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The vertical limbs of a U shaped tube are filled with a liquid of density r upto a height h on each side. The horizontal portion of the U tube having length 2h contains a liquid of density 2r. The U tube is moved horizontally with an accelerator g/2 parallel to the horizontal arm. The difference in heights in liquid levels in the two vertical limbs, at steady state will be
- a)2h/7
- b)8h/7
- c)4h/7
- d)None
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The vertical limbs of a U shaped tube are filled with a liquid of dens...
Given:
a=g/2
Pressure at A
PA=Po+ρgh+(2ρ)g(h−x)=Po+3ρgh−2ρgx
Pressure at B
PB=Po+ρgx
Using
PA−PB=[2ρ(h+x)+ρ(h−x)]a
∴ (Po+3ρgh−2ρgx)−(Po+ρgx)=[3ρh+ρx]×g/2
OR 3ρgh−3ρgx=3ρgh/2+1ρgx/2
OR 3ρgh/2=7ρgx/2 ⟹x=3h/7
∴ Difference in the heights between two columns ΔH=(2h−x)−x=2h−2x
⟹ ΔH=2h−6h/7=8h/7
a=g/2
Pressure at A
PA=Po+ρgh+(2ρ)g(h−x)=Po+3ρgh−2ρgx
Pressure at B
PB=Po+ρgx
Using
PA−PB=[2ρ(h+x)+ρ(h−x)]a
∴ (Po+3ρgh−2ρgx)−(Po+ρgx)=[3ρh+ρx]×g/2
OR 3ρgh−3ρgx=3ρgh/2+1ρgx/2
OR 3ρgh/2=7ρgx/2 ⟹x=3h/7
∴ Difference in the heights between two columns ΔH=(2h−x)−x=2h−2x
⟹ ΔH=2h−6h/7=8h/7
Most Upvoted Answer
The vertical limbs of a U shaped tube are filled with a liquid of dens...
To understand why the difference in heights in liquid levels in the two vertical limbs will be 8h/7, let's analyze the scenario step by step.
1. Initial State:
- The U-shaped tube is filled with a liquid of density r in the vertical limbs up to a height h on each side.
- The horizontal portion of the U tube, which has a length of 2h, contains a liquid of density 2r.
2. Forces Acting on the System:
- The force due to gravity acts vertically downwards on each column of liquid in the vertical limbs.
- The force due to the acceleration g/2 acts horizontally towards the right on the liquid in the horizontal portion.
3. Pressure Difference:
- The pressure at the bottom of each vertical limb is higher than at the top due to the weight of the liquid column.
- The pressure difference between the two vertical limbs is given by the equation: ΔP = ρgh, where ρ is the density of the liquid and h is the height of the liquid column.
4. Balancing Forces:
- In order to maintain equilibrium, the net force on the liquid in the horizontal portion must be zero.
- The force due to gravity is balanced by the force due to the acceleration: F_g = m*g = (2h*r*g) = (2h*(2r)*g/2).
- The force due to pressure difference is balanced by the force due to acceleration: F_p = ΔP*A = (ρgh)*(2h) = (r*g*h)*(2h).
5. Equating Forces:
- Equating the forces due to gravity and pressure difference: (2h*r*g) = (r*g*h)*(2h).
- Simplifying the equation: 2r = h.
6. Calculating the Difference in Height:
- The difference in heights in the two vertical limbs is equal to the height of the liquid column in one vertical limb minus the height in the other.
- Δh = (2h) - h = h.
7. Final Answer:
- Substituting the value of h from equation 5: Δh = 2r = 2*(2h) = 4h.
- Therefore, the difference in heights in the two vertical limbs at steady state is 4h.
- This corresponds to option 'B' in the given answer choices.
By analyzing the forces and pressures acting on the system, we can determine the difference in heights in the two vertical limbs of the U-shaped tube when it is moved horizontally with an acceleration of g/2.
1. Initial State:
- The U-shaped tube is filled with a liquid of density r in the vertical limbs up to a height h on each side.
- The horizontal portion of the U tube, which has a length of 2h, contains a liquid of density 2r.
2. Forces Acting on the System:
- The force due to gravity acts vertically downwards on each column of liquid in the vertical limbs.
- The force due to the acceleration g/2 acts horizontally towards the right on the liquid in the horizontal portion.
3. Pressure Difference:
- The pressure at the bottom of each vertical limb is higher than at the top due to the weight of the liquid column.
- The pressure difference between the two vertical limbs is given by the equation: ΔP = ρgh, where ρ is the density of the liquid and h is the height of the liquid column.
4. Balancing Forces:
- In order to maintain equilibrium, the net force on the liquid in the horizontal portion must be zero.
- The force due to gravity is balanced by the force due to the acceleration: F_g = m*g = (2h*r*g) = (2h*(2r)*g/2).
- The force due to pressure difference is balanced by the force due to acceleration: F_p = ΔP*A = (ρgh)*(2h) = (r*g*h)*(2h).
5. Equating Forces:
- Equating the forces due to gravity and pressure difference: (2h*r*g) = (r*g*h)*(2h).
- Simplifying the equation: 2r = h.
6. Calculating the Difference in Height:
- The difference in heights in the two vertical limbs is equal to the height of the liquid column in one vertical limb minus the height in the other.
- Δh = (2h) - h = h.
7. Final Answer:
- Substituting the value of h from equation 5: Δh = 2r = 2*(2h) = 4h.
- Therefore, the difference in heights in the two vertical limbs at steady state is 4h.
- This corresponds to option 'B' in the given answer choices.
By analyzing the forces and pressures acting on the system, we can determine the difference in heights in the two vertical limbs of the U-shaped tube when it is moved horizontally with an acceleration of g/2.
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B
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The vertical limbs of a U shaped tube are filled with a liquid of density r upto a height h on each side. The horizontal portion of the U tube having length 2h contains a liquid of density 2r. The U tube is moved horizontally with an accelerator g/2 parallel to the horizontal arm. The difference in heights in liquid levels in the two vertical limbs, at steady state will bea)2h/7b)8h/7c)4h/7d)NoneCorrect answer is option 'B'. Can you explain this answer?
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