Name:  Steven Sahyun and Phil Siemens

Institution:  UW-Whitewater

Address:      University of Wisconsin - Whitewater
500 W. Main St.
Physics Department
Whitewater, WI  53190
     

Phone:         262-472-5113

Fax:                  

E-mail:         sahyuns@uww.edu

 

Apparatus Title:  Cage Lab

 

Abstract (50-75 words)

The Cage Lab consists of a metal wire mesh box that can be suspended by a fiberglass rod axle. In order to break the symmetry of the object, a ball of clay is added to one corner of the cage. Upper-level students use this non-ideal pendulum to investigate the center of mass, moment of inertia, principle rotational axes, and precession or nutation of the system when spun about an axis of stability.

Equipment and costs required to construct apparatus:

Item

Source

Part number

Cost

2 (1 pkg.) Model airplane control rod tubes (Carbon Fiber 1/4" diameter).

Trump’s Hobbies

2401 NW Kings Blvd.

Corvallis, OR 97330

541-753-7540

N/A

$25

2 3/4" Binder Clips

Staples

N/A

$2

2  #6 ball bearing fishing swivels (Danielson , Laker, Sampo)

tackledirect.com

 

BX6RZ

$6

Soldered wire mesh, 1/4" weave 14" X 36" (could also use 1/2" weave)

Local hardware store

N/A

$4

Sculpey modeling clay

(14 oz)

Wall Mart

N/A

$10

at least 4 straight, light-weight bamboo skewers

Wall Mart

N/A

$2

2 collars - machined - used to fit the swivel to the rod.

 

N/A

N/A

Total Cost

$49

 

Description: 

Construction information: A rectangular cage is constructed from a sheet of wire mesh by a process of cutting and folding. The shape of the cage is a flattened cube (approximately 25 cm ´ 25 cm ´ 12.5 cm), so that its two largest faces are squares. The mesh is cut from the bulk in the pattern below.  It is folded along the dashed lines and the edges may be “sewn” together with some wire.

The cage is suspended by a “trapeze” consisting of two hollow 1/4-inch carbon fiber rods. Since these rods are also used for model airplane or helicopter control rods, they may be found at some hobby stores. Two rods often come in a single  package and are typically 82 cm long. Cut the rod that is to be the axle approximately 4 cm shorter than the parallel rod to allow for the swivels and for parallel hanging. The rod package comes with end-caps that need to be drilled to accommodate one end of a swivel that acts as the axle bearing. Plastic or aluminum ends may also be machined for this purpose. The end-caps may need to be glued onto the axle for rigidity. It is important to achieve a snug fit between the end cap and the swivel so that the swivel does not become loose during rotation. However, it is also important that the swivel can still be removed from the end-cap to allow the cage to be easily slid onto and off the axle rod. If there is a very tight fit between the end-cap and the swivel it is best to not glue the end-cap.

 

The swivels have ball bearings that allow the axle to freely spin. However, they come with attachment rings that need to be clipped off before inserting into the end-caps. The other ends of the swivels are tied with 35 cm strings to the parallel bar. The strings may be taped to the parallel bar to keep them from sliding.

 

After students perform several initial observations of the cage, a spherical ball is impaled on one corner of the cage. The ball is made from Sculpey clay and is approximately 1/3 the mass of the cage. It is attached to one corner of the cage by pressing the clay ball into the cage.

 

For the initial experiments (center of mass, moments of inertia, etc.) the parallel bar is clamped to a support rod attached to a table. When students investigate the precession, nutation, and axes of stability of the cage system, they suspend the parallel bar by a string that is attached to the supporting clamp.

 

The binder clips are used to clamp and fix the cage to the axle. The clips keep the cage from sliding on the rod and connect the axle to the cage for rotation but can be easily removed when the cage is to be taken off the axle.

 

Experiments:

This apparatus allows determination of the center of mass of the cage and ball system. When the cage is suspended from a horizontal axis, its center of mass must lie in the plane beneath the axis. By determining the intersection of three of these planes, the center of mass is uniquely identified.

 

The axle is attached to the cage along one of the short edges that doesn't include the clay ball. The bob is hung from the same axle, which is then suspended horizontally with help of a clamp stand, so that the cage swings freely from the axle. By sighting along the string, a vertical plane is determined which includes the axle. Students then draw all the lines where this plane intersects the cage with a marker. They repeat this process for two other edges of the cage using a different colored marker each time.

 

The intersection of two planes is a straight line, which pierces the cage at the two points where the lines marked on the cage cross each other. The center of mass lies in both planes, so it must lie along the line of their intersection.  Students insert bamboo skewers into the cage along each of the three lines to identify each of the three pairs of intersecting planes.

 

The coordinates of the center of mass are measured using a fourth skewer to gauge its distance from the three sides of the cage and relate the position to an origin located at one corner of the cage. A long skewer, with a mark at its center, is inserted through the cage and fastened in place with the mark at the center of mass. The other skewers are then withdrawn. The cage is then mounted so that the axle goes through its center of mass.

 

The cage is rotated so that the ball is above the axis and then released. The maneuver is repeated with the ball below the axis, and to the side at the same level as the axle. Students make observations and comments on the process.

 

When students measure the moment of inertia, they suspend the cage by the axle along the short edge farthest from the ball. The axle is fixed horizontally, and the cage is made to swing about this axis. Its period of oscillation is measured by noting the time it takes to swing through many periods. Students repeat this process for four additional edges. They then relate their results to theoretical calculations for a non-ideal pendulum using the Parallel Axis Theorem. From this information, students calculate the moment of inertia about axle and find the moment of inertia about the principle axes through the center of mass for the cage-ball system.

 

After calculating the principle axes for the system, students observe the rotational motion about a principle axis where the rotating system remains in a stable orientation, and then about non-principle axis. For this experiment, the parallel bar is suspended by a string. When the strings are correctly positioned, the assembly is free to rotate about the axis of the axle, about the vertical axis of the string, and over a limited range of horizontal tilt of the axle. A test of the correct positioning is that the apparatus maintains any orientation in which it is placed, within the limits of the trapeze.

 

Measurements of nutation and precession are performed using the cage as a symmetric rotor. The clay ball is removed and the cage is mounted so that its greatest moment of inertia is in the axial direction.  Students then predict and measure the nutation of the cage when it is rotated and given  a slight perturbation. For precession, students slide the cage up the axial rod by a small amount (less than 1 cm). The suspension string is repositioned so that the system maintains equilibrium.  When the cage is again rotated, students then observe, measure, and check with theoretical calculations the phenomenon of precession.

Sketch(es) (computer generated if possible):