A liquid stands at the same level in the u-Tube when at rest what will the difference in height h of the liquid in the two limbs of the u-Tube, when the system is given An acceleration a towards right? take a as area of a cross section and G as the acceleration due to gravity
To determine the difference in height (h) of the liquid in the two limbs of the U-tube when the system is given an acceleration (a) towards the right, you need to consider the forces acting on the liquid in the U-tube.
1. Gravity force: The weight of the liquid in the U-tube creates a downward force due to gravity, given by the formula Fg = m * g, where m is the mass of the liquid and g is the acceleration due to gravity.
2. Buoyant force: The liquid is also subject to a buoyant force, which is the upward force exerted on it by the surrounding fluid. The buoyant force is given by the formula Fb = ρ * g * V, where ρ is the density of the liquid and V is the volume of the liquid displaced.
3. Inertial force: When the U-tube is given an acceleration towards the right, there will be an inertial force acting on the liquid in the right limb. This force is given by the formula Fi = m * a, where m is the mass of the liquid and a is the acceleration.
The net force acting on the liquid in the U-tube is given by the difference between the gravity force, the buoyant force, and the inertial force:
Net force = Fg - Fb - Fi
Now, let's assume that the liquid in the U-tube is at the same level (h) when at rest. This means that the net force acting on the liquid is zero. Therefore:
0 = Fg - Fb - Fi
Substituting the respective formulas, we get:
0 = (m * g) - (ρ * g * V) - (m * a)
Now, let's observe the two limbs of the U-tube separately:
Left limb: The height of the liquid in the left limb is h.
Right limb: The height of the liquid in the right limb is h - Δh (where Δh is the difference in height caused by the acceleration).
Considering the pressure difference between the two limbs, we have:
Pressure in left limb = Pressure in right limb
Given that pressure is given by the formula P = ρ * g * h (pressure = density * gravity * height), we can equate the pressures:
(ρ * g * h) = (ρ * g * (h - Δh))
Rearranging the equation, we get:
(h - Δh) = h
Expanding the equation, we find:
h - Δh = h
Canceling out "h" on both sides, we get:
-Δh = 0
Multiplying both sides by -1, we find:
Δh = 0
This means that the difference in height of the liquid in the two limbs of the U-tube when the system is given an acceleration towards the right is zero. Therefore, there is no difference in height between the two limbs.