EMT-X(UT-226/16)
ÿoˆÁTˆEı YÁPˆy‘] ( B.D.P.)
◊`l˘Á[˝bÔÁ‹ôˆ YÃ[˝›l˘Á ( Term End Examination ) :
◊Qˆ„a∂ëˆÃ[˝, 2015 C L«X, 2016
G◊STˆ ( Mathematics )
V`] Yy ( 10th Paper : Analytical Statics and
Fluid Statics )
Y…SÔ]ÁX f 50
Time : 2 Hours
Full Marks : 50
( ]Á„XÃ[˝ mÃ[˝”±ºˆ f 70% )
( Weightage of Marks : 70% )
Y◊Ã[˝◊]Tˆ C ^UÁ^U =w¯„Ã[˝Ã[˝ LXÓ ◊[˝„`b ]…_Ó ÂVCÃ^Á c˜„[˝*
%£à˘ [˝ÁXÁX, %Y◊Ã[˝¨K˜~TˆÁ A[˝e %Y◊Ã[˝õıÁÃ[˝ c˜ÿôˆÁl˘„Ã[˝Ã[˝ Âl˘‰y X∂ëˆÃ[˝
ÂEı‰ªRÙO ÂXCÃ^Á c˜„[˝* =YÁ„‹ôˆ Y“‰`¬Ã[˝ ]…_Ó]ÁX a…◊ªJÙTˆ %Á‰ªK˜*
Special credit will be given for accuracy and relevance
in the answer. Marks will be deducted for incorrect
spelling, untidy work and illegible handwriting.
The weightage for each question has been
indicated in the margin.
1* Y“„TˆÓEı ◊[˝\ˆÁG ÂU„Eı AEı◊ªRÙO Eı„Ã[˝ ◊X„Ã^ Â]ÁªRÙO V«◊ªRÙO Y“‰`¬Ã[˝
=w¯Ã[˝ ◊VX f
10 × 2 = 20
◊[˝\ˆÁG – %
→ →
→
(Eı) AEı◊ªRÙO a]Tˆ_›Ã^ [˝_„GÁúˆ› P1 , P2,..., Pn ^UÁy‘„]
→
→
→
AEı◊ªRÙO VªRÕÙ [˝ÿô«ˆÃ[˝ A1(r1 ), A2(r2 ),..., An (rn ) ◊[˝≥V«„Tˆ
◊y‘Ã^Á Eı„Ã[˝, Â^FÁ„X B Tˆ„_Ã[˝ O ◊[˝≥V«Ã[˝ aÁ„Y„l˘
B.Sc.-7560-B
[
Y„Ã[˝Ã[˝ YúˆÁÃ^ V–rÙ[˝Ó
2
A1, A2 ,..., An
→ →
→
r1 , r2 ,..., rn
◊[˝≥V«m◊_Ã[˝ %[˝ÿöˆÁX Â\ˆkÙÃ[˝ ^UÁy‘„]
* ÂVFÁX Â^, [˝_„GÁúˆ›◊ªRÙO
O
→
AEı◊ªRÙO
B◊¨K˜Eı YÁPˆy‘] ( Elective )
a]Ã^ f ªV«c˜O H∞RÙOÁ
EMT-X(UT-226/16)
[˝_
R
n → →
→
G =
rk × Pk
k =1
∑
ÂVFÁX Â^,
(i)
A[˝e
AEı◊ªRÙO
◊[˝≥V«„Tˆ
}Á]Eı
-AÃ[˝ a„Ü a]T«ˆ_* AÃ[˝ ÂU„Eı
