The Centrifuge Project

 

A compact DIY centrifuge and camera system enables some interesting experiments for pendulum motion, buoyancy, pseudo-gravity and fluid flow.

Introduction and Theory

Gravity Demonstrations

Many of the common demonstrations in introductory physics involve the effects of gravity. This includes objects sliding down incline planes, trajectories of baseballs, and the oscillation period of a pendulum. These phenomena are all governed by the gravitational attraction from the Earth’s mass, described by the gravitational acceleration $g = 9.8 m/s^2$. Dropping an object on Earth leads to an increase in falling speed of 9.8 m/s (21.9 mph) every second. The force on a mass m under the influence of gravity is F = mg.

In these types of experiments, parameters such as the mass, the angle of incline, the initial velocity, and the length of the pendulum are changed to check the predictions made by the equations of motion. The quantity g is constant. But what if we could also alter that as a parameter, without having to go into space or to another planet? One way to do this is to add Centrifugal Force to gravity by spinning the experiment in a Centrifuge.

Moving in a Circle

An object moving in a circle is constantly changing its direction. Some type of restraining force is needed to keep the ball going in the circle, such as a rope or metal rod in tension. This restraint provides a so-called Centripetal Force. The object experiences the inward pulling tension as if it were being pulled by an outward force, which is called the Centrifugal Force, $F_c$.

If an experiment is performed in a chamber going in a circle of radius R at a velocity v, there will be a combined downward pull of gravity and a radial pull of $F_c = \frac{mv^2}{R}$ that determines its motion. The rotational speed $\omega = \frac{v}{R}$ (measured in radians per second) is often used, giving $F_c = m \omega^2 R.$

Static Pendulum in a Spinning Centrifuge

A static pendulum (restrained from oscillating) in a centrifuge of radius R, rotating at a constant angular speed $\omega$, will cause the pendulum to tilt at an angle $\gamma$ from vertical, like in the amusement park ride photo. The pendulum bob of mass m experiences the sum of the force of gravity (mg), the centrifugal force ($F_c$), and the tension T of the pendulum rod. In the static case, all these forces are exactly balanced, giving us $\tan{\gamma} = \frac{\omega^2R}{g}$.

The faster the rotation or the larger the radius of the centrifuge, the greater is $F_c$ and the tilt angle $\gamma$. The pendulum acts as if there is a net pull of artificial gravity along the tilt angle.

Inclined Plane in a Spinning Centrifuge

A sliding object on an inclined plane (low friction) has very similar physics to a static pendulum in a spinning centrifuge. Instead of the tilt angle of the pendulum, we have the tilt of the inclined plane $\gamma$, this time measured from the horizontal. The object of mass m experiences the force of gravity mg, the centrifugal force $F_c$, and the supporting force N from the ramp. The object will be suspended on the ramp when these forces are exactly balanced. If $F_c$ is increased beyond this balance condition, or the ramp angle is reduced, the object will move uphill.

Oscillating Pendulum in a Static Centrifuge

If the centrifuge is not spinning ($\omega = 0$), then we just have a pendulum that oscillates normally. We can consider two cases of interest: A) The pendulum of length $\ell$ and mass m is constrained to swing along the centrifuge arm. The oscillation of the angle $\gamma$ has a frequency $\Omega = \sqrt{\frac{g}{\ell}}$, dependent only on the pendulum length and the Earth’s gravitational acceleration g. In configuration B), the pendulum swings across the arm. If we constrain the value of $\gamma = \gamma_0$ then the oscillation frequency is $\Omega = \sqrt{\frac{g \cos{\gamma_0}}{\ell}}$, since only the component of gravity along the vertical ($g \cos{\gamma_0}$) affects the pendulum. If $\gamma_0 = 0^\circ$ (oscillation in the vertical plane), we get the same behavior as case A. If $\gamma_0 = 90^\circ$ (oscillation in the horizontal plane), gravity has no influence on the mass and there is no oscillation. 

Oscillating Pendulum in a Spinning Centrifuge (case A)

When both the centrifuge and pendulum are moving, the physics gets more complex. 

In case A, where we constrain the oscillation to be along the centrifugal arm only ($\beta = 0$), the physics shows that the angle $\gamma$ is governed by the equation below (derived using the Lagrangian method). The oscillation frequency $\Omega$ and the amplitude are both a function of $\ell, g, \omega$, and R. This equation can be solved numerically, allowing the angle to be plotted as a function of time for specified values of these parameters.

