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Transcript

Fall 2006 Moment of Inertia Name Section Theory In applying Newton’s Second Law of Motion to rotational motion, it is known that the relation between torque and angular acceleration depends on both the mass and the distribution of that mass; this relationship is known as the moment of inertia. The moment of inertia for discrete distributions of mass is defined as I mi ri2 and for continuous distributions of mass I r 2 dm Fortunately, the moment of inertia has been calculated and expressed in simpler form for a number of regular bodies. These values can be found in any physics textbook. In this experiment, you will use an apparatus that will allow you to measure the moment of inertia of several different bodies dynamically. These experimentally determined moments will be compared to the theoretical moments of inertia. The apparatus, shown below, has a rotating cradle into which various regular objects can be placed. A string is wrapped around a drum under the cradle and runs parallel to the table and over a pulley at the edge of the table. A mass m placed at the end of the string causes the cradle to rotate as the mass descends to the floor. The acceleration of a of the mass can be determined from y 1 2 at 2 (1) where t is the time it takes the mass to fall a vertical distance y. The linear acceleration of the mass is related to the angular acceleration of the cradle by a r (2) where r is the radius of the drum (the string has the same linear acceleration). The rotational analog of Newton’s second law applied to the apparatus gives I (3) where is the torque applied to the drum and I the moment of inertia of the apparatus (cradle + any object in the cradle). But is just rT (4) since the applied force is just the tension T in the string wrapped around the drum. Applying the linear form Newton’s second law to the hanging mass mg T ma (5) gives the last relation needed to derive an expression for experimentally determining I. Starting with Equation 3, then substituting Equations 4, 5, and 2, 151 Page 1 of 6 Fall 2006 g I mr 2 1 a (6) Apparatus Rotational dynamics apparatus, Table clamp, Vertical rod, String, Pulleys, Stopwatch, Hooked masses, Meterstick, Vernier caliper, Metal disk, Metal hoop. Figure 1 Experimental setup. Figure 2 Dynamical relationships, drawn for clarity; as shown in Figure 1, the axis of rotation of the system is actually parallel to the plane of the page. Procedure Moment of Inertia of the Cradle 1. Make sure that the cradle apparatus is level – the feet are adjustable if needed. Rotate the cradle so that the string unwinds from the drum. Measure the diameter of the drum with the vernier caliper and record this value in Table 1. 2. Wind the string back around the drum and place a 20g hooked mass at the other end of the string off the table. You will need to hold the cradle in place once the mass is added or it will begin its descent to the floor. 3. Measure the vertical distance y that the mass m will descend to the floor and record these values. 4. When ready to make a run, let go of the mass and start the timer simultaneously. Stop timing when the mass hits the floor and record the time. 5. Repeat Step 4 4 more times and determine the average time. Remember to keep the distance y constant! 6. Calculate and record the rate of acceleration a with the average time and Equation 1. 7. Calculate and record the moment of inertia of the cradle itself with Equation 6. 151 Page 2 of 6 Fall 2006 Moment of Inertia of the Disk 1. Measure the diameter of the disk and record this value, along with the mass of the disk, in Table 2. 2. Using the dimensions you determined in Step 1, calculate and record the theoretical moment of inertia for the disk. 3. Switch to a 100g hooked mass. As in Procedure 1, time the descent of this mass over a distance y 5 times and record these values, along with the average time. 4. Calculate and record the rate of acceleration a with the average time and Equation 1. 5. Calculate and record the moment of inertia of the disk and cradle combined with Equation 6. Since we are interested in the moment of inertia of the disk alone, subtract the moment of inertia you found for the cradle itself in Procedure 1 and record the moment of inertia of the disk. 6. Calculate and record the percent error in the moments of inertia of the disk. Moment of Inertia of the Hoop 1. Repeat Procedure 2 with the hoop in place of the disk. Record all data in Table 3. Table 1 Moment if Inertia of the Cradle Diameter of the drum (m) ____________________ Descending mass (kg) ____________________ Vertical distance (m) ____________________ Trial Rate of acceleration (m/s2) Time (s) 1 ____________________ 2 3 Moment of inertia of the cradle (kgm2) 4 ____________________ 5 Average time (s) _______________ 151 Page 3 of 6 Fall 2006 Table 2 Moment if Inertia of the Disk Diameter of the disk (m) ____________________ Mass of disk (kg) ____________________ Theoretical moment of inertia of disk (kgm2) ____________________ Descending mass (kg) ____________________ Vertical distance (m) ____________________ Trial Rate of acceleration (m/s2) Time (s) 1 ____________________ 2 3 Moment of inertia of the disk and cradle (kgm2) 4 ____________________ 5 Average time (s) _______________ Moment of inertia of the disk alone (kgm2) ____________________ Percent error ____________________ 151 Page 4 of 6 Fall 2006 Table 3 Moment if Inertia of the Hoop Diameter of the hoop (m) ____________________ Mass of hoop (kg) ____________________ Theoretical moment of inertia of hoop (kgm2) ____________________ Descending mass (kg) ____________________ Vertical distance (m) ____________________ Trial Rate of acceleration (m/s2) Time (s) 1 ____________________ 2 3 Moment of inertia of the hoop and cradle (kgm2) 4 ____________________ 5 Average time (s) _______________ Moment of inertia of the hoop alone (kgm2) ____________________ Percent error ____________________ 151 Page 5 of 6 Fall 2006 Pre-Lab: Moment of Inertia Name Section 1. What is the formula for calculating the moment of inertia (about the central axis) of a solid disk? This will be in terms of the dimensions of the disk. 2. What is the formula for calculating the moment of inertia (about the central axis) of a hoop? This will be in terms of the dimensions of the hoop. 3. What are the SI (mks) units of the moment of inertia? 4. Explain what each of the variables (I, m, r, g, and a) in Equation 6 represents. 5. A uniformly accelerated object descends (from rest) a vertical distance of 132.7cm to the floor in a time of 7.13s. At what rate was it being accelerated? 151 Page 6 of 6