46. Critical depth at a section of a rectangular channel is 1.5 m. The specific energy at that section is
    

1. 1. 0.75 m

   
   
   

   2. 1.0 m

   
   
   

   3. 1.5 m

   
   
   

   4. 2.25 m

   
   
   

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1. View Hint View Answer Discuss in Forum

   y_c = 1.5 m
   

   Ec =	3	yc
   	2

   
   

   =	3	× 1.5 = 2.25 m
   	2

   Correct Option: D

   y_c = 1.5 m
   

   Ec =	3	yc
   	2

   
   

   =	3	× 1.5 = 2.25 m
   	2

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47. A partially open sluice gate discharges water into a rectangular channel. The tail water depth

    in the channel is 3 m and Froude number is	1	. If a free hydraulic jump is to be
    	2 √2

    formed at a downstream of the sluice gate after the vena contracta of the jet coming out from the sluice gae, the sluice gate opening should be (coefficient of contraction C_c = 0.9)
    

1. 1. 0.3 m
      

   
   
   

   2. 0.4 m
      

   
   
   

   3. 0.69 m
      

   
   
   

   4. 0.9 m

   
   
   

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1. View Hint View Answer Discuss in Forum

   y1 =	y2	[-1 + √1 + 8Fr22]
   	2

   
   
   = 0.62 m
   Coeff. of contraction, C_c
   

   Cc =	area of vena contract
   	area of opening

   
   

   ⇒ 0.9 =	0.62 × b
   	y × b

   
   

   ∴ y =	0.62	= 0.69 m
   	0.9

   
   

   Correct Option: C

   y1 =	y2	[-1 + √1 + 8Fr22]
   	2

   
   
   = 0.62 m
   Coeff. of contraction, C_c
   

   Cc =	area of vena contract
   	area of opening

   
   

   ⇒ 0.9 =	0.62 × b
   	y × b

   
   

   ∴ y =	0.62	= 0.69 m
   	0.9

   
   

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48. A stream function is given by Ψ = 2x²y + (x + 1)y²
    The flow rate across a line joining points A(3, 0) and B(0,2) is
    

1. 1. 0.4 units

   
   
   

   2. 1.1 units

   
   
   

   3. 4 units

   
   
   

   4. 5 units

   
   
   

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1. View Hint View Answer Discuss in Forum

   Ψ = 2x²y + (x + 1)y²
   At A (3, 0)
   Ψ_{(3, 0)} = 2(3)² - (0) + (3 + 1)(0)²
   = 0
   At B(2, 0)
   Ψ_{(2, 0)} = 2(0)²(2) + (0 + 1)(2)²
   = 0 + 4 = 4
   Flow rate across line joining A and B = |Ψ_A - Ψ_B|
   = |0 - 4| units

   Correct Option: C

   Ψ = 2x²y + (x + 1)y²
   At A (3, 0)
   Ψ_{(3, 0)} = 2(3)² - (0) + (3 + 1)(0)²
   = 0
   At B(2, 0)
   Ψ_{(2, 0)} = 2(0)²(2) + (0 + 1)(2)²
   = 0 + 4 = 4
   Flow rate across line joining A and B = |Ψ_A - Ψ_B|
   = |0 - 4| units

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49. Cross-section of an object (having same section normal to the paper) submerged into a fluid consists of a square of sides 2 m and triangle as shown in the figure. The object is hinged at point P that is one meter below the fluid free surface. If the object is to be kept in the position as shown in the figure, the value ‘x’ should be
    
    

1. 1. 2 √3
      

   
   
   

   2. 4 √3
      

   
   
   

   3. 4 m
      

   
   
   

   4. 8 m

   
   
   

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1. View Hint View Answer Discuss in Forum

   Take moment of vertical component of hydrastatic force on two section
   
   

   V10 +	2	= V2 ×	x
   	2		3

   
   

   ρ gv1 × 1	= ρgv2 ×	x
   		3

   
   

   ∴ v1 =	= v2	x
   		3

   
   

   (2 × z × w) =		1	x × 2		(w) ×	x
   		2				3

   
   ⇒ x = √12 = 2√3
   

   Correct Option: A

   Take moment of vertical component of hydrastatic force on two section
   
   

   V10 +	2	= V2 ×	x
   	2		3

   
   

   ρ gv1 × 1	= ρgv2 ×	x
   		3

   
   

   ∴ v1 =	= v2	x
   		3

   
   

   (2 × z × w) =		1	x × 2		(w) ×	x
   		2				3

   
   ⇒ x = √12 = 2√3
   

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50. The circulation ‘Γ’ around a circle of radius 2 units for the velocity field u = 2x + 3y and v = 2y is
    

1. 1. – 6π units

   
   
   

   2. – 12π units

   
   
   

   3. – 18π units

   
   
   

   4. – 24π units

   
   
   

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1. View Hint View Answer Discuss in Forum

   ^→V = ui + vj
   u = 2x + 3y , v = -2y
   

   	∂u	= 3 ,	∂v	= 0
   	∂y		∂x

   
   

   circulation =		∂v	-	∂u		× Area
   		∂x		∂y

   
   = (0 - 3) × πR²
   = (-3)π × 2²
   = – 12π units
   

   Correct Option: B

   ^→V = ui + vj
   u = 2x + 3y , v = -2y
   

   	∂u	= 3 ,	∂v	= 0
   	∂y		∂x

   
   

   circulation =		∂v	-	∂u		× Area
   		∂x		∂y

   
   = (0 - 3) × πR²
   = (-3)π × 2²
   = – 12π units
   

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