Problem 29: Radius of Water Dome
1. Determine Horizontal Exit Velocity ($v_x$):
Water exits the gap between the discs horizontally. The area of the opening is the cylindrical surface area of the gap: $A = 2\pi r h$. Using the flow rate $\mu$ (kg/s): $$ \mu = \rho A v_x = \rho (2\pi r h) v_x $$ $$ v_x = \frac{\mu}{2\pi r h \rho} $$
Water exits the gap between the discs horizontally. The area of the opening is the cylindrical surface area of the gap: $A = 2\pi r h$. Using the flow rate $\mu$ (kg/s): $$ \mu = \rho A v_x = \rho (2\pi r h) v_x $$ $$ v_x = \frac{\mu}{2\pi r h \rho} $$
2. Analyze Fluid Motion (Projectile Motion):
Once the fluid leaves the gap, it falls under gravity. Since we neglect capillary effects and air resistance, each fluid element behaves like a projectile projected horizontally.
Once the fluid leaves the gap, it falls under gravity. Since we neglect capillary effects and air resistance, each fluid element behaves like a projectile projected horizontally.
- Horizontal velocity is constant: $v_x$.
- Vertical motion is free fall: $y = \frac{1}{2}gt^2$.
3. Calculate Horizontal Displacement:
At depth $H$ ($y=H$), the time taken to fall is: $$ H = \frac{1}{2}gt^2 \implies t = \sqrt{\frac{2H}{g}} $$ The horizontal distance traveled by the fluid is: $$ x = v_x t = \left( \frac{\mu}{2\pi r h \rho} \right) \sqrt{\frac{2H}{g}} $$
At depth $H$ ($y=H$), the time taken to fall is: $$ H = \frac{1}{2}gt^2 \implies t = \sqrt{\frac{2H}{g}} $$ The horizontal distance traveled by the fluid is: $$ x = v_x t = \left( \frac{\mu}{2\pi r h \rho} \right) \sqrt{\frac{2H}{g}} $$
4. Total Radius of Dome ($R$):
The total radius $R$ at depth $H$ is the initial radius of the disc plus the horizontal expansion $x$. $$ R = r + x $$ $$ R = r + \frac{\mu}{2\pi r h \rho} \sqrt{\frac{2H}{g}} $$
The total radius $R$ at depth $H$ is the initial radius of the disc plus the horizontal expansion $x$. $$ R = r + x $$ $$ R = r + \frac{\mu}{2\pi r h \rho} \sqrt{\frac{2H}{g}} $$
Answer:
$$ R = r + \frac{\mu}{2\pi r h \rho} \sqrt{\frac{2H}{g}} $$