n →
^◊V ∑ Pk
→
≠O
k =1
AEı◊ªRÙO [˝„_Ã[˝ a]T«ˆ_; A[˝e
(ii)
, TˆÁc˜„_ [˝_„GÁúˆ›◊ªRÙO
n →
^◊V ∑ Pk
k =1
→
=O
,
10
TˆÁc˜„_ [˝_„GÁúˆ›◊ªRÙO AEı◊ªRÙO }Á]„EıÃ[˝ a]T«ˆ_*
(F) AEı◊ªRÙO a«b] %Y“aÁ^Ô \ˆÁÃ[˝› a…y V«◊ªRÙO ◊[˝≥V«
ÂU„Eı AEı◊ªRÙO ={∂ëˆ Tˆ„_ %◊\ˆEı„bÔÃ[˝ %W˝›„X
aÁ]ÓÁ[˝ÿöˆÁÃ^ M«ı_‰ªK˜* W˝Ã[˝Á ^ÁEı =Y^«N˛ %l˘ ◊X„_ a…y
•ÁÃ[˝Á G◊PˆTˆ [˝y‘„Ã[˝FÁÃ[˝ a]›EıÃ[˝S y = f ( x ) * ÂVFÁX
Â^, y ◊X‰∂oˆ Y“Vw¯ a]›EıÃ[˝S ◊aà˘ EıÃ[˝„[˝
d 2y
1
=
2
c
dx
1/ 2
2
⎧⎪
⎛ dy ⎞ ⎫⎪
1
+
⎜
⎟
⎨
⎬
⎝ dx ⎠ ⎪⎭
⎪⎩
, Â^FÁ„X
c
AEı◊ªRÙO
Y“ªJÙ_* =Y^«N˛ `Tˆ¤ W˝„Ã[˝, Ac˜O a]›EıÃ[˝S◊ªRÙO a]ÁW˝ÁX
Eı„Ã[˝ ÂVFÁX Â^, [˝y‘„Ã[˝FÁÃ[˝ a]›EıÃ[˝S
y = c cosh (x /c ) *
10
B.Sc.-7560-B
3
EMT-X(UT-226/16)
◊[˝\ˆÁG – %Á
(G) ρ HX±ºˆ ◊[˝◊`rÙ TˆÃ[˝„_Ã[˝ ]„W˝Ó AEı◊ªRÙO aÁ]Tˆ◊_Eı Âl˘y
a+…SÔ\ˆÁ„[˝ ◊X]◊#Tˆ %[˝ÿöˆÁÃ^ %◊\ˆEı„bÔÃ[˝ %W˝›„X
aÁ]ÓÁ[˝ÿöˆÁÃ^ %Á‰ªK˜* Âl˘y◊ªRÙOÃ[˝ ªJÙÁY„Eı‰≥V–Ã[˝ G\ˆ›Ã[˝TˆÁ Â[˝Ã[˝
EıÃ[˝”X A[˝e ÂVFÁX Â^, A◊ªRÙO \ˆÁÃ[˝„Eı‰≥V–Ã[˝ G\ˆ›Ã[˝TˆÁÃ[˝
ÂU„Eı [˝QÕˆ*
AÃ[˝ ÂU„Eı ÂVFÁX Â^, Âl˘y◊ªRÙO a]„[˝„G ◊X∂oˆGÁ]› c˜„_,
ªJÙÁY„Eı≥V–, \ˆÁÃ[˝„Eı≥V–GÁ]› %X«\…ˆ◊]Eı ÂÃ[˝FÁÃ[˝ ◊V„Eı
a]„[˝„G W˝Á[˝]ÁX A[˝e B Â[˝G \ˆÁÃ[˝„Eı‰≥V–Ã[˝ G\ˆ›Ã[˝TˆÁÃ[˝
[˝„GÔÃ[˝ [˝ÓÿôˆÁX«YÁTˆ›*
10
(H) EıÁ„Tˆ¤a›Ã^ %„l˘Ã[˝ a]Á‹ôˆÃ[˝Á_ ◊V„Eı ( X, Y, Z ) =YÁe`
→
◊[˝◊`rÙ AEı◊ªRÙO [˝_ P -AÃ[˝ %W˝›„X ρ HX±ºˆ ◊[˝◊`rÙ
AEı◊ªRÙO TˆÃ[˝_ aÁ]ÓÁ[˝ÿöˆÁÃ^ %Á‰ªK˜* ÂVFÁX Â^, aÁ„]ÓÃ[˝
%Á[˝`ÓEı `Tˆ¤ c˜„[˝
⎛ ∂Y ∂Z ⎞
⎛ ∂Z − ∂X ⎞ + Z ⎛ ∂X − ∂Y ⎞ = 0 .
