RollingAPPARATUSTHEORYDATA COLLECTIONDATA ANALYSISOBSERVATIONSIntroductory MechanicsExperimental Laboratory1RollingGoals: Observe the difference between translational and rotational motion. Use an electronic measuring device to acquire data. Compare sources of error.APPARATUS In this experiment a wheel of radius (R) is rolled down an inclined plane with a raised guide to keep the wheel on the track. The wheel is hollow and has partitions to hold four steel balls. The balls can be relocated in different parts of the wheel to change the moment of inertia (I) while keep the mass (m) of the wheel constant.Two photogates are positioned over the track and can measure the time (t) that the cart passes in the beam at each photogate. The times can be read from the graphing calcula-tor attached to the photogates. The photogates only record the elapsed time between the start and stop of the beam. The experimenter must determine the width (d) of the wheel at the beam to determine the velocity through each gate.THEORY Velocity is the time rate of change of position of an object. If the width of an object and the time it takes to pass a point are both known the velocity is (EQ 1)Angular velocity is the time rate of change of the angle of a rotating object, measured in radians per second (rad/s). For an object that rolls without slipping the angular velocity is related to velocity.(EQ 2)When an object is influenced by a force, the mass of the object provides resistance to the force. That resistance is called inertia. When a force causes an object to rotate, that force is called a torque. Resistance to torque is not only due to mass, but also how far the mass is away from the axis of rotation. The resistance to rotation is called the vdt---=ωvR---dRt-----==2 Rollingmoment of inertia. For an object of radius (R) with all the mass (m) at the perimeter, the moment of inertia is Ip = mR2. For objects with the same mass distributed within that radius, the moment of inertia is less. The more mass closer to the axis of rotation, the smaller the moment of inertia.Objects in motion posess kinetic energy (K). If the object is rolling it has kinetic energy due to the forward motion of its center of mass (KCM) and its rotation (Krot). Transla-tional kinetic energy is based on the mass and velocity, KCM = (1/2)mv2. Rotational kinetic energy is based on the moment of inertia and angular velocity, Krot = (1/2)Iω2. (EQ 3)The form of EQ 3 can use EQ 2 to convert velocity to angular velocity.(EQ 4)As a wheel rolls down a slope the potential energy of gravity (ΔU = mgh) is used to increase kinetic energy and overcome the work done by friction WF. Rolling friction is relatively small, so we should be able to neglect it. If the initial and final angular veloc-ities are ωi and ωf, then the change in kinetic energy is Kf - Ki. The relation between these different forms of energy is(EQ 5)This equation can be rearranged to solve for the moment of inertia (I).(EQ 6)DATA COLLECTION 1. Open the wheel and place the the four steel balls are in the inner positions. Measure and record the mass (m) and radius (R) for the wheel. To get the radius measure the diameter and divide by 2.2. Measure the height of the track at each photogate and take the difference to get the height (h).3. Check that the computer is running the LoggerPro program. 4. Place the wheel at the upper photogate so it just begins to interrupt the beam as indi-cated by the red light. 5. Place a ruler at the base of the wheel along the track and push the wheel through the gate, recording the distance (di) until the the wheel no longer interrupts the beam.6. Repeat steps 4 and 5 for the lower photogate and record the distance (df).7. Start the program, release the wheel from a start position close to the first photogate and record the times from the program (ti and tf).K12---mv212---Iω2+=K12---mR2ω212---Iω2+12---Ipω212---Iω2+==ΔU12---IpI+()ωf2ωi2–()=I2ΔUωf2ωi2–()---------------------------Ip–=Rolling 38. Repeat the measurement from the same starting point a total of ten times. Find the average initial time (ti) and average final time (tf) and the uncertainties on both these using the standard deviation of the mean.9. Repeat steps 7 and 8 using a starting point far from the first photogate.10. Open the wheel and move the balls to the outer positions.11. Repeat steps 7 through 9 with the second arrangement of balls.DATA ANALYSIS 12. Use the mass and radius from step 1 to find the point moment of inertia Ip = mR2 and its uncertainty.13. Use the height from step 2 to find the potential energy ΔU = mgh in joules and its uncertainty.14. Use the average times from step 8 and EQ 2 to determine the initial angular velocity (ωi) and final angular velocity (ωf) and their uncertainties.15. Calculate the moment of inertia and its uncertainty from EQ 6 using the results of steps 12 through 14.16. Repeat steps 14 and 15 for the higher start position times recorded in step 9.17. Find the percentage difference in the moments of inertia measured in steps 15 and 16.18. Repeat steps 14 through 17 for the second arrangement of balls with data collected in step 11.OBSERVATIONS For each of these questions make a prediction, and support your answer by answering the question “Why?”.What are the sources of experimental error in the data measurements? Which one(s) is most important?Consider the difference in results from the two starting positions. How careful do you need to be when starting the wheel? Do the moments of inertia measured for the same arrangement agree within experimen-tal error?Was the assumption about the insignificance of friction sound? What results might point to the occurence of friction?Were the moments of inertia measured for the two arrangements different within exper-imental error?Were the moments of inertia measured for the two arrangements consistent with the idea that mass farther from the axis increases the moment of
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