
Â³äðåäàãóâàâ äîöåíò êàôåäðè ô³çèêè Êîðí³÷ Â.Ã.
Çàòâåðäæåíà íà çàñiäàííi êàôåäðè ô³çèêè, Ïðîòîêîë ¹ 3 âiä 01.12.2008 ð.
LABORATORY WORK ¹ 6. Verification of the HuygensSteiner (parallel axis) theorem
Equipment: apparatus on three of thread suspension, 2 cylinders, vernier caliper. Aim of work: to calculate the moment of inertia of cylinder, which is revolved in relation to an axis which does not pass through its center of mass, experimentally and on the theorem of Steiner. Task to work: 1. To expect the moment of inertia of empty diskIdisk. 2. To calculate the moment of inertia of the system I_{1}, which consists of disk and two cylinders. 3. To calculate the moment of inertia of cylinder I^{’} and I experimentally and by the theorem of Steiner. 4. To make sure of justice of theorem of Steiner.
17.1 Theory Moment of inertia, also called mass moment of inertia or the angular mass is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in basic dynamics. Figure 17.1 The moment of inertia depends on mass, form and sizes of bodies, distribution of mass in relation to the axis of rotation, choice of axis of rotation. The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis. The moment of inertia is always defined with reference to a particular axis of rotation  often a symmetry axis, but it can be any axis, even one that is outside the body. The moment of inertia I_{cm} of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space. The moment of inertia I about any axis parallel to that axis through the center of mass is given by .
Description of device and method of measuring
In this work apparatus is used on three of thread suspension which allows finding experimentally the moments of inertia of simple bodies. A apparatus consists of disk (2), which is revolved in rotatory motion by a drive (6). The handle of drive turns (5) and spring thread (8) intertwines. Than it untwists and passes the impulse of rotatory motion to an overhead disk (7). Metallic filaments (4) a few bend over, and the mass center of the system is displaced along the axis of rotation. The moment of forces of resiliency tries to turn a disk in the state of equilibrium. A lower disk accomplishes rotators oscillations the period of which depends on a moment inertia of the system. As a bar center of disk is displaced, his potential energy changes. Using the law of conservation of mechanical energy, will get a formula which allows to define the moment of inertia of lower disk: , (17.1) where: m_{D}– mass of lower disk, g  acceleration of the free falling, R  distance from the center of disk to the points in which threads are fastened to it, r radius of lower disk, L – length of threads of suspension, Ò – period of oscillations , (17.2) where: τ – time of oscillations, k – amount of oscillations. A formula (17.1) enables to define the moment of disk inertia with the bodies placed on him. In obedience to the theorem of Steiner , (17.3) where: I_{0}– moment of inertia of cylinder in relation to an axis 1 (fig. 17.2à), I moment of inertia of cylinder in relation to an axis 2 (fig. 17.2á), m  mass of cylinder, a  distance between axes (fig. 17.2á). It is possible to calculate the moment of inertia of cylinder in relation to an axis 2 experimentally too: (17.4) where: I_{1}– moment of inertia of the system, which consists of disk and two cylinders, placed on identical distances a from the axis of rotation (fig.17. 2á), I_{D} – moment of inertia of empty disk (fig. 17.2â). The moment of inertia of cylinder I_{0} is determined by the formula: (17.5) where: r_{c}– radius of cylinder. Putting I_{0} in (17.3) and, meaning that , will get . (17.6)
Figure 17.2
Procedure and analysis 1. Reject unloaded disk on 5 small limb’s divisions. 2. Measure time t_{D} of 10 oscillations. 3. Repeat that measurements already two times. 4. Put two cylinders in the disk on equal distances from an axis of rotation. 5. Measure time t of 10 disk’s oscillations with two cylinders. 6. Measure once distance a and cylinder’s diameter d. 7. Measure oscillations period of unloaded disk T_{D} and oscillations period with two cylinders Ò (17.2), using values t_{D} and t. 8. Calculate unloaded disk’s moment inertia I_{D}(1) using values: m_{D} = 2 kg, R= 0,13 m, r =5,5×10^{2 }m^{ }, L= 2,2 m. 9. Calculate with two cylinders disk’s moment of inertia I_{1}(17.1), using instead of m_{D}value m_{D}+ 2 . The mass m is shown on cylinders. 10. Measure cylinder’s moment of inertia I^{’} concerning an axis by experimental method’s means (17.4). 11. Calculate cylinder’s moment of inertia I concerning an axis 2 according to Steiner’s theorem (17.6). Data of measurements and calculations put in the table. 12. Compare received results and , and draw a conclusion about validity of the Steiner’s theorem.
Table 17.1
Control questions
1. Define parameters of rotary movement: the moment of force, inertia, an impulse, angular speed, angular acceleration. 2. Write down work’s calculation formulas: kinetic energy of rotary movement. 3. Deduce the basic equation dynamics of rotary movement. 4. Write down the law: preservation moment of an impulse. 5. Deduce parities for kinetic energy of rotary movement. 6. Write down Steiner’s theorem.
Author: E.V. Rabotkina, lecturer. Reviewer: S.V. Loskutov, professor, doctor of physical and mathematical sciences. Approved by the chair of physics. Protocol ¹ 3 from 01.12.2008 .
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