Comprehension of Moments of Inertia - Report Example

Victoria University College of Engineering and Science NEM2202: Dynamics Name Institution I certify that this report has not been submitted elsewhere for assessment and that it does not contain plagiarism or collusion: Sign here SUMMARY The main aim of this experiment is to apply various different analytical and experimental techniques to help in the comprehension of moments of inertia. Through this experiment, the students get in depth knowledge and information regarding moments of inertia. The experiment will lay its basis on applying the moments of inertia of two wheel and axle units and connection of rod shall be estimated through both the geometric and experimental methods. The results obtained from the experiment are then contrasted to the various values derived from theoretical calculations by use of formula that is got from the experiment. It was indeed a success in conducting the experiment as the aims were fulfilled. It is found that the shape of the body affects the moment of inertia as well as the period of oscillation of a body is affected by the C.O.G. in the event that the center of gravity and the pivot of the body increases then also the periodic time increases. This then implies that the moment of inertia for two objects that have the same mass may be different owing to the difference in the distribution in the mass within the bodies. The experiment gives an insight into the different techniques that are applicable in determining moment of inertia of different objects. Contents SUMMARY 2 1.0INTRODUCTION 1 2.0BACKGROUND 2 3.0EXPERIMENTAL DESIGN 4 3.1EXPERIMENT 1 – WHEEL AND AXLE PENDULUM METHOD: 4 1Figure1: Schematic diagram of wheel axle with attached bob support on knife edge (from lab sheet) 5 3.1.1EXPERIMENT EQUIPMENT AND MATERIALS: 5 EXPERIMENT 2: COMPOUND PENDULUM METHOD: 6 EXPERIMENT EQUIPMENT AND MATERIALS: 7 EXPERIMENT PROCEDURE: 8 4.0METHOD OF ANALYSIS 9 5.0RESULTS 13 5.1EXPERIMENT 1– WHEEL AND AXLE PENDULUM METHOD: 13 5.2EXPERIMENT 2: – COMPOUND PENDULUM METHOD: 14 6.0DISCUSSION 16 1EXPERIMENT 1– WHEEL AND AXLE PENDULUM METHOD: 16 2EXPERIMENT 2: COMPOUND PENDULUM METHOD: 17 7.0CONCLUSION 17 8.0REFERENCES 18 APPENDIX 1: EXPERIMENT 1– WHEEL AND AXLE PENDULUM METHOD: 20 Big wheel and axle 20 Small wheel and axle 20 APPENDIX 2: EXPERIMENT 2: – COMPOUND PENDULUM METHOD: 21 Big end 21 Small end 22 1.0 INTRODUCTION The main aim of this experiment is to apply various analytical and experimental techniques to help in the comprehension of moments of inertia. Through this experiment, the students get in-depth knowledge and information regarding moments of inertia. The experiment will lay its basis on applying the moments of inertia of two wheel and axle units and connection of rod shall be estimated through both the geometric and experimental methods. The results obtained from the experiment are then contrasted to the various values derived from theoretical calculations by use of formula that is got from the experiment. 2.0 BACKGROUND Moment of inertia refers to the capability of an object to resist change that occurs in angular momentum. Usually, it is known to be the rotational analog to mass or even inertia in a form of translational motion. Basically, the moment of inertia (I) is calculated by deriving an appropriate formula from a general formula. That is,…………………. I=R (x2 + y2)dm……………………………………………………………………(1) Usually, the formula is easily got as it has been previously derived for most of the basic shapes and this makes it quite easy to get the moments of inertia. Even the moments of inertia for some of the complex solids are easily calculated in the even that the solids are divided into other basic shapes that have known inertia. According to the law of inertia, it is the tendency of an object to resist change when in motion. In criticizing the thoughts of Aristotle in regards to movements, Copernicus and Galileo came up with the thoughts of inertia. Galileo Galilei is known to be the first state and states that, a boddy moving on a level surface will progressively move in the same direction at a constant speed unless disturbed. The