−
X⎜
⎜
⎟
⎟ +Y⎜
⎟
⎝ ∂x ∂z ⎠
⎝ ∂y ∂x ⎠
⎝ ∂z ∂y ⎠
%ÁÃ[˝C ÂVFÁX Â^, aeÃ[˝l˘› [˝_„l˘‰yÃ[˝ Y“\ˆÁ„[˝, AEı◊ªRÙO
%a]a√º¯ TˆÃ[˝„_Ã[˝ a]ªJÙÁ„YÃ[˝ Tˆ_, a]HX„±ºˆÃ[˝ Tˆ_
10
A[˝e a]◊[˝\ˆ„[˝Ã[˝ Tˆ_ %◊\ˆ~*
2* Y“„TˆÓEı ◊[˝\ˆÁG ÂU„Eı %‹ôˆTˆ AEı◊ªRÙO Y“`¬ ◊X[˝ÔÁªJÙX Eı„Ã[˝
Â^-ÂEıÁ„XÁ ◊TˆX◊ªRÙO Y“‰`¬Ã[˝ =w¯Ã[˝ ◊VX f
6 × 3 = 18
◊[˝\ˆÁG – %
(Eı) X, Y, Z [˝_ ^UÁy‘„] ◊TˆX◊ªRÙO aÃ[˝_„Ã[˝FÁ y = b ,
z = −c ; z = c , x = −a ; x = a , y = −b [˝Ã[˝Á[˝Ã[˝
◊y‘Ã^Á Eı„Ã[˝* ÂVFÁX Â^, TˆÁ„VÃ[˝ AEı◊ªRÙO _◊ï˘ [˝_
B.Sc.-7560-B
[
Y„Ã[˝Ã[˝ YúˆÁÃ^ V–rÙ[˝Ó
EMT-X(UT-226/16)
4
UÁEı„[˝ ^◊V Xa + Yb + Zc = 0 c˜Ã^* _◊ï˘ [˝„_Ã[˝
◊y‘Ã^Á„Ã[˝FÁÃ[˝ a]›EıÃ[˝S ◊XSÔÃ^ EıÃ[˝”X*
(F) a]ÁX CLX W ◊[˝◊`rÙ ªJÙÁÃ[˝◊ªRÙO V⯠]aS\ˆÁ„[˝ ^«N˛ c˜„Ã^
AEı◊ªRÙO Ã[˝∂ëˆa ABCD GPˆX Eı„Ã[˝* Ac˜O GPˆX◊ªRÙO A ◊[˝≥V«
ÂU„Eı M«ı_‰ªK˜ A[˝e AEı◊ªRÙO \ˆÁÃ[˝ W ′ , C ◊[˝≥V«„Tˆ ^«N˛*
AEı◊ªRÙO =„Yl˘S›Ã^ \ˆ„Ã[˝Ã[˝ Eı◊PˆX V⯠•ÁÃ[˝Á AB A[˝e
AD-Ã[˝ ]W˝Ó◊[˝≥V« V«◊ªRÙO ae^«N˛ %Á‰ªªK˜, ^Á„Tˆ TˆÁÃ[˝Á AC-Ã[˝
a„Ü α ÂEıÁ„S XTˆ UÁ„Eı* ÂVFÁX Â^, Eı◊PˆX Vâ¯◊ªRÙOÃ[˝
=YÃ[˝ HÁTˆ ( 4W + 2W ′ ) tan α *
(G) AEı◊ªRÙO [˝GÔ„l˘yÁEıÁÃ[˝ %e` AEı◊ªRÙO [˝w¯ÁEıÁÃ[˝ Zı_Eı
