Problem Statement
Determine the speed of rotation of a cylinder with a diameter of 900 mm when the liquid rises to 500 mm at the sides and leaves a circular space of 300 mm diameter at the bottom uncovered. Taking the liquid as water, calculate the total pressure on the bottom. Also, find the depth when the vessel is stationary.
Solution
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Given Data:
Radius of cylinder: \( r = 0.45m \)
Radius at bottom: \( r_1 = 0.15m \) -
Equation for Parabolic Surface:
The free surface follows a parabolic shape given by:
\( z = \frac{r^2 \omega^2}{2g} \) -
For Points A and B:
\( x + 0.5 = \frac{(0.45)^2 \omega^2}{2g} = 0.0103\omega^2 \) (Equation 1) -
For Points C and D:
\( x = \frac{(0.15)^2 \omega^2}{2g} = 0.001147\omega^2 \) (Equation 2) -
Solving for Angular Velocity:
\( \omega = 7.4 \text{ rad/s}, \quad x = 0.063m \)Converting to RPM:
\( N = \frac{60\omega}{2\pi} = \frac{60 \times 7.4}{2\pi} = 70.6 \text{ rpm} \) -
Calculating Volume of Water Spilled:
\( V_{spilled} = \frac{1}{2} \pi (0.45)^2 (0.5+0.063) – \frac{1}{2} \pi (0.15)^2 (0.063) = 0.1768 m^3 \) -
Calculating Total Pressure on the Bottom:
\( V_r = 0.1412 m^3 \)
\( P_{bottom} = \gamma V_r = 9810 \times 0.1412 = 1385 N \) -
Finding Depth When Stationary:
\( d = \frac{V_r}{\pi r^2} = \frac{0.1412}{\pi (0.45)^2} = 0.22m \)
Physical Meaning
When a cylindrical vessel rotates, the liquid inside moves outward due to centrifugal forces. This causes the liquid to rise along the cylinder’s walls while lowering in the center, forming a parabolic surface. If the rotation is fast enough, part of the bottom may become exposed. This principle is used in devices such as centrifuges and hydrocyclones.
Explanation
The problem involves determining the rotation speed required for the liquid to form a free surface exposing a circular region at the bottom. We apply the equation for parabolic liquid surfaces, use volume calculations to determine the liquid left, and finally compute the total pressure exerted on the bottom of the tank.
The results show that at an angular speed of 7.4 rad/s (or 70.6 rpm), the liquid rises to 500 mm at the sides and forms a 300 mm diameter exposed region at the bottom. The remaining water volume exerts a pressure of 1385 N on the bottom.
