Question

A rectangular car tank 20m long, 4m wide and 3m deep is completely open at the top. If it is initially filled to the top: (a) How much liquid will be spilled if it is given a horizontal acceleration of 0.3g in the direction of its length? (b) What is the maximum hydrostatic force acting on the side opposite the direction of the acceleration. (c) Determine the volume of the remaining water at the tank.

Answer 1

Question: A rectangular car tank 20m long, 4m wide and 3m deep is completely open at the top. If it is initially filled to the top:

(a) How much liquid will be spilled if it is given a horizontal acceleration of 0.3g in the direction of its length?

(b) What is the maximum hydrostatic force acting on the side opposite the direction of the acceleration?

(c) Determine the volume of the remaining water at the tank.

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Solution:

Given:

Tank length, l = 20 m

Tank width, b = 4 m

Tank depth, h = 3 m

Horizontal acceleration, a_x = 0.3g = 0.3 x 9.81 = 2.943 m/s²

Density of water = 997 kg/m³

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Step 1 of 3 │ Amount of water spilled due to 0.3g horizontal acceleration

Tank at rest position

Let θ is the angle which the free surface of the water will make with the horizontal,

Therefore the slope of the free surface (θ) is

tan(θ) = a_x / g = 2.943 / 9.81 = 0.3

θ = tan^-1 (0.3) = 16.7⁰

From Figure (2), the depth of water at the front side h₁ is given by

h₁ = 20 x tanθ

h₁ = 20 x 0.3

h₁ = 6 m

Which is much higher than the height of the tank, that means more than half tank water is spilled due to the acceleration, So the modified figure is  (Refer Figure 3)

Therefore the quantity of water which will spill out of the tank is

Spilled volume = (0.5 x 10 x 3 x 4) + (10 x 3 x 4)

Spilled volume = 60 + 120

Spilled volume = 180 m³ = 1,80,000 liters

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Step 2 of 3 │ Hydrostatic force acting on the side opposite the direction of acceleration.

Maximum pressure acting on the opposite side of the direction of acceleration is

P_AB = ρ x g x h_AB

P_AB = 997 x 9.81 x 3

P_AB = 29341.71 N/m²

Therefore hydrostatic force acting in the side AB of tank

F_AB = Area of pressure triangle x width

F_AB = 0.5 x 29341.71 x 3 x 4

F_AB = 176050.26 N

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Step 3 of 3 │ Remaining volume of water at the tank

The remaining volume of water at the tank is

Remaining Volume = Initial Volume – Spilled volume

Remaining Volume = 240 – 180

Remaining Volume = 60 m³ = 60,000 liters

Final Answer:

• Amount of water spilled due to 0.3g horizontal acceleration 180 m³or 1,80,000 liters
• The hydrostatic force acting on the side opposite the direction of acceleration is 176050.26 N
• The remaining volume of water at the tank 60 m³or 60,000 liters

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