ÂU„Eı AEı◊ªRÙO [˝ÓÁaÁW˝Ô„Eı EıSÔ Eı„Ã[˝ ÂEı‰ªRÙO ÂXCÃ^Á c˜_*
ÂVFÁX Â^, %[˝◊`rÙ %e„`Ã[˝ \ˆÁÃ[˝„Eı≥V–, ÂEı≥V– ÂU„Eı
a
V…„Ã[˝ %[˝◊ÿöˆTˆ, Â^FÁ„X a c˜_ [˝w¯◊ªRÙOÃ[˝ [˝ÓÁa*
8π − 4
◊[˝\ˆÁG – %Á
(H) ={∂ëˆ Tˆ„_ Ã[˝◊l˘Tˆ, AEı◊ªRÙO a]Y“ÿöˆ‰¨K˜V ◊[˝◊`r٠ʪK˜ÁªRÙO
[˝w¯ÁEıÁÃ[˝ ◊ªRÙO=„[˝Ã[˝ ]„W˝Ó ρ A[˝e σ ( ρ > σ ) HX„±ºˆÃ[˝
A[˝e ◊]◊`“Tˆ c˜Ã^ XÁ AÃ[˝Eı] V«◊ªRÙO TˆÃ[˝_ a] Y◊Ã[˝]Á„S
%Á‰ªK˜* TˆÁÃ[˝Á AEıaÁ„U ^◊V ◊ªRÙO=„[˝Ã[˝ %„W˝ÔEı Y…SÔ Eı„Ã[˝,
Tˆ„[˝ ÂVFÁX Â^, TˆÁ„VÃ[˝ aÁW˝ÁÃ[˝S Tˆ„_Ã[˝ ]W˝Ó ◊V„Ã^
^ÁCÃ^Á [˝ÓÁaÁW˝Ô ={„∂ëˆÃ[˝ aÁ„U tan−1 ρρ +− σσ ÂEıÁS
=dY~ Eı„Ã[˝*
B.Sc.-7560-B
EMT-X(UT-226/16)
(I¯) ÂEıÁX Tˆ„_Ã[˝ AEı◊ªRÙO ◊ÿöˆÃ[˝ ◊[˝≥V«Ã[˝ ◊V„Eı, V…Ã[˝„±ºˆÃ[˝ aÁ„U
Y◊Ã[˝[˝Tˆ¤X`›_ AÃ[˝÷Y [˝_ Y“„Ã^Á„GÃ[˝ Zı„_, ◊EıªK«˜
Y◊Ã[˝]ÁS TˆÃ[˝_ B ◊X◊V¤rÙ Tˆ„_Ã[˝ =YÃ[˝ ◊ÿöˆÃ[˝ %Á‰ªK˜*
]«N˛ Tˆ_◊ªRÙO %W˝Ô„GÁ_EıÁEıÁÃ[˝ c˜„_ Tˆ_◊ªRÙOÃ[˝ =YÃ[˝ HÁTˆ
◊XSÔÃ^ EıÃ[˝”X*
(ªJÙ) ρ HX„±ºˆÃ[˝ ◊≈y\«ˆLÁEı◊Tˆ YÁTˆ ABC, σ HX„±ºˆÃ[˝ TˆÃ[˝„_
={∂ëˆ\ˆÁ„[˝ \ˆÁa]ÁX %Á‰ªK˜, ^Á„Tˆ B ◊[˝≥V«◊ªRÙO TˆÃ[˝„_Ã[˝
=Y◊Ã[˝Tˆ„_ UÁ„Eı A[˝e A ◊[˝≥V«◊ªRÙO TˆÃ[˝„_ ◊X]◊#Tˆ XÁ
c˜Ã^* ÂVFÁX Â^, ρ : σ = sin A cos C : sin B .
