Question

a right circular cylindrical tank containing water spins about its vertical axis OO at such a speed that the free surface is a paraboloid ACB. What wil be the depth of water in the tank when it comes to rest?​

Answer 1

Answer:

Kepler's second law states that a planet moves in its ellipse so that the line between it and the Sun placed at a focus sweeps out equal areas in equal times

Answer 2

Answer:

The depth of water in the tank is  parabola.A  right circular cylindrical tank containing water spins about its vertical axis OO at such a speed that the free surface is a paraboloid ACB.

Explanation:

Step : 1 Let the density of water be ρ and the cylinder is rotating with angular velocity w.

Also assume an elimentary cylinder of length x and cross-section A

Mass of water contained in an elimentary cylinder ,   M=$\rho \mathrm{V}=\rho(\mathrm{Ax})$

Centripetal force will act at the centre of mass i.e at  x /2

$\begin{aligned}& \mathrm{F}_{\mathrm{C}}=\mathrm{Mrw}^2=(\rho \mathrm{Ax}) \frac{\mathrm{x}}{2} \mathrm{w}^2 \\& \mathrm{~F}_{\mathrm{C}}=\frac{\rho A \mathrm{x}^2 \mathrm{w}^2}{2}\end{aligned}$

Now   for x direction: $\mathrm{F}_{\mathrm{B}}-\mathrm{F}_{\mathrm{O}}=\mathrm{F}_{\mathrm{C}}​$

$\begin{aligned}& \left(\mathrm{P}_{\mathrm{B}}-\mathrm{P}_{\mathrm{O}}\right) \mathrm{A}=\frac{\rho \mathrm{Ax}^2 \mathrm{w}^2}{2} \\& \left(\mathrm{P}_{\mathrm{o}}+\rho g \mathrm{y}\right)-\mathrm{P}_{\circ}=\frac{\rho \mathrm{x}^2 \mathrm{w}^2}{2}\end{aligned}$

where Po is the atmospheric pressure

$\Rightarrow \mathrm{y}-\frac{\mathrm{w}^2 \mathrm{x}^2}{2 \mathrm{~g}}=0$

which is an equation of parabola.

Step : 2   Any point on a parabola is at an equal distance from both the focus, a fixed point, and the directrix, a fixed straight line. A parabola is a U-shaped plane curve. The topic of conic sections includes parabola, and all of its principles are discussed here.

Step : 3 The parabola's equation may be obtained from its fundamental definition. The location of a point that is equally spaced from a fixed point known as the focus (F) is known as a parabola, and the fixed line is known as the directrix (x + a = 0). Consider the parabola's point P(x, y). Using the equation PF = PM, we can determine the parabola's equation. The directrix's point "M" in this instance is the foot of the perpendicular from point P. Consequently, y2 = 4ax is the derived standard equation of the parabola.

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