Apparatus Competition

2007 AAPT Summer Meeting

Greensboro, NC

 

Var-I-able Roller

 

Brett Carroll

Green River Community College

12401 SE 320th St.

Auburn, WA 98092

 

253-833-9111  x4322

bcarroll@greenriver.edu

 

Abstract

Rolling a disk with its mass concentrated at the center against a similar disc with its mass concentrated in a ring at the edge is commonly used to show the effect of the moment of rotational inertia I on acceleration down a ramp.

 

But it is not always clear to the class that these are equal mass objects, or exactly how the mass distribution differs.  The Vari-I-able Roller allows you to quickly change the mass distribution of identical rollers for a clear comparison.

Construction of Apparatus: 

The disc of the Var-I-able Roller is made from an ABS pipe cap commonly found in hardware stores.  Neodymium magnets are used to hold heavy steel spheres in various positions, and the radius at which the spheres are stationed determines the moment of rotational inertia I and thus the magnitude of rolling acceleration down a tilted ramp.

 

The pipe cap was machined on a lathe to remove excess mass, so that the mass of the steel spheres and their position predominate as factors in the moment of rotational inertia of the Roller.  The height of the cap is reduced by approximately half, as is the thickness of the outer ring.  Holes are also drilled in the top of the cap to further reduce the mass of the cap and its contribution to the moment of rotational inertia of the Roller.

 

Smaller holes are drilled in symmetrical lines around the center of the disc at radii of 1, 2, and 3 cm.  These allow the attachment of the neodymium magnets to the disc by small screws.  A number of steel spheres placed at the various magnet positions complete the physical construction of the Var-I-able Roller.

 

To improve the visual contrast of the spheres against the disc, a coat of paint can be applied over the magnets.  Different colors can be used for this if constructing two or more discs.

Use of Apparatus: 

Rolling objects down an incline is frequently used to show the effect of the moment of rotational inertia on the acceleration of rolling objects.  To simplify the situation it is common to compare objects that have identical diameters and masses, but have different distributions of mass about their centers.  The objects are placed on an incline and released, and their accelerations down the ramp are directly compared. 

A common setup is comparing two cylinders of equal radius and mass, but with different mass distributions.  Commonly one cylinder has a heavy outer ring surrounding a lighter core, while the other has a heavy core surrounded by lighter material.

But it is not immediately obvious to the class that the masses of the two discs are equal, or whether the outer or inner portions contain the greater mass.  Usually the students are simply told that this is the case, with no direct visible evidence other than a variation in color between the inner and outer portions and a mention of their densities.

With the Var-I-able Roller you begin the demonstration with two discs which are identical in every respect.  When the steel spheres of both discs are placed in the same place on the innermost magnets, they will roll down an incline with the same acceleration.

When the spheres on one disc are now moved out to a larger radius, it is obvious to the students that the mass of that disc/sphere system has not changed, only the distribution of the mass around the center.  

Rolling the two discs side-by-side down the ramp now clearly shows the effect of mass distribution on acceleration.  The disc with the spheres placed further out has a greater moment of rotational inertia, and accelerates more slowly than the disc with the spheres in the center.  The students can clearly see that the only difference is the new distribution of the mass in one of the discs, and that the new distribution has changed the acceleration and thus the moment of rotational inertia of the disc

To change the position of the spheres, simply roll them from one of the magnets to another in the same line until you have the layout you want.  To perform the acceleration demonstration listed above the position of the spheres should be symmetrical around the center of the disc.

A different demonstration can be performed by creating a non-symmetric distribution of the steel spheres, e.g. by taking out two of the spheres and placing the third at an outer position.  Released properly, the disc will accelerate unevenly down the ramp with a lurching motion.  That can be used to underscore the importance of mass balance in rotation, and for real-life references to wheel balancing for automobiles and other systems containing rotating parts.

An energy demonstration can be also be done with the non-symmetric distribution of the steel spheres.  If the disc is placed on the ramp with its center of mass on the uphill side, the disc will roll uphill.  The disc will move in whichever direction serves to lower its center of mass.  If the disc is caught shortly after it begins to roll up the ramp, the reason for this odd motion wont be apparent to the students, which can lead into a useful discussion of energy conservation.  Allowing the disc to then go through its full motion upwards and oscillate about its new equilibrium lets the students see that in this case, uphill is actually downhill in terms of gravitational potential energy.

 

 

Parts List

Equipment and costs required to construct apparatus:

Item

Source

Part number

Cost

Neodymium magnets (18)

K & J Magnetics

R822CS-S

15.30

Steel spheres (6)

McMaster-Carr

96455K62

3.59

Pipe cap (2)

Home Depot

 

12.98

4-40 screws, nuts (18)

Home Depot

 

3.64

Board (1)

Home Depot

 

6.75

Total Cost

42.26