3* Y“„TˆÓEı ◊[˝\ˆÁG ÂU„Eı V«◊ªRÙO Eı„Ã[˝ ◊X„Ã^ Â]ÁªRÙO Jª ÙÁÃ[˝◊ªRÙO Y“‰`¬Ã[˝
=w¯Ã[˝ ◊VX f
3 × 4 = 12
◊[˝\ˆÁG – %ı
(Eı) ÂEıÁX [˝ÿô«ˆÃ[˝
A1, A2 ,..., An
◊[˝≥V«„Tˆ ◊y‘Ã^Á`›_
→ →
→
-aeFÓEı a]Tˆ_›Ã^ [˝_ F1, F2,...,Fn aÁ]ÓÁ[˝ÿöˆÁÃ^
%Á‰ªK˜* TˆÁ„VÃ[˝ ◊XÃ[˝„Yl˘ (Astatic) aÁ„]ÓÃ[˝ `Tˆ¤◊ªRÙO Â[˝Ã[˝
EıÃ[˝”X*
(F) ABC ◊≈y\«ˆ„LÃ[˝ A, B, C `›bÔy„Ã^ a◊y‘Ã^ P, Q, R
[˝_ ◊TˆX◊ªRÙOÃ[˝ Y“„TˆÓEı◊ªRÙO B `›bÔy„Ã^Ã[˝ ◊[˝YÃ[˝›Tˆ [˝ÁßÃ[˝
=YÃ[˝ ^UÁy‘„] _∂ëˆ\ˆÁ„[˝ ÂU„Eı ◊≈y\«ˆL◊ªRÙO„Eı
aÁ]ÓÁ[˝ÿöˆÁÃ^ Ã[˝Á„F* Y“]ÁS EıÃ[˝”X Â^,
n
EMT-X(UT-226/16)
(G) AEı◊ªRÙO `N˛ %W˝Ô„GÁ_Eı AEı◊ªRÙO a][˝ÓÁaÁW˝Ô^«N˛ ◊ÿöˆÃ[˝
ÂGÁ_„EıÃ[˝ =YÃ[˝ [˝y‘Tˆ„_Ã[˝ aÁ„U aÁ]ÓÁ[˝ÿöˆÁÃ^ ◊ÿöˆÃ[˝
%Á‰ªK˜* ÂVFÁX Â^, aÁ]ÓÁ[˝ÿöˆÁ◊ªRÙO a«◊ÿöˆ◊Tˆc˜›X*
(H) [˝ÓÁFÓÁ EıÃ[˝”X — "HbÔS ÂEıÁS' C "HbÔS `·«¯'*
◊[˝\ˆÁG – %Á
(I¯) (i) \ˆÁa]ÁX [˝ÿô«ˆÃ[˝ Tˆ_, (ii) \ˆÁa]ÁX [˝ÿô«ˆÃ[˝ a]G– Tˆ_
A[˝e (iii) Y¿[˝TˆÁÃ[˝ a]G– Tˆ_
[˝_„Tˆ ◊Eı Â[˝ÁMıÁ„XÁ c˜Ã^ ?
