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  Record its actual mass.  Hold the disk with your hand to prevent it from spontaneously rotating. 10. Simultaneously release the disk and begin recording data. 11. Allow the mass to descent to the ground.  Stop recording data immediately before the descending mass hits the floor or if the string becomes fully unwound from the spindle. 12. From the PASCO graph, highlight the first few seconds of data. 13. Insert a linear trendline to these highlighted data points. 14. Save a copy of this graph as a PDF. 15. Repeat the previous steps for the following rotating objects:  Horizontal Disk & Ring  Vertical Disk FIGURE 1 - A picture showing the approximate set-up for this experiment. DATA ANALYSIS  Show the following calculations for the Horizontal Disk: o Theoretical Moment of Inertia o Theoretical Moment of Inertia Percent Uncertainty o Experimental Moment of Inertia o Experimental Moment of Inertia Percent Uncertainty  Show the following calculations for the Horizontal Ring o Experimental Moment of Inertia o Experimental Moment of Inertia Percent Uncertainty

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