rotational inertia of an object usually depends on the mass and the arrangement of the mass within a given object. According to a given specifi rule, the more the compact is the mass of the object, the less rotational inertia that the object posses. Different studies have been done covering shapes and their inertia and case in point is a ring an a disk. Usually, the rotational inertia of a ring with unchanging density usually largely depends on the mass. The relationship between the mass and the radii is given by the equation such as Iring=12M(R12+R22). Usually a disk is just but considered to be a ring without inner radius therefore, the inertia of the disk is just a function of the mass and the outer radius and this is given precisely by Idisk= 12MR2. However, it is interesting to note that not every other objects presents itself in an easy way to calculate moment of inertia. In the event that the object has non-uniform density or they have uneven weight distribution, it is then incredibly cumbersome if not at all costs impossible to calculate moment of inertia. This then means that the only means that is available for obtaining the moment of inertia is through experiments. 3.0 EXPERIMENTAL DESIGN In this section, the various experimental design and procudures that are used in determination of the moment of inertia for the different objects are explained. This encompasses the use and application of the different experimental techniques. Proper explanation and justification of the experimental techniques that are used are given. 3.1 EXPERIMENT 1 – WHEEL AND AXLE PENDULUM METHOD: In this experiment the moment of inertia for wheel and axle was determined using wheel and axle pendulum method. The picture of wheel and axle on knife-edge support is shown in Figure1: 1 Figure1: Schematic diagram of wheel axle with attached bob support on knife edge (from lab sheet) When doing this experiment, there is estimation of the moment of inertia for every wheel and axle about their central axis. This is done through careful measurement of the mass and dimensions of each of the wheel and axle. When doing this computation, the effects of th grooves in the axles are ignored. A point to note is that the axles have different radius to the wheel and often considered to be different part (s). Another point that needs a lot of care is the mass of each and every other part. 3.1.1 EXPERIMENT EQUIPMENT AND MATERIALS: In doing the experiment, the following materials and equipment will be handy and they include: 1. Knife-edge pendulum support (pivot) 2. Wheel and axle (two various sizes) 3. Pendulum bob 4. Digital weighing scales 5. A steel ruler 6. A stopwatch 7. Calculator Procedure When doing this experiment there was development of an equation for the moment of inertia. The equation developed was for each wheel and axle that was put about their central axis “O” and had the knife-edge support as the pivot point and the pendulum attached as indicated. Assumption: Treat the mass of the pendulum bar as negligible Apply the parallel axis theorem and assume that small oscilliations (sinθ = θ). Using that equation, one goes ahead to experimentally determine the moment of inertia for every wheel and axle and they do this through attaching the pendulum to every axle and then doing a number of measurements. Some of the measurements that are taken into consideration include: 1. The mass of each wheel and axle 2. The length of the pendulum 3. The radius of the axle 4. The time taken for the small oscilliations Through the application of the derived equation, the moment of inertia for every wheel and axle was established through substitution of the value obtained experimentally into the equation. EXPERIMENT 2: COMPOUND PENDULUM METHOD: In this second method, there was use of compound pendulum method to help in finding the moment of inertia for the connecting rod. The figure is shown below: Figure: Connecting rod (From Lab sheet) EXPERIMENT EQUIPMENT AND MATERIALS: The materials that were used in this experiment are shown in the diagram below. Figure : Picture (4) of the equipment and materials used for the experiment In carrying out this experiment, the materials used included: 1. Connecting rod. 2. Knife stand 3. Calculator 4. Measuring tape 5. Timer 6. Protractor EXPERIMENT PROCEDURE: The steps that are undertaken for this experiment include: 1. Measure the mass of the connecting rod 2. Suspend the rod at one end using the knife-edge and at angles of 30° and 10° then record the average period of oscillations for small oscillations. 