Experiment 8.1 Rigid-body rotation about a moving axis Purpose Study rigid-body

Question

Experiment 8.1 Rigid-body rotation about a moving axis Purpose Study rigid-body rotation about a moving axis. Two solid spheres, two solid cylinders, two thin-walled hollow spheres and two thin-walled hollow cylinders, all of them have different radii and masses. Theory The moments of inertia of various round rigid bodies are given in Fig. 1. 1MR2 Solid cylinder and disk Thin-walled hollow cylinder Thin-walled hollow sphere Solid sphere Figure 1 The moments of inertia of round rigid bodies shown in Fig. 1I can be expressed as: 1-cM R2, where c 2/5 for solid sphere., c- 1/2 for solid cylinder and disk, c - 2/3 for thin-wall hollow sphere, i for thin-wall hollow cylinder, Question: You race various round rigid bodies by releasing them from rest at the top of an inclined pla (Fig. 2). Which of them will reach the bottom of the inclined plane first? The key to this question is the concept of "rigid-body rotation about a moving axis" which states that every possible motion of a rigid body can be represented as a combination of translational motion of th center of mass and rotation about an axis through the center of mass. The kinetic energy ofa rigid body is therefore: K=-M +,cm2uF, where M, ucm, 1cm are the mass, speed of the center of the mass, moment of inertia and angular sp of the rigid body, and -m if the rigid body rolls along a plane without slipping. The total mechanical energy of the rigid body is conserved: Kit U1= K2 + U2 . Because: (a) No work is done by kinetic friction if the rigid body which rolls along a plane without slipping: (b) The effect of rolling friction can be ignored provided that the body and the surface on it rolls are perfectly rigid

Explanation / Answer

For table 3 the formula mentioned should be used.

For Isys : Put m = 4.93g , r = 0.00142m , alpha = 11.6 m/s2 ( Value not clear in image)

For Isys+sphere : Put m = 4.93g+0.130kg , r = .00365m , alpha = 4.41 m/s2

For I sys+disk :  Put m = 14.93g +0.126kg, r = 0.00445m , aplha = 132 m/s2

For Isys+disk+ring : Put m = 14.93g + 0.126kg + 0.466kg , r= 0.0054m ,alpha = 2.96 m/s2

Solving for the above will give you the values for table 3.

Once the table 3 is created, Table 4 will be created.

For Table 5 theoretical values needs to be calculated.

Ihollow sphere = 2MR2/3 , Put values from table 1 ( m = 0.130kg , 0.00365m )

Idisk = MR2 / 2 , ( m = 0.126kg , 0.00445m)

Iring = MR2 ( m = .466kg , r =0.0054m )

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