Lab 10 MOI Handout

MOMENT OF INERTIA EXPERIMENT THEORY The measurement of an object’s rotational inertia, or moment of inertia, is generally dependent upon its geometry and the location of its axis of rotation (relative to its center of mass). For example, the moment of inertia for a ring about its center of mass is given by I com = 1 8 M ( D 1 2 + D 2 2 ) where M is the ring’s mass, D 1 is the ring’s inner diameter, and D 2 is the ring’s outer diameter. Similarly, the moment of inertia of a solid disk about its center of mass is given by I com = 1 8 M D 2 where M is the disk’s mass and D is the disk’s diameter. However, when the same solid disk is rotated about its diameter the moment of inertia becomes instead I com = 1 16 M D 2 The magnitude of the net torque exerted onto an object is directly proportional to its moment of inertia, expressed as the equation ∑ i = 1 1 τ iz = I α z τ 1 z = I α z where α is the object’s angular acceleration. The net torque is only due to the tension associated with a string such that the equation becomes τ T = Iα Using the definition of torque magnitude, this torque from the string can be expressed as τ T = rT sin ( 90 ° ) τ T = ( d 2 ) T where r is the radius of the spindle which the string is wound and d is the spindle’s diameter. Combining these two equations the result becomes 1 2 dT = Iα The tension in the string is created by its connection to a descending mass. Using Newton’s Second Law, along a vertical axis, the tension in the string can be written as T = m ( g − a )

MOMENT OF INERTIA EXPERIMENT where m is the mass of the descending mass, g is the acceleration due to gravity magnitude, and a is the descending mass’ acceleration magnitude. The previous equation now becomes 1 2 d ( m ( g − a ) ) = Iα Using the relationship between linear acceleration and angular acceleration a = rα a = ( d 2 ) α the equation can be simplified into 1 2 d ( m ( g − a ) ) = I ( 2 a d ) I = 1 4 md 2 ( g a − 1 ) PROCEDURE 1. Measure the following variables with its specified device:  Vernier Caliper o Spindle Diameter  Ruler o Disk Diameter o Outer Ring Diameter o Inner Ring Diameter 2. Attach the rotating platform to the spindle of the rotational motion base.  Be sure to tighten the side screw which holds the platform in place. 3. Level the A-base.  Review the procedures from the Centripetal Force experiment. 4. Log into the laptop and open the “PHYS 4A PASCO Files” folder on its desktop. 5. Double-click on the “Photogate with Pulley.cap” file. 6. Turn on the photogate and pair it to your interface.  Verify that you’re not paired to another group’s photogate. 7. Unwind the string such that it hangs over the pulley.  Verify that the string is pulling in a straight-line tangent to the spindle; see the lef side of FIGURE 1 . 8. Attach the solid disk horizontally onto the metal shaf as seen in FIGURE 1 . 9. Attach a 100 g hooked mass onto the string.

MOMENT OF INERTIA EXPERIMENT DATA TABLE 1 DESCENDING MASS’ MASS m±δm ( g ) 100 ± 0.5 SPINDLE DIAMETER d ±δd ( cm ) 3.00 ± 0.05 DATA TABLE 2 Horizontal Disk Horizontal Ring Horizontal Disk & Ring Vertical Disk MASS M ±δM ( kg ) 1.43 ± 0.01 1.44 ± 0.01 1.43 ± 0.01 INNER DIAMETER D 1 ±δ D 2 ( cm ) 10.70 ± 0.05 OUTER DIAMETER D 2 ±δ D 2 ( cm ) 12.50 ± 0.05 DIAMETER D ±δD ( cm ) 22.70 ± 0.05 22.70 ± 0.05 ACCELERATION MAGNITUDE a±δa ( m / s 2 ) 0.0163 ± 0.0002 0.0102 ± 0.0001 0.0263 ± 0.0002

