LVang+-+L20+-+Rotational+Inertia+Demonstrator

Lue Vang  Mr. Kellogg  AP Physics - Pd. 7  18 April 2012  Performed On: 11-12 April 2012  Lab Partners: Brent Keath, Kevin Trinh, Lue Vang

**Rotational Inertia Demonstrator **

This lab will study ways of finding angular acceleration, moment of inertia, and kinetic energy through theoretical calculations and through experimental methods.
 * __Purpose: __**

 **__Background:__**  The rotation inertia demonstrator will spin as weight is attached and released from one of its three pulleys. The falling mass will have a constant acceleration, ensuring a constant force and torque. This being the case, angular acceleration and the likes can be found using rotational kinematic equations or the equation for angular acceleration. In addition, an unbalanced torque from unbalanced forces will cause an angular acceleration. Torque produced by tension on the string of the hanging mass can then be used to calculate moment of inertia by using Newton’s second law or by computing mass and distances from fix axes. Finally, kinetic energy is made of up both translational and rotational kinetic energy, so according to the law of conservation of energy, the potential energy before dropping the mass can be used to determine how much of the total energy is converted to translational kinetic energy and how much is converted to rotational energy.

 The two methods of calculating angular acceleration should produce similar, if not equal, answers. Likewise, the two methods for calculation the moment of inertia for the pulley should equate to similar, if not equal, answers. Finally, for the calculations of energy, the total energy from translational kinetic energy and rotational energy should equal to original potential energy.
 * __<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Hypothesis: __**

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **__Apparatus:__**
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Rotational inertia demonstrator (1)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Stop watch (1)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Mass set (1)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">String (1m)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Meter stick (1)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Scale (1)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Calipers (1)

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Hang a mass from one end of the string and loop the string around one of the rotational inertia demonstrator (RID)’s, three pulleys. Let the mass drop to the floor, and wind it up with 4-6 rotations. Measure the vertical distance of the drop and the time required for the mass to fall so. Use the data to calculate rotational acceleration. <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Find the linear and rotational acceleration of the falling mass. Then, calculate the torque produced by tension. Once found, use the said data to calculate moment of inertia (I). Afterwards, find the total moment of inertia by computing and adding the moment of inertia of all the bodies in the pulley system. Finally, compare the two calculated moments of inertia. <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Measure the mass and height of the hanging mass at starting position to calculate the total potential energy. Next, time the mass’s descent to the floor and use it to calculate the velocity, which will then we used to calculate translational kinetic energy. Similarly, use the angular velocity to find the total rotational kinetic energy of the system.
 * __<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Procedure: __**
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 1: Kinematics of Rotational Motion **
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 2: Determining the Moment of Inertia of the Rotational Inertia Demonstrator **
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 3: Energy of Rotational Motion **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **__Data:__**
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 1: Kinematics of Rotational Motion **


 * <span style="color: #000000; display: block; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px; text-align: center;">Table 1.0: Rotations ||
 * || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Revolutions || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Radians ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"># Rotations || <span style="color: #000000; display: block; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px; text-align: right;">5 || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">10π ||


 * <span style="color: #000000; display: block; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px; text-align: center;">Table 1.1: Time of Fall ||
 * <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Time 1 (s) || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Time 1 (s) || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Time 1 (s) || <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Average Time (s) ||
 * <span style="color: #000000; display: block; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px; text-align: right;">9 || <span style="color: #000000; display: block; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px; text-align: right;">10.2 || <span style="color: #000000; display: block; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px; text-align: right;">9.3 || <span style="color: #000000; display: block; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px; text-align: right;">9.5 ||

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">α = 2(ΔΘ – ω0t)/t2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">α = 2(10 <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">π <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">– 0* <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">9.5 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">)/ <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> 9.5 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">2
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Angular Acceleration: Method 1 **
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">α ≈ 0.6962 rad/s2 **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">a = αr
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Angular Acceleration: Method 2 **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Vertical Displacement: 0.92m

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">a = 2(Δs – v0t)/t2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">a = 2(0.92 – 0* <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">9.5 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">)/ <span style="color: #000000; font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> 9.52 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Pulley Radius: 0.03 m
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">a ≈ 0.0204 m/s2 **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">α = a/r <span style="font-family: 'Cambria Math','serif'; font-size: 16px;">⊥ <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">α = 0.0204/0.03
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">α ≈ 0.68 rad/s2 **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Percent difference (α) = |(difference of α/average α)|*100 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = |(.68-.6962)/((.68+.6962)/2)|*100
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">≈ ****<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> 2.35% **<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = Close enough

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Angular acceleration of rotating apparatus: 0.70 rad/s2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Linear acceleration of falling mass: 0.02 m/s2
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 2: Determining the Moment of Inertia of the Rotational Inertia Demonstrator **
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Moment of Inertia: Method 1 **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Radius of pulley: 0.03 m <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Mass of falling mass: 0.23 kg

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Tension = -(Weight) = -(mg) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= -(0.23*-9.8)
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= 2.254 N **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">τ = Iα <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">I = τ/α = (r <span style="font-family: 'Cambria Math','serif'; font-size: 16px;">⊥ <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">T)/α <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = (0.03*2.254)/0.6962 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **= 0.0971 kg*m2**