3. Measure the distance that is in between the two support points In this case, the periodic time and the properties of the connecting will be used to determine the moment of inertia by use of the equation derived for the connecting rod. 4.0 METHOD OF ANALYSIS When doing the analyses, various mathematical derivations will be applied: In doing the first experiment, some of the mathematical derivations include the following: = mass of wheel this experiment uses Equation of time period of pendulum and parallel axis theorem to obtain the moment of Inertia of wheel. Time period equation: ------------- (3) = moment of inertia about a pivot = length of pendulum Parallel axis theorem ------------------ (4) : The mass : The distance from the center Substituting (3) in (4) ……………………. (5) = mass of the bob = length of pendulum = radius of an axle T = is the average time period of per 10 Oscillation For the Experiment 2: Method 1 the following formulae will be applied: Potential Energy: PE=mgh …………………… (6) Where is: m: Mass of wheel g: gravity h: height Kinetic Energy (Rotational): …………………… (7) Where is: m: Mass of wheel v: velocity Kinetic Energy (Translational): …………………… (8) Where is: I: moment of inertia Angler of velocity PE= mgh = …………………… (9) …………………… (10) Where is: v: velocity r: radius of axle And …………………… (11) Where is: L: length of the plane T: time of traveling from top to bottom of the incline plane Sub (10) and (11) into (9) to get the Moment of Inertia Which will be: …………………(12) To determine the moment of inertia for experiment 2: method 2 - connecting rod by using parallel Axis theorem: Time period pendulum method can work out 5.0 RESULTS 5.1 EXPERIMENT 1– WHEEL AND AXLE PENDULUM METHOD: Table 1 below shows result containing the mass of the wheel and axle, the radius of the axle, the length of the pendulum and the mass of the pendulum. Table 1: The properties of the pendulum bob Mass of the pendulum bob 0.044 kg Length of the pendulum 0.09962 Table 2: Time of oscillation when wheel axle supported on the knife edge Small wheel axle Big wheel axle No ϴ = 10° with 10 oscillations ϴ = 30° with 10 oscillations ϴ = 10° with 10 oscillations ϴ = 30° with 10 oscillations Time Moment of inertia Time Moment of inertia Time Moment of inertia Time Moment of inertia 1 13.26 s 0.0014 13.53 s 0.0015 28.98 s 0.0086 30.21 s 0.0094 2 13.46 s 0.0015 13.48 s 0.0015 28.40 s 0.0082 30.18 s 0.0094 3 13.36 s 0.0015 13.54 s 0.0015 28.28 s 0.0081 30.16 s 0.0093 4 13.30 s 0.0014 13.83 s 0.0016 28.05 s 0.0080 30.05 s 0.0093 5 13.30s 0.0014 13.55 s 0.0015 28.05 s 0.0080 30.10 s 0.0093 Average 13.34 s 0.0014 13.59 s 0.0015 28.35 s 0.0082 30.14 s 0.0093 Table 3: The properties of the wheel axle Big wheel and axle Small wheel and axle Mass 3.3 kg 1.4kg Wheel diameter 0.150 m 0.1 m Axle diameter 0.0125 m 0.0125 m Axle radius 0.00625 m 0.00625m 5.2 EXPERIMENT 2: – COMPOUND PENDULUM METHOD: The length and distance from the center of gravity was measured (See Figure 6). The distance between small end and the centre of gravity 0.14565 m The distance between big end and the centre of gravity 0.05435 m Table 7: The properties of the connecting rod Mass 986 g Length (The distance between small end and big end), as shown in Figure 1 20 cm The small end of connecting rod was supported on the knife-edge. The time taken for displacement of 10° and 30° is presented in Table 9. Table 8: Time taken for different displacement for the connecting rod supported on the knife edge Small end Big end No ϴ = 10° with 10 oscillations Moment of inertia ϴ = 30° with 10 oscillations Moment of inertia ϴ = 10° with 10 oscillations Moment of inertia ϴ = 30° with 10 oscillations Moment of inertia 1 8.58 s 0.0054 8.41 s 0.0043 7.33 s 0.0042 7.25 s 0.0041 2 8.30 s 0.0037 8.50 s 0.0049 7.30 s 0.0042 7.25 s 0.0041 3 8.53 s 0.0050 8.41 s 0.0043 7.35 s 0.0043 7.23 s 0.0040 4 8.13 s 0.0027 8.46 s 0.0046 7.30 s 0.0042 7.23 s 0.0040 5 8.45 s 0.0046 8.53 s 0.0050 7.30 s 0.0042 7.26 s 0.0041 Average 8.40 s 0.0043 8.46 s 0.0046 7.32 s 0.0042 7.24 s 0.0041 The big end of connecting rod was supported on the knife-edge. The time taken for displacement of 10° and 30° is presented in Table 10. 