Posted: September 9th, 2022

Conservation of Angular Momentum

Conservation of Angular Momentum
Introduction
Angular momentum is the quantity conserved in the experiment. The linear momentum is conserved when there is no net external force acting on a body (Abdul-Razzaq & Golubović, 2013). Therefore the angular momentum is conserved when the resultant force is zero. The conservation of angular momentum is dated back when it was applied in the study of planetary motions and the rotation of the bodies (Zhang & Marston, 2011). The angular velocity is a quantity affected by air resistance and should be factored in a while investigating the concept. Angular momentum has diverse applications in the construction of bends in structures like roads, railway lines, or bridges. The objective of the experiment is to analyze the rotation of the disks rotating on a common axis.
Theory
From Newton’s send law of rotational motion
τ ̅=(d L ⃗)/dt, (1)
If (d L ⃗)/dt,=0, then angular momentum is conserved.
Whereby τ is the torque and L is the angular momentum
The angular momentum L is connected by the distance r and the momentum m having a velocity v.
L ⃗=r ⃗×p ⃗=(r ) ⃗×(mυ ⃗ ) (2)
Angular momentum is a vector quantity; therefore, its magnitude is given by
L=mvrsinϕ (3)
Procedure
The experiment was done by the use of air tables and spark timer as shown

Figure 1 Apparatus used in the experiment
Two magnetic pucks were used in the experiment whereby one was fixed and the other left to move freely. The procedure was conducted by leveling the air table by using one non-magnetic puck. The paper was placed on the table, and pucks were set at the top right. The masking tape was used in holding the upper puck to the white paper. The air hoses were attached to both pucks and the launching process using a movable puck towards the fixed puck so that it formed the trajectory. The moving puck was deflected at an angle of about 90°. The points on the trajectory were labeled A, B, and C then connected to form a trajectory. The origin of the coordinate system was defined by basing on the center of the stationary puck. The values of t, r, ϕ v, and Δx, as well as z-component of L, were measured. The mass of the movable pack was also measured.
Results

Figure 2 Trajectory obtained from the experiment.
Mass= 0.726 kg
Table 1 Results obtained in the experiment
t (sec) r(m) ∅ (deg) Δx v(m/s) Lz (kg m2/s)
0.05 0.369 10 0.13 0.15 0.1089
0.1 0.289 23 0.11 0.17 0.1234
0.15 0.277 45 0.19 0.18 0.1307
0.2 0.201 50 0.20 0.20 0.1452
0.25 0.198 83 0.21 0.30 0.2178
0.3 0.194 77 0.25 0.19 0.1379
0.35 0.199 55 0.24 0.15 0.1089
0.4 0.201 45 0.18 0.14 0.1016

Figure 3 Relationship between momentum and time
The angular momentum is conserved in the experiment because the initial momentum from the start of the experiment is almost equal to the final momentum, as shown by the curve.
Table 2 Table showing the results obtained from the experiment when the new point is selected far from the center.
t (sec) r(m) ∅ (deg) Δx v(m/s) Lz (kg m2/s)
0.05 0.50 8 0.17 0.08 0.05808
0.1 0.44 10 0.15 0.10 0.0726
0.15 0.39 15 0.11 0.12 0.0812
0.2 0.34 26 0.18 0.15 0.1089
0.25 0.29 30 0.25 0.22 0.1597
0.3 0.31 31 0.27 0.20 0.1452
0.35 0.35 29 0.29 0.17 0.1234
0.4 0.37 22 0.22 0.13 0.09438

Figure 4 A graph of angular momentum versus time
Analysis
No
The calculation from the two origins did not agree because of the variation of the distance that caused the difference in the angular momentum. The longer distance decreases the angular momentum (Kanarev, 2002).
The first origin had the smallest angle of deflection and was measured from the furthest point.
No
Yes
In the second case, the angular momentum was not fully conserved due to factors such as resistance of contact that introduced the errors in the experiment. Some of the energy got lost, which led to a decrease in the velocity.
The shape of the second graph appears like an oscillatory curve, which suggests that the variation in the variables were not consistently leading to the effect of errors in the experiment.
The slope of the second graph was largest at 0.16 m/s and approximately 0.3 seconds. At this point, there was a maximum velocity, which suggests that the energy was fully conserved, and there was less resistance due to the contact friction.
Conclusion
The objective of the experiment was successively achieved. The change of the distance from the center produced a different curve as compared to the original distance. In the first trial, the angular momentum was fully conserved while in the second trial, it was not fully conserved since there was a loss of some energy. The pick points of the curves indicate the sections where the velocity is high and therefore contributing to high momentum. The source of error in the experiment is due to the friction between the bodies.

References
Abdul-Razzaq, W., & Golubović, L. (2013). Demonstrating the conservation of angular momentum using model cars moving along a rotating rod. Physics Education, 48(1), 42.
Kanarev, P. M. (2002). The Law of Conservation of Angular Momentum. Journal of Theoretics, 4(4), 1-7.
Zhang, L., & Marston, P. L. (2011). Acoustic radiation torque and the conservation of angular momentum (L). The Journal of the Acoustical Society of America, 129(4), 1679-1680.

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