PRE-LAB ROTATIONAL INERTIA OF AN OFF-AXIS DISK a 1 The rotational inertia of a disk rotating about its center of mass is 1cm = MIR2 2 If the disk (or any other rigid body) is mounted rigidly on something else and rotating not merely about an axis passing through its center of mass, but rather about another axis parallel to that axis and at a distance d away from the center of mass, the rotational inertia is rather given by: I = 1m +Md2 (1) where M is the mass of the object and d is the distance between the object's center of mass and the actual axis of rotation. This expression is known as the parallel axis theorem. In the gure below, we demonstrate that if the disk is mounted on a bar (say) as it rotates about its center, then the orientation of the disk changes too as the bar rotates (look at the orientation of the red ducial mark on the disk). This is the physical reason that the 1m appears in equation (1). 1~=1=~= If, on the other hand, the disk is mounted at the same position on the bar, but in a way that lets the disk rotate freely with respect to the bar, then the situation could look more like that below as the bar rotates (note here that the fiducial mark's orientation remains the same, indicating that the disk does not spin as the bar rotates) #3:. In this case the rotational inertia of the system is just that of the bar plus Md2 (no 15m contribution since the disk itself does not rotate in this case). Assuming that d=10.0 cm, M=1.50 kg, and R= 11.0 cm, determine the contributions to the rotational inertia of the whole system due to the off-axis disk for these two different mounting schemes. Last week you saw that by averaging the magnitude of the angular acceleration for the upward and downward motion of the weight providing the torque to the system, you could eliminate the effects of any constant frictional torque on the system. Similarly, by taking one half of the difference between the magnitudes of the two angular accelerations, show that you can determine the magnitude of that frictional torque (once you know the rotational inertia). Using this result, along with the rotational inertia of the platform with the ring, and the upward and downward angular accelerations in that configuration from you lab last week, estimate the frictional torque in this apparatus. Combine this estimate of the frictional torque with the radius of the spool to estimate the weight you would need to just offset this frictional torque (and thereby provide motion of this system with a constant angular velocity)