6.0 DISCUSSION 1 EXPERIMENT 1– WHEEL AND AXLE PENDULUM METHOD: Using the work-energy method, the equation for mass moment of inertia for every wheel and axle was found and then the mass moment of inertia for axle A was then calculated and the results were as follows , and for the second axel B ,. From this, a conclusion is reached and then a comparison made with the other laboratory experiments. It emerges that there is a slight difference and this is due to slight slippage during the wheel and axle that rotates down the plane. The minimization can indeed be ensured through having more friction between plane and axle. 2 EXPERIMENT 2: COMPOUND PENDULUM METHOD: The values that are obtained is indicative that the periodic time obtained when the small end connecting rod is supported is higher in comparison to those of the big end of the connecting rod. This is mainly due to the distances that are between the two points as well as because the center of mass of the rod also vary. The periodic time is more when the small end acts as a fulcrum. 7.0 CONCLUSION It was indeed a success in conducting the experiment as the aims were fulfilled. It is found that the shape of the body affects the moment of inertia as well as the period of oscillation of a body is affected by the C.O.G. in the event that the center of gravity and the pivot of the body increases then also the periodic time increases. This then implies that the moment of inertia for two objects that have the same mass may be different owing to the difference in the distribution in the mass within the bodies. The experiment gives an insight into the different techniques that are applicable in determining moment of inertia of different objects. 8.0 REFERENCES Hunt, H 2012, YouTube - Hugh Hunt, viewed 27 February 2013, Retrived from < http://www.youtube.com/user/spinfun> on 2nd October, 2015. Hunt, H 2012, Dynamics movies: Hugh Hunt, Cambridge University Engineering Department, Cambridge University, viewed 2 October 2015, . Kirkup, L 1994, Experimental methods: an introduction to the analysis and presentation of data, John Wiley & Sons, Brisbane. Levinson, T. R 2005, Dynamics, Theory and Applications, New York: McGraw-Hill. Mason, M. T, 2001, Mechanics of Robotics Manipulation, MIT Press. Walker, D. H 2005, Fundamentals of physics (7th ed.), Hoboken, NJ: Wiley. APPENDICES APPENDIX 1: EXPERIMENT 1– WHEEL AND AXLE PENDULUM METHOD: Big wheel and axle % calculation for the experiment 1 - wheel and axle pendulum method % first i will found the moment of inertia for the big wheel and axle T1=28.98; %time for 1 return at 10 degree T2=28.40; %time for 2 return at 10 degree T3=28.28; %time for 3 return at 10 degree T4=28.05; %time for 4 return at 10 degree T5=28.05; %time for 5 return at 10 degree Tv=28.35; %time for average return at 10 degree T10=[T1 T2 T3 T4 T5 Tv]./10; % period of time at 10 degree T11=30.21; %time for 1 return at 30 degree T22=30.18; %time for 2 return at 30 degree T33=30.16; %time for 3 return at 30 degree T44=30.05; %time for 4 return at 30 degree T55=30.10; %time for 5 return at 30 degree Tvv=30.14; %time for average return at 30 degree T30=[T11 T22 T33 T44 T55 Tvv]./10; % period of time at 30 degree r=0.00625; %raduis for the axle in meter mb=0.044; % mass of the bob in kg mw=3.3; % mass of the wheel and axle in kg lb=0.0996; % the mass of bob in kg g=9.81; % the gravity in m/s^2 % we need to find the moment of inertia by using this equation I10=((T10./(2*pi)).^2).*(mb*g*lb)-(mw*(r^2))-(mb*(lb^2));% moment of inertia at 10 degree I30=((T30./(2*pi)).^2).