MOMENT OF INERTIA EXPERIMENT Horizontal Disk Theoretical Moment of Inertia I com = 1 8 M disc D 2 I com = ¿ 0.0092 Theoretical Moment of Inertia Percent Uncertainty δD D ¿ ¿ ¿ 2 ( δ M disc M disc ) 2 + ¿ ∆ I com = √ ¿ 0.0005 0.227 ¿ ¿ ¿ 2 ( 0.01 1.43 ) 2 + ¿ √ ¿ ∆ I com = ¿ 0.73% Experimental Moment of Inertia I = 1 4 md 2 ( g a − 1 ) I = ¿ 0.0135 Experimental Moment of Inertia Percent Uncertainty

MOMENT OF INERTIA EXPERIMENT Horizontal Disk & Ring Theoretical Moment of Inertia I com = 1 8 M ring ( D 1 2 + D 2 2 ) + 1 8 M disc D 2 I com = ¿ 0.014 Theoretical Moment of Inertia Percent Uncertainty δ D 1 D 2 ¿ ¿ δ D 2 D 2 ¿ ¿ δD D ¿ ¿ ¿ 2 ( δ M ring M ring ) 2 + ¿ ∆ I com = √ ¿ 0.0005 0.107 2 ¿ ¿ 0.0005 0.125 2 ¿ ¿ 0.0005 0.227 ¿ ¿ ¿ 2 ( 0.01 1.44 ) 2 + ¿ √ ¿ ∆ I com = ¿ 1.59%

MOMENT OF INERTIA EXPERIMENT Experimental Moment of Inertia I = 1 4 md 2 ( g a − 1 ) I = ¿ 0.0216 Experimental Moment of Inertia Percent Uncertainty δd d 2 ¿ ¿ δa a ¿ ¿ ¿ 2 ( δm m ) 2 + ¿ ∆ I = √ ¿ ∆ I = ¿ 3.60%

MOMENT OF INERTIA EXPERIMENT δd d 2 ¿ ¿ δa a ¿ ¿ ¿ 2 ( δm m ) 2 + ¿ ∆ I = √ ¿ ∆ I = ¿ 5.126%

MOMENT OF INERTIA EXPERIMENT Vertical Disk Theoretical Moment of Inertia I com = 1 16 M disc D 2 I com = 0.00460 Theoretical Moment of Inertia Percent Uncertainty δD D ¿ ¿ ¿ 2 ( δ M disc M disc ) 2 + ¿ ∆ I com = √ ¿ ∆ I com = ¿ 0.73% Experimental Moment of Inertia I = 1 4 md 2 ( g a − 1 ) I = ¿ 0.00837 Experimental Moment of Inertia Percent Uncertainty

MOMENT OF INERTIA EXPERIMENT Horizontal Ring 0.00487 ± 1.41% 0.0081 ± 5.126% 66.32% Vertical Disk 0.00460 ± 0.73% 0.00837 ± 3.47 % 81.95%

MOMENT OF INERTIA EXPERIMENT Answer the following questions below using complete sentences: 1. As the descending mass was descending, it began to rotate with an increasing speed. How did this phenomenon affect the PASCO data? - This phenomenon caused the acceleration to increase (positive), thus creating a constantly increasing slope. Therefore, the data began to linearly increase in a constant manner. 2. Overall, were your experimental moment of inertia’s accurate or inaccurate? Justify your answer. - Given that our experimental moment of inertia was 3.47%, and given that we define an “accurate” inertia to be that with a percentage less than or equal to 15%, our experimental moment of inertia was accurate. 3. Identify the independent variable(s) for this experiment.  Be specific and use proper vocabulary . - The independent variables for our experiment were variables that we could change/influence. These variables included the various objects we used (eg. The ring, the disk, etc.) which differed in their masses and spatial orientation. 4. Identify the fixed variable(s) for this experiment.  Be specific and use proper vocabulary . - Our fixed variable which remained unchanged and constant throughout our experiment was gravity. 5. Identify the dependent variable(s) for this experiment.  Be specific and use proper vocabulary .