MEM 331 – Elastic and Inelastic analysis of a Ballistic Pendulum and Determination of the Rotational Moment of Inertia 03/05/2015 Department of Mechanical Engineering and Mechanics Drexel University Kristap Kancans Mark Kauf John Simpson Abstract This experiment is conducted using a ballistic pendulum to explore the differences between elastic and inelastic collisions. The setup includes a pendulum, a launcher; serving as a gun and two small balls (steel and plastic); serving as bullets. The equipment is used in two different experiments; for the first set­up, inelastic collision experiment, the ball is launched into the pendulum and the maximum angular displacement is measured. For the second experiment, the setup emulates the closest to an elastic collision. When the ball is launched, the maximum rotational velocity and angular displacement is measured. These measurements were used to calculate the kinetic energy and conservation of momentum before and after collisions. Introduction The purpose of the experiment is to study the dynamics of inelastic and elastic collisions through the analysis of a ballistic pendulum. A ballistic pendulum is a device designed to measure a bullet’s momentum. The first portion of the experiment involves an inelastic collision used to calculate the velocity of a projectile launcher through the measurement of maximum angular displacement, conservation of momentum and energy considerations. The ball, or bullet, is shot from the launcher and is embedded in the bob of the pendulum. The pendulum reacts by swinging up to a particular height. In this reaction, momentum is conserved and can be represented by the following equation: mbV o = M V [Eq 1] In equation 1, the mass of the ball multiplied by the muzzle velocity is equal to the total mass (ball and pendulum) multiplied by the velocity of the catcher (ball and pendulum bob). The linear kinetic energy of the bob and ball after the collision can be modeled using potential energy. This potential energy is expressed using the following equation: 2
1
2 MV
= M gh [Eq 2] Combining equation 1 and equation 2, the muzzle velocity of the launcher can be calculated. The equation for muzzle velocity becomes: m
V o = (1 + mpb)√2gh [Eq 3] For the second portion of the experiment, elastic collisions were modeled using the same system. A rotational momentum of the ball was measured before and the rotational momentum of the ball and pendulum system combined was measure after. This is used to investigate the kinetic energy and gravitational potential at maximum height. Angular momentum is modeled by: L=⃗
r×⃗
p [Eq 4] When perpendicular this relation reduces to: L = mbrV o [Eq 5] Where mb represents the mass of the ball and r is the distance from the center of the ball to rotational centroid of the pendulum rod. Using an axis of symmetry, the angular momentum is: Lo = I ωo [Eq 6] Angular velocity can be modeled as: ω=
r⃗
×⃗
v
|⃗
r|2
[Eq 7] Which reduces to the following when perpendicular: v = ωr [Eq 8] Lastly the angular kinetic energy of the pendulum system is found as: K o = 12 Iω2o Procedure Part 1: Ballistic Pendulum Inelastic Collision The first part of the experiment used a launcher, a C­clamp, a table clamp, a mounting rod, a plastic ball, a steel ball and a rotary motion sensor (RMS). Software named DataStudio was used in the experiment. The set­up is shown in Figure 1. The pendulum was screwed to the RMS, allowing it to swing freely. Figure 1: Pendulum Setup The plastic ball was then loaded into the launcher in the “short” position. DataStudio was prepared to display the angular position and the program was run to collect data. Next, the ball was launched and caught in the pendulum catcher. After the pendulum swung back, data collection was stopped. In the angular position graph in DataStudio, the maximum angular displacement was labeled and recorded. This procedure was repeated for each ball in short and medium ranges. Part 2: Find the Center of the Mass This part had the same setup as part 1, but with only the steel ball at medium speed. This time the ball was launched and the pendulum was caught before it swung back. The pendulum was then removed from the RMS and placed at the edge of a table with the pendulum shaft perpendicular to the edge and the counterweight hanging over the edge. The pendulum was pushed out until it barely balanced on the edge of the table. Finally, the distance from the center of rotation to the center of mass was measured. The same procedure was repeated for a plastic ball. Part 3: Ballistic Pendulum Elastic Collision The setup of part 3 was similar to part 1 but the pendulum‘s catcher was facing away from the launcher. The DataStudio file “Elastic_collision.ds” provided was opened. The experiment was performed same as in part 1 and the maximum displacement and initial angular velocity were recorded. This procedure was repeated for three trials using the steel ball were and the average value was calculated from those results. Results
To learn the differences between elastic and inelastic collisions using a ballistic pendulum, Part 1 of this experiment focused on using a ballistic pendulum apparatus to explore inelastic collisions. The maximum angular displacement was recorded for short and medium ranges of plastic and steel balls and are shown in Table 1. Part 2 of this lab involved finding the center of mass for each ball which was found to be 37.09 cm for steel and 36.62 cm for plastic as shown in Table 2. The third part was to discuss the effects of an elastic collision. The maximum angular displacement and angular velocity were taken throughout the experiment. The values discussed can be seen in Table 3 below. Table 1: Maximum Angular Displacement Table 2: Center of Mass Table 3 ­ Angular displacement, initial angular velocity, and period Table 4: Initial Values ­ Part 3 Table 5: Calculated Values ­ Part 3 Discussion For part one of the experiment, or the inelastic collisions, the percentage of energy loss was calculated to be 62.55% and 87.99% for steel and plastic respectively. The values were the same for both short and medium measurements. The energy loss in these calculations is due to energy lost as heat when the ball is embedded in the pendulum bob. No change was noted in momentum when the velocity of the launcher was increased. This observation was consistent for both steel and plastic. As was noted before, the change in velocity did not affect the ratio for either steel or plastic. The percentage of energy lost in the elastic collision was calculated to 82.31%. This is a significant amount of energy loss. Since this is not a perfectly elastic collision, energy will be lost. The energy lost during the elastic collision observed through this experiment was due to heat loss. This energy loss due to the transfer of heat occurs when the bullet strikes the pendulum bob. The potential energy observed was higher than the initial kinetic energy. This should be expected considering the design of the experiment renders the elastic collision assumption invalid. The moment of inertia was larger than the other two moments of inertia. Better handling of the set­up of the pendulum could attribute to a more accurate moment of inertia. If the bullet strikes the bob unevenly this will throw off the calculation. Conclusion This experiment focused on fundamental principles associated with elastic and inelastic collisions. The main source of error for this experiment could be ascribed to the pendulum catcher. Even slight misalignment of the catcher could vary the results of the experiment. Additionally, the results gathered shows that much of the energy within the system was lost during the collision as heat. Overall, this experiment demonstrated the fundamental principles behind the conservation of angular and linear momentum as well as the conservation of energy. It effectively allowed to measure high speed collisions which exemplify what test engineers go through when designing weapons, canons, and other high speed launchers.