Moment of Inertia of a Flywheel - Lab Report Example

MOMENT OF INERTIA OF A FLYWHEEL Abstract The aim of the experiment was to establish the moment of inertia of a flywheel by using a combination of linear and circular motion equations. The experiment involved hanging a mass on the fly wheel via a spindle and the mass was left to drop freely through a distance of 1m thus providing the necessary force to accelerate the flywheel. The mass was increased from five kilogram to 20kg in step of 5kg. Having taken the relevant measurement including time of covering 1m drop and the radius of spindle; the torque created by the hanging mass Pr and the angular acceleration were calculated. It was necessary to have at least for runs for each mass so as to reduce the error in timing. Plotting Pr against  the moment of inertia was obtained from the gradient of graph as 0.552kgm2 while the value obtained theoretically through calculation was  kgm2. The experiment was concluded as being a success because the graph between Pr and  exhibited a linear relationship as expected and that the theoretical and graphical values were found to be close. . Introduction When a body is under linear motion, they will be moving in a straight line though the may not be moving at a constant speed or with constant acceleration. A trolley moving on a frictionless plane gives a good example of linear motion where there is constant velocity. A high density object such as a stone or a piece of metal that is under going free fall has a motion that is very close to uniform acceleration (Tipler P.A. 2003). In real life some of the areas where circulation motion is encountered is the leisure park equipment that includes wave swinger and the pirate ship. When vehicles are negotiating corners, some house hold equipments including salad dries, washing machines among others (Knudsen, M.& Hjorth, G. ,2000). In this experiment the aim is establish the moment of inertia of a flywheel by using a combination of linear and circular motion equations. Theory for linear motion Linear motion involves the kinematics of bodies under uniform velocity or where the non-uniform velocity may be involved but the acceleration is uniform in nature. As we had seen in the introduction a good example of this is where we have free fall of objects where gravity is ideally the only force acting on the falling object. The objects will be under acceleration that is caused by the force of gravity which near the earth surface is relatively constant at about 9.81m/s2 (Nainan K. ,2009). Air resistance could have some influence by for compact and relatively heavy objects and with distance being short enough not to result to very high speeds then air resistance will be negligible (Resnick, R. & Halliday, D.,1966). The equation that are dealt with under linear motion are i.  ii.  iii.  iv.  Where v= final velocity, u = initial velocity, t= time taken, s= the distance covered a = acceleration Theory behind circular motion When objects are subjected to circular motion, the velocity will be changing constantly even when there speed remain constant and thus the objects will be under constant acceleration for example in the casing involving the whirling of a mass attached to the end of a string. With the direction of the object constantly changing it means the velocity will be changing because velocity is a vector quantity which will change when there is change in direction even when the magnitude is kept constant. The direction of motion of the mass will be changing constantly in the direction of the centre of circle described by the path the mass is tracing. This a manifestation of the mass undergoing constant acceleration in the direction of the centre of circle , and the acceleration is what is referred to as centripetal acceleration  is given by the expression Figure 1 Moment of inertia Moment of inertia for a flat disc is given by Experiment set up and quantities M=mass of disc R=radius of disc m=falling mass kg h=distance fall m t=time of fall of mass s d=diameter of spindle R=radius of spindle, m (d/2) a=acceleration of mass m/s2  =angular velocity rad/s  =angular acceleration rad/s2 P=tension in cord N g =is gravitational accn 9,.81m/s2 Method Relating formulae The centripetal force F=ma Considering the inertia of a flywheel  (1) Where  frictional torque Moment of inertia Newton law applied to falling mass  (2) From  Acceleration  (3) Thus putting (3) into (2)  (4) From (v=u+at) and using the expression for acceleration a given by equation 3b , the speed of mass m when it hits the ground is (4) Since  and substituting for v using (5) the angular velocity  of the flywheel when mass hits ground is  (6)  (7) Substituting for  in (7) using (6)  ( 8) Rewriting equation (1) gives Results Theoretical moment of inertia The theoretical moment of inertia is given by Where M = 59.31lb= = 59.31x0.453592=26.9kg R=0.19m Table 1 summarizes the results obtained in the experiment. It can be seen that for the first mass of 5 kg the time taken to cover the 1m distance range between 23.23s to 25.67. The average time was 24.3s for this mass. The heaviest mass used was 20kg and the time to cover 1m for this mass ranged between 12.0 and 12.7 with the average coming to 12.3625. Calculating angular acceleration The calculation of  involved the use of the equation So for the 5kg mass t=24.3s Calculating P The tension in the string P was calculated from the eq  (equation 4) For mg= 5 0.001726 Pr for the same mass = 0.001726x0.01893=0.094617 Using the same procedure the result were as shown in table 1 The table shows that the angular acceleration increased from 0.178923 to 0.6913 as the mass was increased from 5kg to 20 kg. The product of tension P and radius of spindle r increased from 0.094617to 0.378095. When the Pr was plotted against the angular acceleration  a linear relationship graph was obtained. From the graph it can be seen that the gradient of the graph is 0.551 which meaning the moment of inertia of the flywheel is 0.551 kgm2. The y-intercept is -0.01 which represents the frictional torque. Weight(N) Time 1 Time 2 Time 3 Time 4 Average Pr 0 0 0 0 0 0 0 0 5 24.6 25.67 23.7 23.23 24.3 0.178923273 0.094617 7.5 19.10 19.75 19.37 19.33 19.3875 0.281083764 0.141898 12.5 14.68 15.15 15.35 15.24 15.105 0.463060698 0.236414 15 14.15 14 14.35 14.06 14.14 0.528421601 0.28366 17.5 13.0 13.10 13 12.96 13.015 0.623721963 0.330876 20 12 12.5 12.7 12.25 12.3625 0.691300343 0.378095 Discussion From the experiment it has been seen that increasing the mass attached to spindle resulted in the mass moving at a faster speed and covering the distance of 1m within a shorter time. This is because the frictional force does not increase with the increasing pass and thus as the mass increase the downward force increases considerably resulting into increased acceleration. With the timing having at least 4 values the error was considerably reduced. When Pr was plotted against the angular acceleration  a straight line graph was obtained. This was in agreement with what was expected. The gradient of the graph was found to be 0.551 kgm2which represent the moment of inertia of the flywheel. The y-intercept of the graph was -0.01 which is the frictional torque of the system. The negative sign indicates that this friction resists the toque Pr that is causing the mass to accelerate downwards. The theoretical moment of inertia was found to be  is slightly lower but close enough to the graphically obtained value. It is because of the frictional torque causing resistance to the motion of the flywheel that attempt is made to reduce it by lubricating the bearing in addition to use of bearings being in itself a precaution of reducing friction. Conclusion From the results it can be concluded that the experiment could successfully be used in finding the moment of inertia of a flywheel. The theoretical value was found to be close to the value that was obtained graphically. Having a number of values taken is an important in elimination errors more so where a person is used to start of stop the watch. References Knudsen, M.& Hjorth, G. (2000). Elements of Newtonian mechanics: including nonlinear dynamics (3 ed.). Springer. p. 96. ISBN 3-540-67652-X Nainan K. (2009).,Gravitation Resnick, R. & Halliday, D. (1966), Physics, Chapter 3 (Vol I and II, Combined edition), Wiley International Edition, Library of Congress Catalog Card No. 66-11527 Tipler P.A.& Mosca G. (2003). "Physics for Scientists and Engineers", Chapter 2 (5th edition), W. H. Freeman and company: New York and Basing stoke, Nainan K. (2008).Varghese, Hypothesis on MATTER (second edition). 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