Final Answer:
The equation of the free surface of a fluid-filled cylinder, rotated by an axis parallel to its main axis and passing through a point on its circumference, is a parabola. The shape of the free surface is determined by the rotation, resulting in a concave upward curve resembling a parabolic form.
Step-by-step explanation:
When a fluid-filled cylinder is rotated about an axis parallel to its main axis and passing through a point on its circumference, the resulting free surface takes the shape of a parabola. To understand this, consider the dynamics of the rotating fluid. The centripetal force acting on the fluid elements causes them to move outward, forming a parabolic curve. Mathematically, this can be expressed using the equation of a parabola.
Let (y) represent the radial distance from the axis of rotation to a point on the free surface, and (x) denote the axial distance along the main axis. The equation of the parabolic free surface can be expressed as (y = a x2 + b x + c), where (a), (b), and (c) are constants determined by the specific conditions of the system.
To find these constants, boundary conditions and fluid mechanics principles must be applied. The parabolic shape emerges as a result of balancing gravitational forces, centripetal forces, and fluid pressure. Through rigorous mathematical analysis and integration, the constants can be determined, providing a complete description of the fluid's free surface in the rotated cylinder scenario.
In conclusion, the rotation of a fluid-filled cylinder about an axis parallel to its main axis, passing through a point on its circumference, leads to a parabolic free surface. This phenomenon arises from the interplay of various forces and can be precisely described through mathematical modeling and analysis.