Moment of Inertia of Wheels and Axle Units and Conrod Using Different Methods - Coursework Example

MOMENT OF INERTIA OF WHEELS AND AXLE UNITS AND CONROD USING DIFFERENT METHODS Student Name Course Date Abstract This report, presents an analysis of the moment of inertia on two wheels and axle units and that of the conrod using different mathematical methods . The mathematical methods used in this case were the pendulum method, compound pendulum and the inclined plane. Based on this methods, the moment of inertia these bodies under consideration was obtained and then compared with that obtained from the geometrical methods. From this analysis, it was found that the pendulum method was applied by making several assumptions that led the situation to behave like the parallel axis method. In addition to that, it was found that the deviation of the moment of inertia from the pendulum method was so immense for small masses while for large masses, the deviation was negligible. Table of Contents Abstract 2 MOMENT OF INERTIA OF WHEELS AND AXLE UNITS AND CONROD USING DIFFERENT METHODS 4 1.0 Introduction 4 2.0 Background 4 2.1 Pendulum Method 5 2.2 Inclined Plane Method 5 2.3 Geometrical Method 6 2.4 Compound Pendulum 7 2.5 Parallel Axis Theorem 7 3.0 Experimental Design 7 Instruments Required 8 3.1 Pendulum method 8 4.0 Results 9 5.0 Discussion 14 6.0 Conclusion 15 References 16 MOMENT OF INERTIA OF WHEELS AND AXLE UNITS AND CONROD USING DIFFERENT METHODS 1.0 Introduction A rotating body about its axis requires a torque in order to change its direction or rate of motion. In this case, the amount of torque needed to alter the motion is directly proportional to the change in the angular velocity of the body in consideration. In addition to this, the constant of proportionality is an important factor to consider for rotating bodies when analysis of the body under consideration is being taken. This constant has the units of mass and length. It is commonly known as the moment of inertia. With regard to that, this constant depends on the mass and the shape of the body. With respect to this, there are different methods of determining the moment of inertia; pendulum inclined plane and the final methods. In this paper, an analysis of the moment of inertia of two wheels and axle units and the Conrod is done using two different methods for each case. 2.0 Background The moment of inertia is the measure of the resistance of the bodies to rotate about a given axis. This resistance of the motion of the body is affected by the size, mass and shape of the body under consideration. With respect to that, the further the distance of the object from the axis, the larger the moment of inertia. This moment of inertia is given by equation 1:Where; From the equation, it is observed that the larger the radius the larger the moment of inertia given that the mass is constant. The following are the various methods of analysing the moment of inertia that were used in this experiment. 2.1 Pendulum Method A pendulum is a simple machine that rotates about the fulcrum. This machine rotates at an angle in an a repeated manner. When this body rotates, it takes a certain amount of time to reach its maximum position before it returns. All this time through which the pendulum takes to make a complete revolution is called a period. Equation 2 gives the relationship between the moment of inertia of the rotating body and the periodic time. 2.2 Inclined Plane Method This method is used in calculating the moment of inertia of bodies that are rolling down an inclined plane. In this case, the wheel and axial of the body are considered. For better performance of the method, the force that resists motion is neglected. Therefore, the moment of inertia is obtained from the amount of work. In this case, the law of the conservation of energy is the most fundamental thing to be put into consideration, potential energy changes to kinetic energy When bodies are released to fall from inclined planes, the bodies experiences the moment of inertia. This moment of inertia is easily determined by the inclined plane method. Equation 3 shows the relationship between the potential energy and kinetic energy while equation 4 shows the relationship between the potential energy and the components of kinetic energy. That is; Potential Energy = Kinetic Energy But, the kinetic energy comprises of the rotational energy and translational energy. With respect to that, equation 3 expands to; Equation 4 reduces to equation 5 that gives the equation of the moment of inertia as shown below. 2.3 Geometrical Method This method is used in determining the moment of the inertia of a body by considering the weight of the wheels and the axle. Through this, the moment of the inertia is obtained as shown below in equation 6. In this case, 2.4 Compound Pendulum This method utilizes the knife as the point of rotating while the object under investigation in terms of its moment of inertia is suspended vertically and it rotates in normal forces of gravity with small amplitudes. Equation 7 shows the moment of inertia of a body under the compound pendulum method. 2.5 Parallel Axis Theorem This method operates under the principle of rotating mass in which the axis of rotation is at the centre of the mass. In this case, the moment of inertia is expressed in terms of the partial value of moment of inertial and that due to change in mass. Equation 8 shows the equation of the moment of inertial of a rotating object. Where is the moment of inertia of the axis passing through the centre of mass, m is the mass of the object and d is the distance between the axes parallel to each other. In this case, the angle of oscillation is small enough to allow the assumption of . 