# Moment of Inertia

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Momentof Inertia

Momentof inertia describes the rotational inertia which is an analog ofmass for the linear motion. This appears in rotational motiondynamics, and must be specified with respect to the chosen axis ofrotation. This experiment was carried out to determine the moment ofinertia. This involved three main actions. This include determinationof

1. The period of one oscillation for the disk and all other objects placed on the disk

2. Geometrical parameters of upper and lower disks and distance between them (cm)

3. Geometric parameters of the ring and cylinder respectively (cm)

Objective

Theexperiment aimed at studying the rotational inertia and determininghow different shapes of masses as well as different masses behavewhen compared based on their inertia. Specifically, the experimentcarried out helped in determining the differences in inertia betweena ring and a disk. Increasing forces were used to reduce the angularacceleration for both the ring and the disk. The differences inacceleration were then measured to help in determining the resistanceof the disk and the ring to rotational motion. Thereafter, the radiusof the masses were compared and how the torque applied related toangular acceleration. A predictable force was achieved using thegravitationalacceleration of 9.8ms-1.

Theresults presents the inertia of the disk and the ring by dividingapplied torque by the resulting acceleration. The two values will becompared and the percentage error determined.

Theresults obtained are presented in Tables 1-3 as shown below.

Table1: Theperiod of one oscillation for the disk and all other objects placedon the disk

 Items Mass of the Disk MD=478g Mass of the Ring, M1=195.16g Mass of the cylinders 2M2 = 201.17g Preset Count Data 10 10 10   Time t/s         1 14.118 14.509 14.455 2 14.133 14.486 14.626 3 14.038 14.531 14.559 4 14.025 14.051 14.6 5 14.09 14.541 14.519           Average Time 14.099 14.525 14.552   Average Period 1.408 1.453 1.456

Table2:Geometricalparameters of upper and lower disks and distance between them (cm)

 D1 H a b Items 1 14.9 48.3 12.45 6.55     2 14.8 47.4 12.5 6.6 7.22 3.79 3 14.85 48.58 12.55 6.53     Average 14.85 48.09 12.5 6.56

Table3: Geometricparameters of the ring and cylinder respectively (cm)

     Din Dout D cylinder D ring 2d = Dring-Dcylinder Items     1 11.237 12.02 2.5 14 11.499 2 11.311 12.034 2.502     3 11.376 12.032 2.502       Average 11.308 12.02866667 2.501333333

Discussion

Accordingto the law of inertia, an object in motion tends to resist changes.According to Galileo Galilei, a body moving on a level groundcontinues to move in constant speed at the same direction provided itis not disturbed. Newton backed-up Galilei by stating that a bodywill remain at rest or in motion. The force that resists motion isreferred as inertia and forms the Newton’s first law.

Therotational inertia for an object depends on mass and arrangement ifmass within it. A more compact object have a reduced rotationalinertia. A ring and a disk were studied in this experiment. Rotationof a ring that has constant density depends on mass and outer andinner radius. Relationship between mass and radii is expressed usingthis equations

Sincea disk/cylinder is like a like without the inner radius, its inertiais a function of the mass and its expression is:

Experimentallyand inertia is found by applying a known torque to the body anddividing it by resulting acceleration

Themass of the disk, Mo=478g,mass of the ring M1=195.16g and the mass of the cylinder, 2M2=201.17g. The moment of Inertia calculated is as shown below.

Comparisonof Moments of Inertia

 Experimental Theoretical Iring Icylinder Idisk

Forceis a product of mass and acceleration. Torque is the moment ofinertia multiplied by the angular acceleration. The moment of inertiameasures the resistance in changing angular velocity of a body.Therefore, moment of inertia is dependent on mass and itsdistribution with respect to axis of rotation. For solid disk, themass is distributed equally between the circumference and the centerof the circle, expressed as.Therefore, the moment of inertia of the ring is higher than that ofthe disk since the mass is far from the rotational axis.

Forthe solid disk, sections of the object closer to the rotational axismoves slower than the parts closer to circumference. This impliesthat the average velocity is half that of maximum velocity.Therefore, the moments of inertia for the parts of the object closerto axis of rotation remains considerably lower than the moment ofinertia for the parts of the object closer to circumference.

Onthe other hand, the ring has its mass concentrated on the narrowband. Distance from axis of rotation to the center of narrow band isequivalent to the radius of the ring. Therefore, the velocity of themass of the ring is approximately equal to tangential linear velocityof the particles at the circumference of the ring.

Therefore,the differences in inertia is because of the differences in linearvelocities f average particles in both ring and the disk.

ReferenceList

Kopal,Z. (1972). Moments of Inertia of the Lunar Globe, and Their Bearingon Chemical Differentiation of Its Outer Layers. TheMoon4(1-2), pp. 28-34.