(ªJÙ) %◊\ˆEıbÔ ±ºˆÃ[˝S g W˝–”[˝Eı c˜„_, z =¨JÙTˆÁÃ^ a„]Áb’
[˝ÁÃ^«]⯄_Ã[˝ ªJÙÁY Â[˝Ã[˝ EıÃ[˝”X*
(ªK˜) L_ •ÁÃ[˝Á \ˆ◊Tˆ¤ AEı◊ªRÙO %W˝Ô„GÁ_EıÁEıÁÃ[˝ [˝Á◊ªRÙO, %X«\…ˆ◊]Eı
%[˝ÿöˆÁÃ^ Ã[˝ÁFÁ %Á‰ªK˜* ÂVFÁX Â^, [˝Á◊ªRÙOÃ[˝ =YÃ[˝ _◊ï˘
HÁTˆ, ={∂ëˆ [˝ÓÁaGÁ]› Tˆ„_Ã[˝ AEı◊V„EıÃ[˝ =YÃ[˝
%X«\…ˆ◊]Eı HÁ„TˆÃ[˝ π mS c˜„[˝*
(L)
a
[˝ÓÁaÁW˝Ô ◊[˝◊`rÙ AEı◊ªRÙO aÃ[˝” [˝w¯ÁEıÁÃ[˝ ◊ªRÙO=[˝, Âa◊ªRÙOÃ[˝
Y◊Ã[˝◊W˝Ã[˝ %„W˝ÔEı Y◊Ã[˝]ÁS TˆÃ[˝_ •ÁÃ[˝Á \ˆ◊Tˆ¤¯ %Á‰ªK˜* Âa◊ªRÙO
={∂ëˆ [˝ÓÁ„aÃ[˝ ªJÙÁÃ[˝◊V„Eı a]„EıÏ◊SEı Â[˝„G H«Ã[˝„Tˆ YÁ„Ã[˝*
Y“]ÁS EıÃ[˝”X, ÂEıÏ◊SEı Â[˝G 2ag -AÃ[˝ LXÓ TˆÃ[˝_◊ªRÙO
a„[˝]Áy (just) V«\ˆÁ„G ◊[˝\ˆN˛ c˜Ã^*
P : Q : R = a : b : c.
B.Sc.-7560-B
[
Y„Ã[˝Ã[˝ YúˆÁÃ^ V–rÙ[˝Ó
2
B.Sc.-7560-B
3
EMT-X(UT-226/16)
EMT-X(UT-226/16)
(b)
( English Version )
4
A heavy uniform inextensible string is
suspended from two given points and is in
1.
equilibrium in a vertical plane under the
Answer any two questions taking one from each
Group :
action
10 × 2 = 20
→ →
→
→ →
→
that
the
show that y satisfies the equation
d 2y
are acting on a rigid body at points
r1 , r2,..., rn
Supposing
form y = f ( x ) with a proper choice of axes,
A system of coplanar forces P1 , P2 ,..., Pn
→
→
→
A1(r1 ), A2 (r2 ),..., An (rn )
gravity.
equation to the curve can be put in the
Group – A
(a)
of
1
=
2
c
dx
respectively, where
1/ 2
2
⎧⎪
⎛ dy ⎞ ⎫⎪
1
+
⎜
⎟ ⎬
⎨
⎝ dx ⎠ ⎪⎭
⎪⎩
,
where c is a parameter.
are the position vectors of
Solve
this
equation
with
appropriate
A1, A2 ,..., An with respect to a point O in the
conditions to show that the equation to the
plane of the force. Show that the system
curve is y = c cosh (x /c ) .
→
can be reduced to a single force R at O
together with a couple
→
G =
n →
→
∑ rk × Pk .
k =1
n →
→
Hence show that (i) if
Pk ≠ O , then the
k =1
∑
system reduces to a single force only; and
n →
→
(ii) if
Pk = O , the system reduces to a
k =1
∑
couple only.
B.Sc.-7560-B
10
[
Y„Ã[˝Ã[˝ YúˆÁÃ^ V–rÙ[˝Ó
10
Group – B
(c)
A plane area, completely immersed in a
liquid of density ρ , is in equilibrium under
gravity. Find the depth of the centre of
pressure of the area and show that it is
greater than the depth of the centre of
gravity.
Hence show that if the plane is lowered
uniformly, then the C.P. approaches to the
horizontal through C.G. with uniform
velocity which is inversely proportional to
the square of the depth of C.G.
10
B.Sc.-7560-B
EMT-X(UT-226/16)
(d)
EMT-X(UT-226/16)
A fluid of density ρ is in equilibrium under
inclined at an angle α to AC. Show that the
→
a force P whose components are ( X, Y, Z )
parallel to a set of Cartesian axes. Show
that
the
necessary
condition
of
equilibrium is
thrust
(c)
under conservative field of forces, the
surfaces of equipressure, of equidensity
2.
