## Finding the moment of inertia of a Vex wheel using parallel axis theorem

May 17, 2015

The new Vex game “nothing but net” might involve rotating shooter wheels.  We know that if all the mass of a wheel was located on its rim then the moment of inertia about its rotating axis  (I_rim) would be

I_rim = r^2 * m   where m is the mass of the wheel and r = radius of wheel.   But we know that the wheels actually  have mass that is unevenly distributed along the radius so the moment of inertia I_wheel will be less than I_rim.

Easy experiment to determine I_wheel if we know its mass.

We can determine I_wheel experimentally using the parallel axis theorem and the dynamics of a pendulum.

Parallel axis theorem says that any object that is rotated about an axis parallel to and a distance , d, from an axis going through the centroid of the object will add an amount =  m*d^2 to the moment of inertia about its centroid.  I.e.

I_parallel = I_centroid + m*d^2 .

Suppose we now swing the mass, m,  about the parallel axis like a pendulum  using just the torque from gravity pulling on the mass.      It is easy to show that the period, T , of the pendulum is related to the distance , d, and the moment of inertial , I_parallel, by the following formula:

T = 2*pi*sqrt(I_parallel/(d*m*g))     .   g is gravitational constant and assumes swing angles smaller than say 10 degs from the lowest point on the pendulum path.

If we measure the period of the pendulum we can rearrange the equation and find I_parallel

I_parallel = T^2*d*m*g/(2*pi)^2

Once we have I_parallel, we can now use the parallel axis theorem to determine I_wheel.

I_wheel = I_parallel –  d^2*m  =  d^2*m * (T^2*g/d/(2*pi)^2 -1)

(This assumes  that the string has negligible mass relative to the mass of the wheel)

Vex 5 in Wheel experiment:

Given….r_wheel = 2.5 inches ,

wheel mass,   m = 180 gm (0.180  kg)

The pendulum is created by suspending the wheel with a thread 2.75 inches from its center so

d =. 07 m (approx. 2.75 inches)

The average period  T = .668 sec

I_wheel  = d^2*m*(T^2/d*9.8/(6.28)^2 -1)

= d^2*m*( .248*T^2/d -1)

= .07^2*.180*(.248*.668^2/.07 -1) = 0.00051 kg m^2

Equivalent radius with a rim only mass r_e

r_e = sqrt( I_wheel/m) = .0533 m ( 2.1 inches)

This  means that the wheel behaves as if the mass if located at 84 %  of the radius of the wheel which one could almost guess by looking at it.