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. me 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. MR MR2 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. I can be expressed as: 1-c M R:, where c 2/5 for solid sphere, c/2 c 1 for thin-wall hollow cylinder for solid cylinder and disk, c 2/3 for thin-wall hollow sphere, Question: You race various round rigid bodies by releasing them from rest at the top of an inclined plane (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 the center of mass and rotation about an axis through the center of mass. The kinetic energy of a rigid body is therefore: KMvmtiLcma where M, 'en ? are the mass, speed of the center of the mass, moment of inertia and angular speed of the rigid body, and if the rigid body rolls along a plane without slipping The total mechanical energy of the rigid body is conserved: K, +U, K +U Is along a plane without slipping (b) The effect of rolling friction can be ignored provided that the body and the surface 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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