10
(d)
Answer any three questions taking at least one
from each Group :
6 × 3 = 18
Forces X, Y, Z act along the three straight
lines y = b , z = −c ; z = c , x = −a ; x = a ,
y = −b respectively. Show that they will
a b c
+ +
=0.
X Y Z
Find the equation of the line of action of the
resultant.
Four equal rods each of weight W form a
(e)
have a single resultant if
(b)
rhombus ABCD with smooth hinges at the
joints. The frame is suspended by the end A
and
a
weight
W′
attached
rod
is
A square hole is punched out of a circular
lamina, the diagonal of the square being a
radius of the circle. Show that the centre of
gravity of the remainder is at a distance
a
from the centre of the circle whose
8π − 4
diameter is a.
Group – B
A small uniform circular tube, whose plane
is vertical contains equal quantities of
fluids whose densities are ρ and σ ( ρ > σ ) ,
A given volume of liquid is at rest on a fixed
plane under the action of a force, to a fixed
point in the plane varying as the distance.
If the free surface be hemispherical, find
the thrust on the plane.
(f)
A triangular lamina ABC of density ρ floats
in a liquid of density σ , with the plane
vertical, the point B being in the surface of
A stiffening rod of negligible weight joins
the liquid and the point A not immersed.
the middle points of AB, AD keeping these
Show that ρ : σ = sin A cos C : sin B .
[
to
stiffening
C.
B.Sc.-7560-B
is
the
and do not mix. If they together fill half the
tube, show that the radius passing through
the common surface makes with the
ρ−σ
vertical an angle tan −1
.
ρ+σ
Group – A
(a)
in
( 4W + 2W ′ ) tan α .
⎛ ∂Y ∂Z ⎞
⎛ ∂Z − ∂X ⎞ + Z ⎛ ∂X − ∂Y ⎞ = 0 .
−
X⎜
⎜
⎟
⎟ +Y⎜
⎟
∂
z
y
∂
⎝ ∂x ∂z ⎠
⎝ ∂y ∂x ⎠
⎝
⎠
Show also that for a heterogeneous fluid
and of equal potential energy coincide.
2
Y„Ã[˝Ã[˝ YúˆÁÃ^ V–rÙ[˝Ó
B.Sc.-7560-B
3
3.
EMT-X(UT-226/16)
Answer any four questions taking two from each
Group :
EMT-X(UT-226/16)
(f)
A
number
→ →
→
F1, F2,..., Fn
(b)
(
n
)
of
coplanar
acting at A1, A2 ,..., An
forces
(g)
an
isothermal
and is kept with its base horizontal. Show
of a
that the resultant thrust on the bowl is
body are in equilibrium. Find the condition
π
for their Astatic equilibrium.
portion on one side of a vertical diametrical
Three forces P, Q, R acting at the vertices
plane.
(h)
times the horizontal thrust on the
A thin circular tube of radius a contains
opposite sides respectively of the triangle
liquid, filling half of its circumference, and
ABC, keep it in equilibrium. Show that
turns about the vertical diameter with
P : Q : R = a : b : c.
uniform angular velocity. Prove that the
A solid hemisphere rests in equilibrium
liquid will just separate, if the angular
same radius. Show that the equilibrium is
velocity is
unstable.
Explain
'Angle of friction' and 'Cone of
friction'.
Group – B
(e)
in
A hemispherical bowl is filled with water
with the curved surface on a fixed sphere of
(d)
pressure
due to gravity g is constant.
A, B, C along the perpendicular to the
(c)
the
atmosphere at a height z when acceleration
3 × 4 = 12
Group – A
(a)
Find
4
What are meant by (i) Plane of floatation,
(ii) Surface of floatation and (iii) Surface of
buoyancy ?
B.Sc.-7560-B
[
Y„Ã[˝Ã[˝ YúˆÁÃ^ V–rÙ[˝Ó
B.Sc.-7560-B
2g
.
a