*(mb*g*lb)-(mw*(r^2))-(mb*(lb^2));% moment of inertia at 30 degree I10 = 0.0086 0.0082 0.0081 0.0080 0.0080 0.0082 I30 = 0.0094 0.0094 0.0093 0.0093 0.0093 0.0093 Small wheel and axle % calculation for the experiment 1 - wheel and axle pendulum method % first i will found the moment of inertia for the small wheel and axle T1=13.26; %time for 1 return at 10 degree T2=13.46; %time for 2 return at 10 degree T3=13.36; %time for 3 return at 10 degree T4=13.30; %time for 4 return at 10 degree T5=13.30; %time for 5 return at 10 degree Tv=13.34; %time for average return at 10 degree T10=[T1 T2 T3 T4 T5 Tv]./10; % period of time at 10 degree T11=13.53; %time for 1 return at 30 degree T22=13.48; %time for 2 return at 30 degree T33=13.54; %time for 3 return at 30 degree T44=13.83; %time for 4 return at 30 degree T55=13.55; %time for 5 return at 30 degree Tvv=13.59; %time for average return at 30 degree T30=[T11 T22 T33 T44 T55 Tvv]./10; % period of time at 30 degree r=0.00625; %raduis for the axle in meter mb=0.044; % mass of the bob in kg mw=1.4; % mass of the wheel and axle in kg lb=0.0996; % the mass of bob in kg g=9.81; % the gravity in m/s^2 % we need to find the moment of inertia by using this equation I10=((T10./(2*pi)).^2).*(mb*g*lb)-(mw*(r^2))-(mb*(lb^2));% moment of inertia at 10 degree I30=((T30./(2*pi)).^2).*(mb*g*lb)-(mw*(r^2))-(mb*(lb^2));% moment of inertia at 30 degree I10 = 0.0014 0.0015 0.0015 0.0014 0.0014 0.0014 I30 = 0.0015 0.0015 0.0015 0.0016 0.0015 0.0015 APPENDIX 2: EXPERIMENT 2: – COMPOUND PENDULUM METHOD: Big end % calculation for the experiment 3 - connecting rod method m=0.986; % mass of the connecting rod in kg g=9.81; % gravity in m/s^2 h=0.05435; % the hight from the big end to the center of gravity in meter % time for the big end at 10 degree t101=7.33; % time number 1 for the big end at 10 degree t102=7.30; % time number 2 for the big end at 10 degree t103=7.35; % time number 3 for the big end at 10 degree t104=7.30; % time number 4 for the big end at 10 degree t105=7.30; % time number 5 for the big end at 10 degree t10v=7.32; % time number average for the big end at 10 degree T10=[t101 t102 t103 t104 t105 t10v]/10; % time for the big end at 30 degree t301=7.25; % time number 1 for the big end at 30 degree t302=7.25; % time number 2 for the big end at 30 degree t303=7.23; % time number 3 for the big end at 30 degree t304=7.23; % time number 4 for the big end at 30 degree t305=7.26; % time number 5 for the big end at 30 degree t30v=7.24; % time number average for the big end at 30 degree T30=[t301 t302 t303 t304 t305 t30v]/10; % now find the moment of inertia for all conditions I10=((T10/(2*pi)).^2)*(m*g*h)-(m*h^2); % moment of inertia for the big end at 10 degree I30=((T30/(2*pi)).^2)*(m*g*h)-(m*h^2); % moment of inertia for the big end at 30 degree I10 = 0.0042 0.0042 0.0043 0.0042 0.0042 0.0042 I30 = 0.0041 0.0041 0.0040 0.0040 0.0041 0.0041 Small end % calculation for the experiment 3 - connecting rod small end method m=0.986; % mass of the connecting rod in kg g=9.81; % gravity in m/s^2 h=0.14565; % the hight from the small end to the center of gravity in meter % time for the small end at 10 degree t101=8.58; % time number 1 for the small end at 10 degree t102=8.30; % time number 2 for the small end at 10 degree t103=8.53; % time number 3 for the small end at 10 degree t104=8.13; % time number 4 for the small end at 10 degree t105=8.45; % time number 5 for the small end at 10 degree t10v=8.40; % time number average for the small end at 10 degree T10=[t101 t102 t103 t104 t105 t10v]/10; % time for the small end at 30 degree t301=8.41; % time number 1 for the small end at 30 degree t302=8.50; % time number 2 for the small end at 30 degree t303=8.41; % time number 3 for the small end at 30 degree t304=8.46; % time number 4 for the small end at 30 degree t305=8.53; % time number 5 for the small end at 30 degree t30v=8.46; % time number average for the small end at 30 degree T30=[t301 t302 t303 t304 t305 t30v]/10; % now find the moment of inertia for all conditions I10=((T10/(2*pi)).^2)*(m*g*h)-(m*h^2); % moment of inertia for the small end at 10 degree I30=((T30/(2*pi)).^2)*(m*g*h)-(m*h^2); % moment of inertia for the small end at 30 degree I10 = 0.0054 0.0037 0.0050 0.0027 0.0046 0.0043 I30 = 0.0043 0.0049 0.0043 0.0046 0.0050 0.0046 Read More