# PHYSICS 120/130

## II. Theory

First, you must use Newton's Laws, applied to translational and rotational motion, in order to derive the relationship between moment of inertia and the parameters measured experimentally.

If the radius of the vertical shaft is R and the tension in the cord is T, then the torque applied to the shaft is simply TRsin(q) = TR (since T and R are perpendicular) and Newton's Second Law for the motion of the shaft is

t = TR = Itotal a ,                                                           (4)

where Itotal is the total moment of inertia of the rotating system, which we will split up into two parts,

Io  is the moment of inertia of the shaft and threaded rod (which will only be measured) and Iadded is the contribution to the moment of inertia due to the masses attached to the threaded rod, theoretically  given by

which simplifies if both radii are the same.

#### I0                                                          I1

Note: r1 and r2 are the distances from the center of each mass to the center of the vertical shaft.

Newton's Second Law for the motion of the weight hanger is

(6)

where T is the tension in the cord, m is the mass of the weight hanger and a is the acceleration of the weight hanger.  Downward has been chosen to be the positive direction (as usual).

Since they are moving together, the linear acceleration a of the weight hanger and the angular acceleration a of the shaft are related by

a= a R                                                                        (7)

Combining equations (4), (6), and (7) gives

 Itotal = I0 + Iadded = mR˛[(g/a)-1]  =    mR˛[(g/Ra)-1]                                        (8)

We can use this equation, along with measurements of a =a/R, to experimentally determine the moments of inertia.