Chapter 10, Problem 79P

Figure 10.32 shows an apparatus used to measure rotational inertias of various objects, in this case spheres of varying masses M and radii R. The spheres are made of different materials, and some are hollow while others are solid. To perform the experiment, a sphere is mounted to a vertical axle held in a frame with essentially frictionless bearings. A spool of radius b = 2.50 cm is also mounted to the axle, and a string is wrapped around the spool. The string runs horizontally over an essentially frictionless pulley and is tied to a mass m = 77.8 g. As the mass falls, the string imparts a torque to the spool/axle/disk combination, resulting in angular acceleration. The mass of the string is negligible, but the combination of axle and spool has non-negligible rotational inertia I₀ whose value isn’t known in advance. In each experimental run, the mass m is suspended a height h = 1.00 m above the floor and the rotating system is initially at rest. The mass is released, and experimenters measure the time to reach the floor. Results are given in the tables below. Determine an appropriate function of the time t which, when plotted against other quantities including M and R, should yield two straight lines—one for the hollow spheres and one for the solid ones. Plot your data, establish best-fit lines, and use the resulting slopes to verify the numerical factors 2/5 and 2/3 in the expressions for the rotational inertias of spheres given in Table 10.2. You should also find a value for the rotational inertia of the axle and drum together.

FIGURE 10.32 Problem 79

Sphere mass M (g)	783	432	286	677	347
Sphere radius R (cm)	6.25	3.86	9.34	9.42	9.12
Fall time t (s)	2.36	1.22	2.72	3.24	2.91

Sphere mass M (g)	947	189	821	544	417
Sphere radius R (cm)	6.71	5.45	6.55	4.67	9.98
Fall time t (s)	2.75	1.41	2.51	1.93	3.47