This set of Fluid Mechanics test focuses on “Hydrostatic Force on Plane Area – 2”.

1. The greatest and the least depth of a circular plate of 4 m diameter from the free surface of water are 3m and 1 m respectively as shown. What will be the total pressure in (kN) on the plate?

a) 123

b) 185

c) 246

d) 308

View Answer

Explanation: Total liquid pressure on the lamina = F = γyA, where γ = specific weight of the liquid, y = depth of centroid of the lamina from the free surface, A= area of the centroid. Now, γ = 9.81 * 10

^{3}N / m

^{3}; y = 1 + 3 – 1 / 2 – 2m, A =

^{π}⁄

_{4}* 4

^{2}= 4π m

^{2}. Hence, F = 246.55 kN.

2. The greatest and the least depth of a circular plate of 4 m diameter from the free surface of water are 3m and 1 m respectively as shown. What will be the depth (in m) of it’s centre of pressure?

a) 1.125

b) 1.25

c) 2.125

d) 2.25

View Answer

Explanation: The depth of the centroid y and the centre of pressure y

_{CP}are related by:

where I= the moment of inertia and A = area and θ = the angle of inclination of the lamina to the horizontal. Now,

y = 1 + 3 – 1 / 2 = 2, I =

^{π}⁄

_{64}* 4

^{2}= 4π, A =

^{π}⁄

_{4}* 4

^{2}= 4π, sin θ =

^{1}⁄

_{2}Thus, y

_{CP}= 2.125m.

3. The highest and lowest vertices of a diagonal of a square lamina (each side equal to 4m) are 1 m and 3 m respectively as shown. What will be the water force (in kN) on the lamina?

a) 78

b) 118

c) 157

d) 196

View Answer

Explanation: Total liquid pressure on the lamina = F = γyA, where γ = specific weight of the liquid, y = depth of centroid of the lamina from the free surface, A= area of the centroid. Now, γ = 9.81 * 10

^{3}N / m

^{3}; y = 1 + 3 – 1 / 2 = 2m, each side of the lamina =

Hence, F = 156:96 kN.

4. The highest and lowest vertices of a diagonal of a square lamina (each side equal to 4m) are 1 m and 3 m respectively as shown. What will be the depth (in m) of it’s centre of pressure?

a) 1.08

b) 1.58

c) 2.08

d) 2.58

View Answer

Explanation:

where I = the moment of inertia and A= area and θ = the angle of inclination of the lamina to the horizontal. Now, y = 1 + 3 – 1 / 2 = 2m, each side of the lamina = =8; sin θ = 3-1/4 =

^{ 1}⁄

_{2}. Thus, y

_{CP}= 2.08m.

5. A square lamina (each side equal to 2m) is submerged vertically in water such that the upper edge of the lamina is at a depth of 0.5 m from the water surface. If the pressure on the surface is 12 bar, what will be the total water pressure (in kN) on the lamina?

a) 39

b) 59

c) 64

d) 71

View Answer

Explanation: Total liquid pressure on the lamina = F = γyA, where γ = specific weight of the liquid, y = depth of centroid of the lamina from the free surface, A= area of the centroid. Now, γ = 9.81 * 10

^{3}N / m

^{3}; A = 2 * 2 = 4 m

^{2}. Hence, F = 63.65 kN.

6. A container is lled with two liquids of densities ρ1 and ρ2 up to heights h_{1} and h_{2} respectively. What will be the hydrostatic force (in kN) per unit width of the lower face AB?

a) ρ2g(h_{1} + h_{2})

b) ρ2g(h_{1} + \(\frac{h_2}{2}\))

c) ρ2g(\(\frac{ρ_1}{ρ_2}\)h_{1} + \(\frac{h_2}{2}\))

d) ρ2g(\(\frac{ρ_1}{ρ_2}\)h_{1} + \(\frac{h_2}{2}\))^{2}

View Answer

Explanation: Total liquid pressure on the lamina = F = γyA, where γ = specific weight of the liquid, y = depth of centroid of the lamina from the free surface, A= area of the centroid. Now,

7. A container is lled with two liquids of densities ρ and 2ρ up to heights h and eh respectively. What will be the ratio of the total pressure on the lower face AB and on the upper face BC?

a) 1 : 1

b) 3 : 1

c) 2 : 1

d) 3 : 2

View Answer

Explanation: Total liquid pressure on the lamina = F = γyA, where γ = specific weight of the liquid, y = depth of centroid of the lamina from the free surface, A= area of the centroid.