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Pulleys: <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">I = 0.00058 kg*m2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Thin Rods: <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Mass: 0.0748 kg <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Length: 0.345 m <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">I per rod = 1/3ML2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = (1/3)0.0748*0.345^2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> ≈ 0.003 kg*m2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">I for four rods = 4(1/3ML2) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = 4((1/3)0.0748*0.345^2) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> ≈ 0.012 kg*m2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Movable Mass: <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Mass: 0.018 kg <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Distance from Center (radius): 0.02125 m <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> I per movable mass = 1/2MR2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> =1/2(0.018*0.021252) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> ≈ 0.009*0.021252 kg*m2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> I for four movable masses = 4(1/2MR2) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = 4(1/2(0.018*0.021252)) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> ≈ 0.0000162 kg*m2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> ΣI = I (pulley) + I (rods) + I (movable masses) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = 0.00058 + 0.012 + .000016
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Moment of Inertia: Method 2 **
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">≈ ****<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> 0.013 kg*m2 **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Percent difference (α) = |(difference of α/average α)|*100 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = |(.0971-.013)/((.0971+.013)/2)|*100
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">≈ ****<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> 152.77% **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Falling Mass: <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Mass: .23 kg <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Vertical displacement: .9m <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Potential energy = mgh <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = .23*9.8*.9 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **= 2.0286 J** <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Fall time: 9.53 s <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Average velocity = d/t <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = 0.9 /9.53 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> ≈ 0.094 m/s <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Final velocity = 2vavg <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = 2(0.094) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **≈ 0.189 m/s**
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 3: Energy of Rotational Motion **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Translational Kinetic Energy: <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> KEf = (1/2)mv2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = (1/2)(.23*0.1892) <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **=.0041 J**

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Pulley: <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> Radius: 0.03 m <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> ωf = v/r <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = 0.189/0.03 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> I = 0.0971
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= 6.3 rad/s **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Rotational Kinetic Energy: <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> KE = (1/2)Iω2 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= (1/2)0.0971*6.32
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= 1.93 J **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">KE (total) = KEtrans + KErot <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> = 0.0041 + 1.93 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> **= 1.9341 J**

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Percent difference (E) = |(difference of E/average E)|*100 <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">= |((2.0286-1.9341)/((2.0286+1.9341)/2)|*100
 * <span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">≈ ****<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;"> 4.77% **

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">In experiment 1, the angular acceleration obtained from kinematic equations is about 0.70 rad/s2 while the one from using (α = a/r) is about 0.68 rad/s2. The percent difference between the two is 2.35%, so the two results are not significantly different since a small change as +0.03 rad/s2 to 0.68 rad/s2 would result in a large of a decrease as about 5%. In fact, that means the results are very close, so the data suggests that methods 1 and 2 agree within the uncertainty of measurement.
 * __<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Analysis: __**

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">In experiment 2, the two calculated moments of inertia should be of close values, and they are. The calculated method resulted in a moment of inertia of 0.971 kg*m2 while the experimental method produced a moment of inertia of 0.013 kg*m2. The percent difference between the two methods is 152.77%, so according to the lab results, it is method 1 and 2 do not agree within the uncertainty of measurement.

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">The calculative method resulted in a total energy of about 2.03 J while the experimental method produced a value of about 1.93 J. In comparison, the two are only different by a percentage of about 4.77%, thus methods 1 and 2 produce values that agree within the uncertainty of measurement.

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">All in all, the lab appears to be a relative success. Ideally, the calculations should end up with close results, but experiment 2’s did not.
 * __<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Conclusion: __**

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 1 was pretty good with only a 2.35% difference. The only plausible source of inaccuracy may be from the measurements taken. Also, friction may skew the data a tad. Still, the percent difference assures that the two values from the two different methods agree within the uncertainty of measurement.

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 2 resulted in a unexpectedly high percent difference. This may be due to a inadequate collection of data because since the equation for the moment of inertia of disks is (1/2)MR2, the radius of the disks were collected for “distance from the center” in method 2. It is unclear whether the lab wanted the radius of the disk or the distance of the center of mass of the disk from the center of the the system. Nonetheless, substituting a reasonable value of .02125m – 0.355m would’ve still ended up with a moment of inertia with a large percentage difference from method 1’s moment of inertia. The percent difference would’ve still been over 100%.

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Experiment 3 ended up with a good percent difference of only 4.77%, is very accurate. Again, like experiment 1, the difference may be a result of slightly inaccurate measures and/or the loss of energy through friction. Still, the low percent difference indicates that the results agree within the uncertainty of measurement.

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Seeing as the calculations of the proceeding experiment depends on the calculations of the experiment preceding it, miscalculation from one section of this lab will greatly result in miscalculations in the experiment after.

<span style="font-family: 'Trebuchet MS','sans-serif'; font-size: 16px;">Overall, experiment 1and 3 turned out relatively well, but experiments 2 was not successful. To increase the chances of success in the future, it would be wise to take better notes when collecting data or be more specific in the lab directions. Pictures, diagrams, and/or video documentation would be most useful as well.