Question

I am spinning a bucket of water about its vertical axis with angular velocity \(\Omega\). Show that, once the water has settled in equilibrium (relative to the bucket), its surface will be a parabola. (Use cylindrical polar coordinates and remember that the surface is an equipotential under the combined effects of the gravitational and centrifugal forces.)

Short answer

The surface of the water forms a parabola described by \\(z = \frac{V_0}{mg} - \frac{1}{2g}\Omega^2r^2\\).

Step-by-step solution

Understanding the Forces

We need to consider two main forces acting on the water in the bucket: the gravitational force and the centrifugal force due to rotation. Gravitational force acts downward, while the centrifugal force acts radially outward in the rotating frame.

Choosing the Coordinate System

We use cylindrical polar coordinates \(r, \theta, z\) where \(r\) is the radial distance from the rotation axis, \(\theta\) is the angular coordinate, and \(z\) is the height above a reference plane. The bucket rotates about the vertical axis (z-axis).

Expressing the Potential Energy

The potential energy involves both gravitational and centrifugal components. Gravitation potential energy is \(V_g = mgz\) and centrifugal potential energy is given by \(V_c = \frac{1}{2}m\Omega^2r^2\), where \(m\) is the mass of the water element.

Setting the Total Potential Energy

Since the surface of the water is an equipotential, the total potential \(V = V_g + V_c = mgz + \frac{1}{2}m\Omega^2r^2\) must be constant for any point \(r, z\) on the surface.

Equating the Potential Energy Terms

Set the potential energy to a constant value, \(V_0\). This gives: \(mgz + \frac{1}{2}m\Omega^2r^2 = V_0\). Solving for \(z\), we have \(z = \frac{V_0}{mg} - \frac{1}{2g}\Omega^2r^2\).

Identifying the Parabolic Surface

The equation \(z = \text{constant} - \frac{1}{2g}\Omega^2r^2\) represents a parabola. The water's surface, described by this equation, is thus a parabola opening downward along the \(z\)-axis due to the \(-r^2\) term.

Key concepts

Angular Velocity

Angular velocity is an important concept when dealing with rotation, such as spinning a bucket of water. It describes how fast an object is rotating around an axis. In our case, the rotation occurs around the vertical axis. Angular velocity is denoted by \( \Omega \) and is measured in radians per second. Here are some key points to remember about angular velocity:

• It is a vector quantity, meaning it has both a magnitude and a direction.
• The direction is determined by the right-hand rule, pointing along the axis of rotation.
• It describes the rate at which an angle is swept out by a rotating object.

In cylindrical coordinates, angular velocity becomes extremely useful. It affects how forces, such as gravitational and centrifugal forces, interact to shape the water's surface. When the bucket rotates steadily, the angular velocity causes the centrifugal force to push water outwards. This interplay results in the water forming a parabolic shape in equilibrium.

Centrifugal Force

The centrifugal force comes into play when an object is in a rotating frame, like our bucket of water. It acts perpendicular to the axis of rotation, pointing outward. This force can be thought of as the inertia of an object moving in a curve, trying to "escape" the curve.Consider the following about centrifugal force:

• It is proportional to the square of the angular velocity: \( F_c = m \Omega^2 r \), where \( r \) is the radial distance from the center.
• It is a fictitious force experienced in a non-inertial frame—appears due to rotation rather than any physical interaction.
• In our rotating bucket scenario, this force pushes the water outwards, affecting the balance between the gravitational pull and the centrifugal push.

When water in the bucket spins, the centrifugal force increases with the square of the distance from the center. In equilibrium, this force acts alongside gravity to help shape the surface of the water, forming a parabola in cylindrical coordinates. The interplay of centrifugal force and angular velocity reveals how dynamic and fascinating rotations can be.

Gravitational Potential Energy

Gravitational potential energy is fundamental to understanding how systems like our spinning bucket interact with the surrounding forces. It is the energy an object has due to its position in a gravitational field, which in most cases, is Earth’s field.Gravitational potential energy is described by:

• Formula: \( V_g = mgh \), where \( m \) is the mass, \( g \) is the gravitational acceleration, and \( h \) is the height.
• It acts downward and affects how the shape of the liquid surface aligns with gravity.
• In a gravitational field, this energy seeks to minimize itself, contributing to equilibrium conditions.

In the context of the rotating bucket, gravitational potential energy combines with centrifugal energy to determine the surface of the water. As water settles, the balance between these two types of potential energy dictates its final shape—a downward-opening parabola resting along its equipotential. By understanding how gravitational potential energy interacts with forces in rotation, one can appreciate how equilibrium is achieved in a rotating system.