Finding the Shape of the Surface of Water Rotating in a Bucket
By: Marco Tavora Ph.D.
Tags: Science, Math

In this article, I will consider the problem of finding the shape of the free surface of the water inside a half-full bucket rotating with constant angular speed. Since I will work with systems moving with speeds much smaller than the speed of light, relativistic effects can be safely neglected. The derivation will be based on the Newtonian laws of motion, first enunciated in Newton’s magnum opus, the Principia.

The Principia was first published in 1687, followed by two expanded editions in 1713 and 1726. An American edition appeared in 1846. Page 83 contains the statement of “Newton’s Laws of Motion,” as shown below.

This article is based on Knudsen and Hjorth, which I will henceforth call KH.

Figure 1: The portrait of Isaac Newton facing the title page of the Principia and, on page 83, “Newton’s Laws of Motion.”

Kinematics in Accelerated Reference Frames

What are the equations of motion of a body, expressed in terms of coordinates in the reference frame S, moving arbitrarily with respect to the inertial reference frame I?

Our first goal will be to find Newton’s second law expressed in the moving frame S. Note that any motion of S can be obtained by a combination of a translation of the origin of I and a rotation of S around an axis crossing its origin.

Figure 2: The system of coordinates S, moving arbitrarily with respect to the inertial system I (figure based on KH).

We first call R the position of a body of mass m with respect to I and r the position of the body with respect to S. To find Newton’s second law of motion in S coordinates we first need to obtain the relation between the acceleration in I and S.

The velocity and acceleration vectors of m measured in the inertial coordinate frame are:

Equation 1: The velocity and acceleration vectors of m measured in the inertial coordinate frame I.

The velocity and acceleration vectors of m measured in the moving coordinate frame are:

Equation 2: The velocity and acceleration vectors measured in the moving coordinate frame S.

Note that, since we are using Newton’s mechanics, the measure of time does not change from I to S.

As shown in Fig. 2, the position of the body in I can be written as:

Equation 3: The position of the body in the reference frame I.

Differentiating Eq. 3 twice we obtain, after some algebra, the following expression:

Equation 4: The decomposition of the total acceleration relative to S.

In this equation, a is the acceleration of the body with respect to S, the three terms on the right-hand side are, respectively:

The relative acceleration between I and S.

The acceleration of the body relative to the inertial frame I if the body is at rest at an instantaneous position (x, y, z) in S (this is the so-called comoving acceleration).

The Coriolis acceleration.

Figure 3: In the inertial reference frame (top), the black ball moves in a straight line. However, due to the Coriolis and centrifugal forces present in the rotating reference frame (bottom) the observer (red dot) who sees the object as following a curved path (source).

The three components of a have the following form:

Equation 5: The expression for the three terms in Eq. 4.

We can re-write the second and third lines of Eq. 5 using ω and dω/dt where ω is the angular velocity of the frame S:

Equation 6: The comoving and Coriolis accelerations written in terms of the angular velocity and its time derivative.

Figure 4: Directions of the centrifugal and Coriolis forces in the S frame.

The second term (with an extra minus sign) in the comoving acceleration (the first line of Eq. 6) can be written as:

Equation 7: Centrifugal acceleration.

Dynamics in Accelerated Reference Frames

We will now express Newton’s second law as seen by an observer in the reference frame S. Following KH, we assume that the force acting on the body is the same in both reference frames. Hence:

which gives us Newton’s second law in S:

Equation 8: Equation of motion (Newton’s second law) of the body as observed from S.

For Newton’s second law to be valid in S, we interpret the last four terms on the right-hand side of Eq. 8 as fictitious forces.

The Rotating Bucket

We now consider a bucket half full of water rotating with angular velocity ω about its symmetry axis. To find the shape of the surface of the water we proceed as follows.

Figure 5: In the inertial reference frame I, the bucket is rotating. Eventually, the water comes to rest relative to the bucket. In the non-inertial reference frame S that also rotates with angular velocity ω, the bucket and the water are at rest (based on this source).

In the non-inertial reference frame S that also rotates with angular velocity ω, the bucket is at rest. After some time the water will come to rest with respect to the bucket.

Consider a water volume element, with mass m. The forces acting on it are:

The gravitational force

The centrifugal force

The force originating from the pressure gradient

Since the mass is at rest in the rotating coordinate system S, the sum of these forces must be zero in S:

Equation 9: Equilibrium equation for the water volume element in the non-inertial reference frame.

The double product in the second term can be rewritten as:

The components of the equilibrium equation are, therefore:

Equation 10: Components of the equilibrium equation in the non-inertial reference frame S.

After integration, we obtain:

Equation 11: The pressure P obtained after integrating Eq. 12.

For a surface with constant pressure, Eq. 11 gives us:

Equation 12: The surface of the water is a paraboloid.

We conclude that the shape of the water in the bucket is a paraboloid.