Nae: Partner(s): Date: Angular Moentu 1. Purpose: In this lab, you will use the principle of conservation of angular oentu to easure the oent of inertia of various objects. Additionally, you develop a qualitative feeling for oent of inertia. 2. Theory: For a syste in the absence of any net external torque, the angular oentu, L, is conserved (i.e. reains constant). This is the principle of conservation of angular oentu. In the case of two rotating bodies that undergo collision, conservation of angular oentu says: L before = L after ( I 1 " 1 + I 2 " 2 ) before = ( I 1 " 1 + I 2 " 2 ) after (1) 3. Procedure: In this experient, you will exaine a collision between an object having a known oent of inertia, a solid disk, and an object with an unknown oent of inertia. Fro this, you will deterine the oent of inertia of the unknown object using the principle of conservation of angular oentu. To do this, you will set a turntable (the disk) into rotation with an angular velocity ω 0, and hold the unknown object just above it. If we call the turntable object #1, and the unknown object #2, the angular oentu before the collision is: L before = I 1 " 0 (2) If the unknown object is suddenly dropped onto the rotating turntable so that the two objects rotate with a coon angular velocity ω f then: L after = ( I 1 + I 2 )" f (3)
Figure 1: Depicting a rotational collision Applying the principle of conservation of angular oentu to this scenario yields Equation (4). I 1 " 0 = ( I 1 + I 2 )" f (4) This can be solved for the unknown s oent of inertia about the axis of rotation: " 1 - " 2 I 2 = I 1 " 2 (5) You can find the oent of inertia of the turntable fro static easureents and the forula fro your text for the oent of inertial for a disk. Then, by easuring the initial and final angular velocities of the rotating objects, you ll be able to find a value for the oent of inertia of the unknown object. This ethod can be used to deterine the oent of inertia for any object and is, consequently, very useful when the geoetry of the object would ake a theoretical calculation difficult. In this experient, however, you will use three siple objects as unknowns: a slab, hollow cylinder and a dubbell. In this way you can copare your experiental results with theory. However, don t look up the oents of inertia of these objects in your textbook just yet. You ll ake estiates in the lab. 101
Before you start (and before you consult your textbook), try to estiate a value for the oents of inertia for the objects used in this lab. Object Estiate ( kg 2 ) Slab Dubbell Hollow Cylinder Later, when you copare your initial estiate to your easureent, it should help you to retain soe sense of nuerical agnitude for oents of inertia. As described above, you will set a turntable into rotation with an angular velocity ω 0, and hold the object with an unknown oent of inertia above the turntable. Once the rotational speed of the turntable has been easured at least twice, you will drop the object. In order to copare your oent of inertia results to those presented in your textbook, you need to drop the unknown object onto the very center of the turntable. If it is dropped off-center, the oent of inertia you easure is still the oent about the axis of rotation - but is not the oent about any axis of syetry of the unknown object itself, which is what is presented in your text. Your results will depend upon how well you align and drop the objects so that they are centered over the axis of rotation. It s up to you to find a good technique for doing this. [Does height perspective help or hurt?] Ask for any equipent you ight need (plub bobs, levels, etc.). A page of circular graph paper ight be useful (perhaps you d cut a hole in it?). You ight wish to have each person in the group try dropping the objects five ties without aking any easureents, just to see who s best at dropping the objects into alignent. Note: The dropped object and the turntable ust be at rest with respect to each other; otherwise the value that you easure for ω 2 is not the sae as in Equation (3). 102
To easure the angular velocity, we will again use the Data Studio software and a photogate using the Photogate sensor. Attach a paper flag to the turntable. The photogate can easure the tie it takes the paper wedge to pass through a photogate and fro this, you can calculate the average angular velocity, ω = θ / t, of the disk during its passage through the photogate. With this in ind, what factors influenced your choice for the size of the paper wedge that you will be using in this experient? The flag passes through the photogate a distance r fro the axis of rotation. The width of the flag at this point is w. In the space below, use trigonoetry to find the angle, θ, subtended by the flag at the point it passes through the photogate: Your data can be recorded below. Turntable M d1 = R d1 = M d2 = R d2 = k I d1 = kg 2 I d2 = kg 2 I turntable = I d1 +I d2 = kg 2 k 103
Slab Hollow Cylinder M = kg a = b =! 1 = rad/s! 2 = rad/s M = kg R 1 = R 2 =! 1 = rad/s! 2 = rad/s Dubbell M = kg r =! 1 = rad/s! 2 = rad/s Experiental ( kg 2 ) Theoretical ( kg 2 ) % Error Slab Dubbell Hollow Cylinder 104
4. Questions: i. Coent on the observed oent of inertia, both in coparison to theory and your own estiates. ii. Are the collisions in this experient elastic or inelastic? Explain carefully, using both equations and sentences. iii. Explain how it is possible for two objects with the sae total ass and sae radius to have very different oents of inertia. iv. In this experient we assued that no net external torque acted on our syste; yet, if you set the disk rotating it will eventually decelerate and coe to rest. Try to deterine where this decelerating torque coes fro and what its nuerical value is. A separate experient ay be required to accoplish this. Describe the design of your experient, any results you obtain and conclusions. 105
5. Initiative: 106
6. Conclusions: 107