3.0 Experimental Design The wheel and axle experiment was done in two different parts: Two Wheel and Axle components of different Dimensions were used in one part Removable bob weight threaded to suit both Wheel & Axle components Inclined plane Connecting Rod Retort Stand Instruments Required The following pieces of equipment were required to conduct all of the experiments Digital weighing scale A steel ruler A stop watch A Vernier calliper Wheel and Axle Experiment 3.1 Pendulum method For the case of the pendulum method, moment of inertia for each individual wheels was obtained using the pendulum equation discussed in the background section. With respect to the arrangement of the system, parallel axis theorem was used in order to determine the moment of inertia of the bodies about the point of rotation, the point where the knife edge and the axle were in contact. In order to perform this experiment, an angle of 10 degrees was chosen for testing the response of the system when it is released. By choosing 9 oscillation as the point of analysis, the total time of the oscillation was recorded. In addition to this, the following was done: The masses of the objects were determined by weighing the objects on a digital weighing machine that had a resolution of ±1 gram. The mass and dimensions of the pendulum bob were also obtained. The diameter, radius and thickness of each wheel and axis was obtained too. Desired period of oscillations at small angles for both wheel and axis units were recorded fort analysis 4.0 Results Tables 1 through 3shows the dimension and the masses of the wheels used and the axles. Table 1: Dimensions of the Small Wheel and axle; Diameter, radius and Length Diameter d (mm) Radius r (mm) Length L (mm) Small wheel 100.00 50.00 20.30 Axle 12.50 6.25 149.65 Table 2: Dimensions of the large Wheel and axle; Diameter, radius and Length Diameter d (mm) Radius r (mm) Length L (mm) Large wheel 150 75 23 Axle 12.60 6.30 152.75 Table 3: Dimensions and mass of the Wheels; Thickness and Mass Thickness (mm) mass (kg) Small wheel 20.35 1.36 Large wheel 23 3.274 From the above dta of the wheels and axles used, the following was deduced. Volume Equation for Axle Volume Equation for wheel Volume ratio Mass of wheel and Axle Based on the above calculation, table 4 shows the results of the moment of inertia using the geometrical method. Geometrical Method Table 4: Moment of inertia of the small and large wheels based on the Geometrical Method Moment of inertia () Small wheel 0.0015 Large wheel 0.00908 In addition to that, table 5 shows the results of the oscillation of the pendulum for the small wheel while table 6 shows the results of the pendulum oscillation of the for the large wheel and axle. In addition, table 7 shows the Table 5: Results of small oscillations of the pendulum with the small wheel Set Oscillations Time, t (s) 1 9 11.50 2 9 11.40 3 9 11.48 Table 6: Results of small oscillations of the pendulum with the large wheel Set Oscillations Time, t (s) 1 9 25.57 2 9 26.12 3 9 26.19 Table 7: Inclined plane using small wheel Set Mass (kg) Radius of axel (mm) Gravity Height (m) Time, t (s) 1 1.36 6.25 9.81 0.206 5.55 2 1.36 6.25 9.81 0.206 6.12 3 1.36 6.25 9.81 0.206 6.23 Table 8:Inclined plane using large wheel Set Mass (kg) Radius of axel (mm) Gravity Height (m) Time, t (s) 1 3.275 6.25 9.81 0.206 9.20 2 3.275 6.25 9.81 0.206 8.50 3 3.275 6.25 9.81 0.206 9.09 On the side on the connecting rod, table9 shows the results of the experiment when the connecting was used in determining the moment of inertia of the rod at different times. Table 10 alos, shows the results of oscillations about the gudgeoned end (small hole) of the connecting rod Table 9: The Dimension and the mass of the connecting rod used in the Experiment Position Diameter (mm) Radius (mm) Gudgeon End (d1) 17.35 8.675 Big End (d2) 51.45 25.725 Length ‘h’ 1 (mm) 140 Length ‘h’ 2 (mm) 59.7 Length ‘l’ (mm) 199.71 Mass (kg) 0.971 Table 10: Results of oscillations about the gudgeoned end (small hole) of the connecting rod Set Oscillations Time, t (s) 1 9 7.53 2 9 7.20 3 9 7.15 As a result of this, table 11 shows the results of the moment of inertia on the different methods used in comparing the moment of inertia with the geometrical method. Table 11: Theoretical inertia values for individual components Method Small Wheel Big Wheel Connecting Rod Pendulum 0.0017 0.01 0.00364 Inclined Plane 0.0015 0.0085 Geometrical 0.0015 0.00908 5.0 Discussion Pendulum In this case, the system was treated to be the normal pendulum. This necessitated the use of the general equation in determining the moment of inertia od the bodies under consideration. Table 11 shows the results for this. In addition to that, the mass of the bar of the pendulum was neglected during the calculation of the moment of the inertia of the bodies. This neglected parameter and frictional forces might have led to the deviation of the moment of inertia obtained using this method. Incline Plane On the case of the inclined plane method, work energy method was used in the determination of the moment of inertia. The result of the moment of inertia of both the small and the large wheels were as shown in table 11. In this case, the expectation was that the larger wheel to have the large moment of inertia while the small wheel had to be small, this was due the size and the mass of the wheel as it was depicted in the background section. Inaccuracies within this experiment may be caused by slippage of the wheel whilst rolling down the inclined surface; however, these inaccuracies were ignored to simplify the experiment. 6.0 Conclusion The main purpose of the experiment was to determine the moment of inertia of the connecting rod and the wheels and axle units. This was to be determined using the mathematical approach and then comparing the results by that of the geometrical method. In addition to that, different shapes and the size of the loads were compared too. Based on the above objectives, it was found that the inaccuracies of the moments of inertai fro the smaller wheel were large as compared to that to the larger wheel. That is, ±0.0002 mm and ±0.001 mm respectively With respect to that, it can be concluded that the large with high frequencies have large the moments of inertia. But this depends of the mass distribution and the centre of axis of the object under consideration References 1. Riley, W 1996, Engineering Mechanics Dynamics, 2nd edn, John Wiley and Sons, Brisbane Parallel axes theorem on moment of inertia : Ok Physics!. 2014. Parallel axes theorem on moment of inertia : Ok Physics!. [ONLINE] Available at: http://okphysics.com/parallel-axes-theorem-on-moment-of-inertia/. [Accessed 15 April 2014]. Read More