Moment of Inertia of a Flywheel - Lab Report Example

Moment of Inertia of a Flywheel Name Date Abstract We performed an experiment to determine the moment of inertia of a flywheel by accelerating it using a falling mass. This began with measuring the radius, circumference, the time and the distance covered to the ground by the mass attached to the flywheel. Different masses and corresponding time taken to cover the distance were measured and recorded on a table. This was done a number of times and the average time taken by each mass to accelerate to the ground was recorded. The velocity, V and angular acceleration, α were calculated with the help of the excel software before plotting a straight line for the moment of inertia of the flywheel. The moment of inertia of the flywheel was found to be 0.483 kgm2 and the frictional torque was found to be 0.0174 Nm. The moment of inertia that was calculated was found to be 0.4855 kgm2. The percentage error was 0.51%. This means that the torque obtained from the experiment was less than the expected value basically due to errors encountered. This experiment has facilitated in developing understanding of the dynamics of the rotational systems, error analysis and comparing the theoretical values to those obtained from the experiment. (200 words) Objective To determine the moment of inertia of a flywheel by accelerating it using a falling mass Theory and methodology The apparatus that was used are normally used to measure the moment of inertia for flywheel with radial arms in which the masses are attached. The torque that rotates the flywheel was obtained by attaching a hanging mass on a string wrapped around the flywheel (Meriam & Karaige, 2007). The radii of the axle and the flyhweel (r, R), the mass of the flywheel and the mass to be hung on the string were measured and recorded. The string was wound around the axle until the mass was at a height of 1m above the ground like as shown in the figure below. Then the system was released to allow the mass to fall and rotate the axle and the flywheel. The stop watch was used to measure the time taken by the mass to cover the vertical height. When the hanging mass was released, it accelerated to the ground. Different masses was placed at a time and the time taken to descent to the ground were measured. With the help of the excel software, the moment of inertia of the flywheel was determine by plotting a straight line. The theoretical value for the moment of inertia was also estimated using the formula for the flywheel of a solid disc, in order to check the accuracy of the experiment results. The equation for the angular acceleration of the flywheel: α = a / r. The acceleration (a) is determined by timing (t) the mass as it falls through a measured distance (d). When starting from rest, the mass falls a vertical distance to the floor in time with constant linear acceleration (Sawhney, 2009). Results Table for average time Weight (Kg) 1 2 3 4 Average 0.5 22.26 21.89 21.92 21.70 21.94 1.0 15.32 15.57 15.85 15.42 15.54 1.5 13.23 13.50 12.67 12.90 13.08 2.0 11.86 11.00 11.40 11.50 11.44 2.5 10.40 10.62 9.90 10.20 10.28 D = 38 mm Height = 1 m Radius of the flywheel, R = 0.19 mm Mass = 59.31 lbs = 26.9 Kg The theoritical moment of inertia for the flywheel is given by Other calculations For a rotating object about a given axis, a the relationship between the torque applied, T, the angular acceleration and the moment of inertia of the object is given by the following equation. T = Iα, thus, Where Tf is the torque caused by the the frictional forces acting on the flywheel, P is the tension in the cord, d is the vertical distance for the fall and I is the moment of inertia Therefore, the moment of inertia, I is the extent to which a body tend to resist change in its rotational motion and the torque is angular acceleration, α. The formulas used for the falling masses included the following Applying newton’s second law on the falling mass: mg – P = ma For acceleration, a = 2h/t2 Thus P = mg – 2mh/t2 The speed of mass when it hits the ground using the formula v = u +at, is V = 2h/t The angular acceleration, w = v/r = 2h/rt Angular acceleration α = 2h/t2r From the above equations, P.r = Iα + Tf Table for torque and angular acceleration Mass (Kg) Height, h (m) Average time (s) Traveling speed V (m/s) Angular velocity w (rev/s) Tension in the cord, P Radius of the spindle, r Pr Angular acceleration α, (rads) 0.5 1 21.94 0.09116 4.79777 4.902923 0.019 0.093156 0.218677 1 1 15.54 0.12870 6.77369 9.801718 0.019 0.186233 0.435887 1.5 1 13.08 0.15291 8.04764 14.69747 0.019 0.279252 0.615263 2 1 11.44 0.17483 9.20133 19.58944 0.019 0.372199 0.804312 2.5 1 10.28 0.19455 10.2396 24.47769 0.019 0.465076 0.99607 The graph for torque against angular acceleration is show below From the graph, the Moment of Inertia of the flywheel = 0.483 kgm2 The frictional torque = 0.1742 Nm Discussion In this experiment, the moment of inertia of the flywheel and its spindle, as well as frictional torque was obtained. The basic arrangement for this experiment is shown below. The moment of inertia of the flywheel which consists of a large uniform disc that is attached to an axle at its center is caused to rotate about the an axis when a torque is applied due to the mass on the string wrapped around the axle. The disc rotates about the central axis as the mass move linearly downwards (Serway& Jewett, 2013). The forces acting on the mass are tension and weight, mg. by applying newton’s second law of motion we get mg – T = ma, thus T = mg – ma The flywheel with a moment of inertia, I rotates due to the torque applied on the axle by the string with the tension, T. the torque is the product of the force applied on the lever arm. Thus the torque is Pr. Where r is the radius of the axle. The mass accelerate through the distance h at a specific time and is calculated as shown below. The point on the outer radius, r of the axle travels the same linear distance that the mass, m travels on the vertical direction. The acceleration, velocity and angular displacement are also related to its linear displacement and the acceleration and the velocity of the outer radius by a given ratio. Specifically, the angular and linear acceleration are related as α = a/R. The theoriticallly, the system is treated as a uniform disc and ignore the effects of the friction in the calculation of the moment of inertia in the flywheel and is calculated as I = MR2. The moment of inertia obtained from the experiement was 0.483 kgm2 and the theoritical value was 0.4855 kgm2. The difference may have resulted from errors during the calulations such as using incorrect procedure or arrithmetical errors. The errors may also have resulted from the experimental procedure that included the mass taking less time to reach the ground after being pushed. The reaction time when taking the values may also have contributed to the error. However, this error was minimised by taking 3 sets of data and calculating the average. Conclusion In this experiment, the moment of inertia of a flywheel has been found to be 0.483 kgm2. This was obtained by accelerating different masses to the ground through a vertical distance. The time taken to reach the ground was measured and recorded in a table to be used to calculate acceleration and the torque. The gradient of the straight line curve obtained is equivalent to the moment of inertia of the flywheel. The theoretical value of the moment of inertia was found to be 0.4855 kgm2 and the frictional torque was found to be 0.0174 Nm. The sources of errors included incorrect procedure or arrithmetical errors, and the errors resulting from the experimental procedure that included the mass taking less time to reach the ground after being pushed. The reaction time when taking the values may also have contributed to the error. However, this error was minimised by taking 3 sets of data and calculating the average. This experiment has facilitated in developing understanding of the dynamics of the rotational systems, error analysis and comparing the theoretical values to those obtained from the experiment. References [1] Meriam J.L., & Karaige. L.G. (2007). Engineering Mechanics: Statics, 6th edition. Wiley. [2] Serway, R. A., & Jewett, J. W. (2013). Principles of physics: A calculus-based text. Boston, MA: Brooks/Cole, Cengage Learning. [3] Sawhney, G. S. (2009). Mechanical experiments and workshop practice. New Delhi, International Pub. House. [4] Rao, J. S., & Dukkipati, R. V. (1992). Mechanism and machine theory. New Delhi, New Age International. Read More