Alfred Leung

California State University

Department of Physics and Astronomy

Long Beach, CA 90840-3901

Phone:     (562) 985-4923

Fax:         (562) 985-7924

E-mail:    afleung@csulb.edu

 


AAPT Apparatus Competition, Entry Description

Introductory Laboratory Category

 

Apparatus Title:  Radius of gyration

 

Abstract

A circular object rolls down an elevated incline and then falls to the ground.  Its radius of gyration can be determined by its initial height, height and range of the fall.  The determination relies on many basic concepts, such as rolling without slipping, relationship between translation and rotational kinetic energies, conservation of energy, and equations of motion for free fall.

 

Description

 

The radius of gyration of an object is a defined quantity used to quantify the moment of inertia which is a very important parameter in the study of rotational motion.  For symmetric objects the moment of inertia is calculated using calculus.  For other objects the calculation is more difficult.  However, the moment of inertia of any object can be defined as Mk2 where M is the mass of the object and k is the radius of gyration.  The physical meaning of k becomes apparent by considering the moment of inertia of a particle.  If a particle of mass M rotates around an axis which is at a distance d from the particle, its moment of inertia is Md2, which has the same form as Mk2.  Consequently, the radius of gyration of an object represents the distance between the axis of rotation and the point where all the mass of the object appears to be concentrated.  The aim of the experiment is to measure the radius of gyration of a sphere and a barrel.  The barrel is made by cutting two parallel and identical sections from a sphere (Fig. 1). 

           

PhotoStacker, (C) Copyright 1992 Ultima Electronics Corp, U.S.A.

 

         Figure 1.   The barrel and sphere.

 

 

 

Since the relationship between the radius of the gyration and the radius of a sphere is known exactly, the measurement of the radius of gyration of a sphere is used to determine the accuracy of the experiment.  The barrel is treated as an unknown.  It challenges students to make accurate measurements to determine its radius of gyration.

 

In the experiment the radius of gyration of the sphere is determined first.  The sphere is initially at rest and then allowed to roll down an incline placed on top of a table.  The end of the incline is made horizontal to ground to simplify the analysis.  After the sphere leaves the incline, it falls to the ground.  The following derivations relate the radius of gyration (k) of the sphere or any other circular object to the initial height on the incline, the height and range of the free fall.   Energy is conserved when the sphere rolls down an incline without slipping.  Then,

                                   

                                                Mgh = ½ Mv2 + ½ Mk2w2 + W                                 [1]

 

where M is the mass of the sphere, h the distance from the bottom of the sphere to the top of  the table, v the velocity at the bottom of the incline, w the angular frequency at the bottom of the incline, and W is the work done by friction between the sphere and the incline.  W can be expressed as MgD. Since there is no slipping, v = rw where r is the radius of the object, Eq. 1 becomes 

 

 

                                                2g(h - D)r2  =  v2 ( r2 + k2 )                                         [2] 

 

 

When the sphere leaves the incline, it has a horizontal speed of v and proceeds to fall to the ground.  The range of the fall,  R = vT  where T is the flight time of the free fall.  If the height of the table is denoted as H, H  = ½ gT2.  By eliminate T, we obtain

 

 

                                                            v2 =  gR2/(2H)                                                 [3]

 

 

Substituting v2 into Eq. 2, we have

 

 

                                                            2(h - D)r2  = R2( r2 + k2 )/(2H)

 

Solving for k, we have

 

 

                                                            k2 =  r2 [4(h - D)H/R2 – 1]                              [4]

 

 

Equation 4 holds for spheres and other circular objects.  All quantities on the right side of Eq. 4 are measured in the experiment to determine the radius of gyration k.

 

 

Equipment and costs required to construct apparatus

 

Item

Source

Cost

Pine board: 1Ó x 10Ó x 8Õ

Home Depot

www.homedepot.com                                                                                                                                                                                                                                                                     

   $9.15

Whitewood: 1Ó x 4Ó x 12Õ for $2.65

(only one half was used)

Home Depot

   $1.33

Tempered hardboard: 2Õ x 4Õ x 0.125Ó for $2.69.

(only one half was used)

Home Depot

   $1.35

Franklin, Model No. 3313.  A set of 8 laquered wooden balls, 90 mm in diameter, for $19.99. 

(Only two were used)

Big 5 Sporting Goods

   $5.00

 

Total Cost

$16.83

 

 

Experimental Setup and Procedure

 

An incline is made of wood and its incline angle is determined to be 20.3o.    To make the incline surface uniform a sheet of tempered hardboard is glued onto the incline.  The hardboard continues beyond the end of the incline.  The section of the hardboard that extends beyond the incline is made horizontal (Fig. 2).   The sphere for the experiment is a 9.0-cm diameter lacquered wooden ball.  To make the barrel two parallel and identical sections are cut from another wooden ball with a milling machine (see Fig 1).   The surface of both objects is made rough by sandpaper.

 

 

 

Figure 2.  The incline

 

 

Measurements with the sphere with a radius of 4.50 cm provide h = 29.5 cm, D = 0.75 cm, H = 93.6 cm, and R = 87.4 ± 0.59 cm (N = 10).   According to Eq. 4 k is calculated to be 2.88 cm.  The theoretical value of k is 2.846.  The percent error of the measurement of k is 1.2%.