To tackle this problem, we need to understand how the rotation of the cylindrical vessel affects the height of the water inside it. When the vessel rotates, the water experiences a centrifugal force that causes it to rise along the sides of the cylinder. This effect is crucial in determining how much of the bottom area will become uncovered when the angular speed is increased.
Understanding the Forces at Play
In a rotating system, the centrifugal force acts outward from the axis of rotation. For a liquid in a cylindrical vessel, this force causes the liquid to rise along the walls. The height of the liquid at the edge of the cylinder can be calculated using the balance of forces. The pressure at the bottom of the liquid column must equal the pressure due to the centrifugal force at the edge.
Initial Conditions
Initially, the angular speed is w, and the height of the liquid is H. The radius of the vessel is R. The centrifugal force can be expressed as:
- Pressure due to liquid column: P = ρgH
- Centrifugal pressure at the edge: P_c = ρ(Rω²)
Setting these equal gives us:
ρgH = ρRω²
From this, we can derive the height of the liquid:
H = (Rω²)/g
Effects of Increasing Angular Speed
Now, when the angular speed is increased to 2w, we need to find the new height of the liquid. The new centrifugal pressure at the edge becomes:
P_c' = ρR(2w)² = 4ρRω²
Setting the new centrifugal pressure equal to the pressure due to the liquid column gives:
ρgH' = 4ρRω²
From this, we can derive the new height:
H' = (4Rω²)/g
Calculating the Uncovered Area
To find out how much area of the bottom becomes uncovered, we need to compare the new height H' with the original height H. The difference in height will determine how much of the bottom is exposed:
H' - H = (4Rω²)/g - (Rω²)/g = (3Rω²)/g
This difference indicates that the height of the liquid has increased significantly, and we can determine the volume of the liquid that is now above the original height.
Final Area Calculation
The area of the bottom of the cylinder is given by:
A = πR²
To find the area that becomes uncovered, we need to calculate the volume of liquid that corresponds to the height difference:
Volume = Area × Height Difference = πR² × (3Rω²/g)
Since this volume corresponds to the area that is now uncovered, we can express the uncovered area as:
Uncovered Area = Volume / Height = (πR² × (3Rω²/g)) / H
Conclusion
In summary, when the angular speed is increased to 2w, the area of the bottom that becomes uncovered can be calculated based on the change in height due to the increased centrifugal force. The uncovered area will depend on the specific values of R, ω, and g, but the key takeaway is that the increase in angular speed significantly affects the height of the liquid and, consequently, the area that becomes exposed.