8. A container is lled with two liquids of densities ρ and 2ρ up to heights h and eh respectively. What will be the ratio of the depths of the centres of pressure of the upper face BC and the lower face AB?

a) 1 : 2

b) 3 : 4

c) 2 : 3

d) 3 : 2

View Answer

9. A gate of length 5 m is hinged at A as shown to support a water column of height 2.5 m. What should be the minimum mass per unit width of the gate to keep it closed?

a) 3608

b) 4811

c) 7217

d) 9622

View Answer

Explanation: To keep the gate closed, moment due to weight of the gate should be balanced by the moment due to the hydrostatic force.

where m = mass of the plate, θ = angle of inclination to the horizontal, F

_{hyd}= hydrostatic force on

the plate, x = distance of the point of action of F

_{hyd}from the hinge point =

^{2}⁄

_{3}* 5 =

^{10}⁄

_{3}

F

_{hyd}= γyA, where γ = specific weight of the liquid = 9.81 * 10

^{3}y = depth of the centre of pressure from the free surface = 2.5/2 = 1.25 and A = 5 * 1. Substituting all the values in the equation, we get m = 9622.5g.

10. A large tank is lled with three liquids of densities ρ1, ρ2 and ρ3 up to heights of h_{1}, h_{2} and h_{3} respectively. What will be the expression for the instantaneous velocity of discharge through a small opening at the base of the tank? (assume that the diameter of the opening is negligible compared to the height of the liquid column)

a) \(\sqrt{2g(h_1 + h_2 + h_3)}\)

b) \(\sqrt{2g(\frac{ρ_1h_1+2ρ_2h_2+3ρ_3h_3}{ρ_1})}\)

c) \(\sqrt{2g(\frac{ρ_1h_1+\frac{1}{2}ρ_2h_2+\frac{1}{3}ρ_3h_3}{ρ_2})}\)

d) \(\sqrt{2g(\frac{ρ_1h_1+ρ_2h_2+ρ_3h_3}{ρ_3})}\)

View Answer

11. A large tank of height h is filled with a liquid of density ρ. A similar tank is half-filled with this liquid and other-halffilled with another liquid of density 2ρ as shown. What will be the ratio of the instantaneous velocities of discharge through a small opening at the base of the tanks? (assume that the diameter of the opening is negligible compared to the height of the liquid column in either of the tanks)

a) 2 : 3

b) 2 : √3

c) √2 : 3

d) √2 : √3

View Answer

12. A large tank of height h is half-filled with a liquid of density ρ and other half-filled with a liquid of density 4ρ. A similar tank is half-filled with a liquid of density 2ρ and other-half filled with another liquid of density 3ρ as shown. What will be the ratio of the instantaneous velocities of discharge through a small opening at the base of the tanks? (assume that the diameter of the opening is negligible compared to the height of the liquid column in either of the tanks)

a) 1 : 1

b) 1 : 2

c) 2 : 1

d) 1 : 3

View Answer

13. A large tank is filled with three liquids of densities ρ, 2ρ and 3ρ up to a height of ^{h} ⁄ _{3} each. What will be the expression for the instantaneous velocity of discharge through a small opening at the base of the tank? (assume that the diameter of the opening is negligible compared to the height of the liquid column)

a) \(\sqrt{gh}\)

b) \(\sqrt{2gh}\)

c) \(2\sqrt{gh}\)

d) \(4\sqrt{gh}\)

View Answer

14. A large tank is filled with three liquids of densities ρ, 2ρ and 3ρ up to heights of ^{h} ⁄ _{6}, ^{h} ⁄ _{3} and ^{h} ⁄ _{2} respectively. What will be the ratio of the instantaneous velocity of discharge through a small opening at the base of the tank in this case to that if the container is filled with the liquid of density ρ only? (assume that the diameter of the opening is negligible compared to the height of the liquid column)

a) \(\sqrt{2} : \sqrt{3}\)

b) \(\sqrt{4} : \sqrt{3}\)

c) \(\sqrt{7} : \sqrt{4}\)

d) \(\sqrt{7} : \sqrt{3}\)

View Answer

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