$\frac{d^2\gamma}{dt^2} = -\frac{g}{\ell} \sin{\gamma} + \frac{R}{\ell} \omega^2 \cos{\gamma}+ \omega^2 \cos{\gamma} \sin{\gamma}$

The last term is small compared to the other two. The first two terms cancel when $\tan{\gamma} = \omega^2 \frac{R}{g}$, where the oscillation ceases as the rod tension, $F_c$ and gravity all balance.

Below are plots of oscillation angle $\gamma (t)$ for $\ell$ = 4.9 cm, R = 0.5m, $\gamma (0) = 10^\circ$ for various values of spin frequency $\omega$.

At $\omega = 0$ (static centrifuge), the pendulum oscillates from $-10^\circ$ to $+10^\circ$ at a frequency of $\Omega = \sqrt{\frac{g}{\ell}}$ = 14.4 rad/s. As $\omega$ increases, the angle range decreases while $\Omega$ remains essentially constant until all the forces balance as $\tan{\gamma} = \omega^2 \frac{𝑅}{g}$. Beyond that, $\Omega$ and the oscillation range increase. The angle $\gamma$ oscillates from $\gamma (0)$ to almost vertical (note the scale change in the plots). In this regime, $F_c$ dominates over gravity.

Oscillating Pendulum in a Spinning Centrifuge (case B)

In case B, we constrain the oscillation to be across the centrifugal arm at a fixed tilt $\gamma_0$, and the physics shows that the angle $\beta$ for $\gamma = \gamma_0$, has an oscillation frequency $\Omega$ for small values of $\beta$:

$\Omega^2 = \frac{g}{\ell} \cos{\gamma_0} + \frac{\omega^2 R}{\ell} \sin{\gamma_0} - \omega^2 {\cos{\gamma_0}}^2$

The last term is small compared to the other two. In this geometry, the gravitational and centrifugal forces add in the plane of the pendulum oscillation. Along the pendulum rod, there is an effective pull force mg’, where $g’ \approx g \cos{\gamma_0} + \omega^2 R\sin{\gamma_0}$. The tilt is fixed in the experiment at $\gamma_0$ defined by $\tan{\gamma_0} = \frac{\omega^2 R}{g}$.

Plots are shown below for $\ell$ = 4.9 cm, R = 0.5m, vs. $\omega$, fixing $\tan{\gamma_0} = \frac{\omega^2 R}{g}$.

At $\omega = 0$ (static centrifuge), the pendulum oscillates at a frequency of $\Omega = \sqrt{\frac{g’}{\ell}}$ = 14.4 rad/s, where g’ = g. As $\omega$ increases $\Omega$ remains constant until $F_c$ dominates over gravity for pendulum oscillation (g’ > g). The tilt angle $\gamma_0$ in this case increases from $0^\circ$ to $80^\circ$ as $\omega$ approaches 10 rad/s.

The Apparatus

What follows is how to make a centrifuge with the objects just discussed, and to capture a video in the reference frame of the object.

Note: Most of the design and fabrication for this system is owed to hobby machinist Keith.

Centrifuge Design

The motor base is bolted to three 1” square Aluminum channels with 1/8” thick walls (acting as horizontal tripod legs), each 20” long, tapped for a ¼” bolt at the base of the motor. The bolts can be loosened so the 3 legs can collapse for storage. The opposite end of the legs have holes to hold vertical rods for the surrounding cylindrical shield and also include leveling bolts. The purpose of the shield is to help prevent contact with the spinning arm or any loose objects. It is 38” in diameter, 2” wider than the centrifuge arm, and is made of three sections of 1/16” thick white vinyl, 15” high. The end of each section wraps around a support post. Further details are discussed in the following images.

A screen made from 3 lengths of vinyl sheeting wrap around vertical rods, secured at the top by pin clamps. 

Screen Details

A white LED acts as a point light to help count revolutions in the video images, to measure the rotation frequency $\omega$:

Balancing the Centrifuge

The motor shaft should be at the center of mass of the centrifuge arm (made from a 1” square aluminum channel). To achieve this, the arm is detached from the motor shaft and balanced on a rod in a separate fixture. Once the experiment and camera objects are in place, a counterweight is loaded with fender washers and positioned to balance the arm. The experiment platform and the counterweight are then locked down with bolts using nuts inside the channel. The arm is then reattached to the motor shaft.

