Abstract Moment of inertia is defined as the measure of an object’s resistance to changes to its rotation. Also defined as the capacity of a cross section to resist bending. The purpose of this experiment is to differentiate the moment of inertia of a disk and a ring in two different approaches, one of which is to rotate at the center and the other is to rotate along its diameter.

I. Introduction Rotational motions involve more than linear motions. Inertia gives us an idea of how hard or easy the object is accelerated. The more inertia it has, the harder it is accelerated and decelerated. In the linear motion, inertia corresponds to its mass as a point object. In practice, an object has an extended shape as a three dimensional sense. Especially, when the object rotates, the shape plays a more important role and it needs more description for the inertia besides its mass.

II. TheoryWhen an object at rest is set

into rotation about some axis, it has a tendency to keep rotating at some angular speed measured in radians/sec. This tendency is called the rotational inertia and is characterized by a physical quantity called the moment of inertia, I, of the object. Moment of inertia is the rotational counterpart of inertial mass in linear motion.

Hence the kinetic energy of a rotating object is:

KE = ½ I2 (1) In the experiment we set the object rotating by attaching it to a hanging weight and allowing the weight to fall. Before dropping the weight, the energy is all potential energy, that of the hanging weight. At the instant the weight hits the floor the energy is all kinetic energy, both in the rotating body and in the falling object. Ignoring friction and the rotational kinetic energy of the pulley (which is reasonable to do since both the object and pulley rotate with little friction and the pulley’s moment of inertia is extremely small compared to that of the object), we can apply the Law of Conservation of Energy:

mgh = ½ mvf2 + ½ If

2. (2)

The velocity (v) of the hanging weight is related to the angular velocity () of the rotating object by v = r where r is the radius of the drum on the object. The final velocity is related to the average velocity, and the average velocity can be found from the distance dropped (h) and the time (t) for the drop: vavg = (vf + vi) / 2 ,

and vavg = h / t, so that (with vi = 0, i.e., starting from rest) vf = 2h / t. We can now substitute for both vf

and f in Eq. (2) and solve for I in terms of the measurable quantities m, g, h, t and r :

I = mr 2 [ (gt 2 / 2h) - 1 ] (3)

The above equation will allow us to find the moment of inertia experimentally for any object. We will compare our results for a uniform, solid disk and a uniform ring with those derived from theory. Using the definition of moment of inertia, I = r 2 dm , one can show that theory predicts

Idisk = ½ MR2 (4)Iring = ½ M(RIN

2+ ROUT2) (5)

where M is the mass of the disk or ring, R is the radius of the disk, RIN

is the inner radius of the disk, and ROUT is the outer radius of the disk. (Note: the “+” sign is correct in Eq. (5)!)

III. Methodology

A. Setup

B. Materials The materials used for the experiment are as follows:1 set rotating platform1 pc disk1 pc ring1 pc photogate1 pc smart timer1 pc smart pulley1 set weights 1 pc mass hanger1 pc vernier caliper

C. Procedure First and foremost, before starting the experiment it would be needed to set-up first the equipment to be used so make the experiment to be smooth. Attach the mounting rod to the smart pulley and photogate head. Connect the mass hanger to a thread and loop it around the cylinder along the vertical shaft. Be cautious and place the disk on the center vertical shaft and connect the smart timer to the photogates then plug it to a power source which is 220V.

Now we move on to the first part, The determination of the moment of inertia of a disk and ring. The ring was placed on the disk then using a vernier caliper, measure the diameter of the shaft to which the thread is wound. Dividing the diameter by 2, we get its radius. It is said that in order to overcome kinetic friction a mass is added on the pan and let it drop with a constant speed. Furthermore, the mass should not be more than 40 grams. Add more mass and record its acceleration by setting the smart timer to ACCEL,LINEAR PULLEY mode. Solve for the experimental value of the moment of inertia of the ring and the disk using I=[m(g-a)r^2]/a . Repeat this process by increasing the mass added two more times and then compute for the percent difference.

Removing the ring from the setup, and doing the exact same procedure as the earlier. We should be able to determine the moment

of inertia of a disk rotated about the center. We measured the moment of inertia of ring by getting the difference between the moment of inertia of the disk and the ring combined and moment of inertia of disk instead of repeating what had been done on the previous part.

For the last part of the experiment which is the determination of the moment of inertia of a disk rotated about its diameter. To start, remove the disk from the vertical shaft and insert the D-shaped hole on the disk to the shaft. Now, same procedures must be done throughout just a different rotating body.

IV. References

Principles of Physics 9th edition by Halliday, Resnick, Jearl Walker

Laboratory Manual

http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

http://en.wikipedia.org/wiki/Moment_of_inertia