» H€î¨\ úf”s 9 7˜' ~A_ ~½Ó†¾Ó\"f 7˜' ~B_ ~½Ó†¾Óܼ–Ð r„½+É M: š¸

 É

r’<HZOgË:\ _ #Œ &ñ #Œ tH~½Ó†¾Ó`¦tH¢¸ 
 _

 7˜'sïH 7˜'YL_ &ñ_\¦Ér. s\¦Ö¸6 xK 0

Au 7˜' ~rõ j˵ 7˜' ~F_ 7˜'YL“ ~τ = ~r × ~F–Ð [taË>

Fig. 7. (Color online) The explanatory diagram of the equilibrium state of the rotary plate by the two torques

~

τ and ~τR.

jË

µ ~τ_ ß¼l τ = rF sin θs“¦ š¸Ér’<HgOZË:`¦ :ŸxK ~½Ó

†

¾Ó`¦&ñ½+É Ãº e”. ¢¸ôÇ [taË>j˵ ~τ_ éߖ0A N·m–Ð"f {9 s


 \-t_ éߖ0A °ú tëߖ, ~½Ó†¾Ó`¦Ÿí†<Ê HÉr 7áx À

Ó_ Óüto|¾Óe”`¦ “d” > ôÇ.

Ó ü

t^‰ ¨î+þA©œI\ e”l 0AK"fH Óüt^‰\ Œ•6 x H

—

¸ŽH j˵_ ½+Ës 0s ÷&#Q ôǍH ›¸|  sü@\ Óüt^‰ _

 #Q‹" e”__ &h\ @/ôÇ —¸ŽH [taË>j˵_ ½+Ës 0s ÷&

#

Q ›¸| s €9כ¹ . Fig. 4_ (a)_ ©œS!\"f r„»¡¤ O\ @/ôÇ [taË>j˵\ _ #Œ r„ H r„óøÍs Fig. 4_ (b)ü< °ú “Ér¨î+þA©œI\ e>” l 0AK"fHr„»¡¤ O\

@

/ôÇ ¢¸ 
_ [taË>j˵ ~τ_Rs €9כ¹  9, s[þt_ ›'a>H

~

τ + ~τ_R = 0“ [taË>j˵\ _ôÇ Óüt^‰_ ¨î+þA›¸| \ ÂÒ½+ËK



 †<Ê`¦CÖ¦ú e”. Õªo“¦ [taË>j˵ ~τR¦\l H0A u

 7˜' ~r^R\ jµ Ë7˜' ~FR\¦•6Œ x #Œ r„óøÍ_ ¨î+þA©œI

\

¦ ›¸Œ•½+É Ãº e”. sQôÇ ¸1ցlx`¦ :Ÿx# Œ [taË>j˵ ~τ_R

¢

¸ôÇ ~τ_R = ~r_R× ~F_R–Ð
è­q ú e”ܼ 9, r„óøÍs ¨î +þ

A©œI\¦ sÀÒ> H [ta>jË˵ ~τR = ~rR× ~FR [taË>j˵

~

τ = ~r × ~F_ ß¼lü< °ú “¦ Õª ~½Ó†¾Ós ìøÍ@/H [taË>j˵

~

τ_R_ 5Åq$í`¦sK½É Ã+º e”> ôÇ. ¢¸ôÇ Fig. 7\"f ˜Ð#Œ Å

ҍH ü< °ú s [taË>j˵ ~τ`¦sÀҍH 0Au 7˜' ~rõ j˵ 7˜ '

 ~F sÀҍH¨î'Ÿ+þA_ €&h rF sin θü< ß¼l °ú 

“

¦, ~½Ó†¾Ós ˆz¼–ÜÐ ìøÍ@/~½Ó†¾Ó“ [taË>j˵ ~τ_R¦`›¸Œ•ôÇ. s

\

¦ ~τR = rRFRzˆ–Ð
?/“¦, j˵ ~FR_ Œ•6 x&h_ 0Au 7˜ '

 ~r^R\ úf”s ÷&•¸2Ÿ¤ j˵ ~FR¦\Œ•6 x •¸2Ÿ¤ôÇ. Õªo

“

¦ [taË>j˵ ~τ_R¦\l H jµ ~ËF_R\¦0Au 7˜' ~rR\ úf”s

÷

&•¸2Ÿ¤Õª ~½Ó†¾Ó`¦›¸• Œ#Œ ~τ = ~r× ~F = −rRFRz“ˆ›'a>

\

¦s6 x #Œ [taË>j˵ ~τ = ~r × ~F_ ß¼l\¦½¨½+É Ãº e”H> í

ߖ'‘ܼ–Ð 6 x½+É Ãº e”. Õªo“¦ Fig. 2\"f "îr)a t Y

U ¼1Ï_ U´s“ r„»¡¤ O\"f j˵ ~F_ Œ•6 x‚õ_ úf”

o r sin θH ¿º [taË>j˵_ ¨î+þA©œI rF sin θ = r^RFR\

"

f F = FR“ âĺ r sin θ = rRe”`¦†<Æ_þv½+É Ãº e”.