WiFi Camera

Video: centrifuge motor at low speed

Experiment Fixtures

Pendulum Oscillation

Experiments

Static Pendulum in a Centrifuge: Effect of Buoyancy

Video: Glass bead in water

Video: Wood bead in water

Video: Glass + Wood bead in water

In the above videos, two pendula, one made from a glass bead and one from a wooden bead pivot about a bolt in a sealed plastic chamber filled with water. This was attached to the end of one arm of the centrifuge along with a wireless video camera. The plane of rotation is parallel to the centrifuge arm. As the centrifuge spin frequency $\omega$ is increased, the glass bead responds to $F_c$ and gravity, tilting at the angle $\tan{\gamma} = \frac{\omega^2 𝑅}{g}$. The wood bead pendulum floats to the top before the spinning starts, then tilts toward the center of the centrifuge as $\omega$ is increased. When both pendulums are used, they tilt opposite to one another. When this experiment is repeated in air, the wood bead pendulum behaves the same as the glass bead pendulum. 

When the centrifuge is spinning, the total force on the water and beads points along the angle $\gamma$, as if an effective gravity pointed in that direction. When an object is immersed in the spinning water, the pressure will change along this angle, making the buoyant force also along $\gamma$, causing the wood bead to float opposite that direction.

Pendulum Type A (oscillate along the centrifuge arm) 

In the videos below a pendulum of length 𝑙 = 4.9cm was placed into the centrifuge of radius R = 40.4cm. The oscillation of the pendulum was constrained to be along the centrifuge arm, held in the start position by friction. Once the rotation speed stabilized, a mechanical impulse released the pendulum. A frame-by-frame analysis was used to measure the rotation frequency $\omega$ (noting the LED embedded in the screen), pendulum oscillation frequency $\Omega$ and final rest angle $\gamma_0$. The rotation frequency was adjusted using a Variac transformer.

As the rotation speed increases, the final rest angle and the oscillation frequency increase. The oscillations are dampened by friction, especially for faster oscillations.

Videos:

        Centrifuge $\omega$ = 1.51 rad/s                         Centrifuge $\omega$ = 4.16 rad/s

         Centrifuge $\omega$ = 6.73 rad/s                         Centrifuge $\omega$ = 9.15 rad/s

Measurements of the final rest angle $\gamma_0$ vs. the centrifuge rotation frequency $\omega$ were compared to the prediction $\tan{\gamma} = \frac{\omega^2 𝑅}{g}$, as seen in the plot. The results show reasonable agreement given the quality of the video capture.

Measurements of the pendulum oscillation frequency $\Omega$ vs. the centrifuge rotation frequency $\omega$ were compared to the prediction from the equation of motion. The results show reasonable agreement given the quality of the video capture. The frequency increases as $F_c$ dominates the gravitational force.

Pendulum Type B (oscillate across the centrifuge arm) 

In the videos below a pendulum of length 𝑙 = 4.9cm was placed into the centrifuge of radius R = 40.4cm. The oscillation was constrained to be across the centrifuge arm, tilted at an angle ϒ0 along the arm, held in the start position by a small hook. Once the rotation speed stabilized, a mechanical impulse released the pendulum. A frame-by-frame analysis was used to measure the rotation frequency $\omega$ (noting the LED embedded in the screen), and pendulum oscillation frequency $\Omega$. The rotation frequency was adjusted using the Variac transformer.

Centrifuge $\omega$ = 2.05 rad/s 

Centrifuge $\omega$ = 4.63 rad/s 

Centrifuge $\omega$ = 12.0 rad/s

The pendulum tilt angle $\gamma_0$ was set to the calculated value at each centrifuge spin speed $\omega$. The video images (looking along the centrifuge arm) were analyzed to determine the centrifuge frequency $\omega$ and the pendulum oscillation frequency $\Omega$. 

The measurements were compared to the calculation, as shown in the plot, for two trials. The results show reasonable agreement given the quality of the video capture. At the highest spin speed the effective gravitational acceleration g’ = 5.6 g.

Sand Timer in a Centrifuge

Video at 2x speed

A 1-minute sand timer was placed horizontally in the centrifuge at R = 38cm from center. The final spin frequency was measured at $\omega$ = 8.98 rad/s. The sand transferred horizontally in 33 seconds. The tilt angle of the sand as it empties into the left chamber is about 73°, close to that of a liquid or a pendulum at that spin speed. Note the video uses a playback speed of 2x. 

Ball on a ramp in a centrifuge

In the video below a steel ball is placed at the bottom of a ramp tilted upwards from the center of the centrifuge. For each ramp angle, there is a rotation speed creating $F_c$ that overcomes the pull of gravity, causing the ball to roll uphill. As previously shown, for a ramp at an angle $\gamma_0$ from the horizontal, the threshold rotation frequency $\omega$ is derived from: $\tan{\gamma_0} = \frac{\omega^2 𝑅}{g}$. There may also be some slight friction from the ramp that delays rolling. This device (as well as the previous pendulums) can be used to measure $\omega$ from the tilt angle. 

Video of ball on a ramp as rotation speed is increasing then decreasing after time mark 00:15.0