Angular Momentum Handout Lab

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT INTRODUCTION In this lab we will seek to determine if angular momentum is conserved during two experiments. In the first experiment we will determine the moment of inertia of a square mass at two positions. We will then experimentally determine the angular velocity of the mass as it slides from position 1 to position 2. The second experiment involves an “inelastic” collision between a spinning disc and a metal ring. Using the moment of inertia determined in last weeks lab we will find the angular momentum of the spinning disc before and after the metal ring is added to the disc. THEORY Every particle has an associated angular momentum, which can be defined as follows ⃗ l = ⃗ r × ⃗ p where ⃗ r is the radial vector describing the particle’s position, relative to the axis of rotation, and ⃗ p is the particle’s linear (or tangential) momentum. When the particle experiences no net external torques then its angular momentum is conserved such that ⃗ l 0 = ⃗ l f where ⃗ l i is the particle’s initial angular momentum and ⃗ l f is the particle’s final angular momentum. In the case where there are multiple particles, the conservation of angular momentum becomes ∑ i = 1 N ⃗ l i 0 = ∑ i = 1 N ⃗ l i ⃗ l 10 + ⃗ l 20 + … + ⃗ l ¿ = ⃗ l 1 f + ⃗ l 2 f + … + ⃗ l Nf and when the system of particle behaves as a rigid body then the equation can be written as ⃗ L 0 = ⃗ L f I 0 ⃗ ω 0 = I f ⃗ ω f I 0 ω 0 = Iω where ω 0 is the rigid body’s initial angular speed, I 0 is the rigid body’s initial moment of inertia, I is the rigid body’s final moment of inertia, and ω is the rigid body’s initial angular speed. A mass that’s connected to a rotating apparatus via a string will accelerate downwards, towards the ground, with an acceleration given by Newton’s Second Law of motion ∑ i = 1 2 F iy = ma y

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT T y + W y = m a y T − W =− ma T = W − ma = m ( g − a ) where T is the magnitude of the tension in the string, m is the attached object’s mass, g is the magnitude of gravity’s acceleration, and a is the magnitude of the attached object’s acceleration. This tension from the string exerts a non-zero net torque onto the middle pulley of the rotating object’s spindle with a magnitude is given by the equation ∑ i = 1 1 τ iz = I α z τ Tz = I α z rT sin ϕ = Iα ( 1 2 d ) T sin ( 90 ° ) = Iα 1 2 dT = Iα where d is the diameter of the middle pulley, I is the rotating object’s moment of inertia, and α is the angular acceleration magnitude of the rotating object. Substituting these previous two equations into each other we get back 1 2 dm ( g − a ) = Iα The magnitude of the tangential acceleration of an object is defined to be a t = αr sin ϕ a t = α ( 1 2 d ) sin ( 90 ° ) a t = 1 2 αd Combining the previous two equations, we obtain the following equation I = 1 2 m ( g − a ) d α

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT 19. Repeat the descending mass trial.  A copy of this graph doesn’t need to be saved. 20. Remove the square mass from the rotating platform. 21. Attach the other square mass, with a string wrapped around it, to the position touching the screw at the 20.5 cm position.  Unwind the string before sliding it into the top groove.  The string must be oriented towards the center post.  Do not lock this square mass into place. 22. Loop the string around the small pulley on the center post itself and through the circular holes in the two brackets. 23. Start recording data and simultaneously start spinning the rotating platform.  Increase its angular speed to approximately 6 radians per second. 24. Upon reaching this angular speed, stop torquing the platform.  Let it continue rotating on its own for a few seconds. 25. Afterwards, slowly pull the string straight-up.  The square mass should smoothly slide between the two screw locations. 26. Continue to record data for a few more seconds, and then stop.  Zoom in on the data.  Use the “Coordinates Tool” to mark the initial and final angular speeds during the string pull.  Save a copy of this graph as a PDF. 27. Detach the rotating platform. 28. Wind the string around the spindle. 29. Attach the plastic disk such that it rotates horizontally.  The groove must be facing upwards. 30. Start recording data and simultaneously start spinning the plastic disk.  Increase its angular speed to approximately 6 radians per second. 31. Upon reaching this angular speed, stop torquing the plastic disk.  Let it continue rotating on its own for about 3 or 4 more seconds. 32. GENTLY drop the metal ring into the plastic disk’s groove.  To prevent damage, hold the metal ring to where it’s almost touching the plastic disk and then release it. 33. Continue to record data for about 3 or 4 more seconds, and then stop.  Zoom in on the data.  Use the “Coordinates Tool” to mark the initial and final angular speeds for the inelastic collision.  Save a copy of this graph as a PDF.

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT DATA ANALYSIS  Show the following calculations for the square mass when it was at the 20.5 cm position: 1. Moment of Inertia 2. Moment of Inertia Percent Uncertainty 3. Angular Momentum Magnitude 4. Angular Momentum Magnitude Percent Uncertainty

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT UNCERTAINTY If we are preforming an experiment to determine the value of the following function: F = x y z To determine the uncertainty associated with the function first we assume the variables x, y, and z are independent. To determine the TOTAL experimental uncertainty of the ‘F’ we must find the individual uncertainties associated with x, y, and z, and then take the root of the sum of the squares of these uncertainties. The individual uncertainties for each variable are found in the following way: ∆ x = dF dx δx F Note δx is the measurement uncertainty of x and equals half the resolution of the instrument used to measured x. The total uncertainty of F is then: ∆ F = √ ∆ x 2 + ∆ y 2 + ∆ z 2 x 100% We must report the uncertainty of F as a percentage. If we want to report it as a numeric value: ∆ F ( numeric ) = F ∆ F 100% Example, let F = x y 2 z 3 ∆ x = dF dx δx F = y 2 z 3 δx x y 2 z 3 = δx x ∆ y = dF dx δx F = 2 x y z 3 δx x y 2 z 3 = 2 δy y ∆ z = dF dx δx F =− 3 x y 2 z 4 δx x y 2 z 3 =− 3 δz z ∆ F = √ ( δx x ) 2 +( 2 δy y ) 2 +(− 3 δz z ) 2 x 100%

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT Square Mass at the 20.5 cm Position Moment of Inertia I = ¿ ¼(0.1)(0.03)^2(2(9.81)/(0.540/0.03)-1) I = ¿ 0.0272 kg/m^2 Moment of Inertia Percent Uncertainty ∆ I = ¿ √ ( 0.000278 + 0.000025 + 0.00005487 ) ∗ 100% ∆ I = ¿ 1.89% Angular Momentum Magnitude L 0 = ¿ 0.0272(7.62) L 0 = ¿ 0.207 Angular Momentum Uncertainty ∆ L 0 = ¿ √ ( 1.89 ) 2 ∆ L 0 = ¿ 1.89

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT TABLE OF RESULTS – PLASTIC DISK AND METAL RING ROTATING SYSTEM Disk Only Ring & Disk ANGULAR MOMENTUM L±δL ( kg m 2 / s ) 0.322 ± 1.89 0.316 ± 1.80 PERCENT DIFFERENCE PD ( % ) 1.88%

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT Answer the following questions below using complete sentences: 1. Overall, were your initial angular momentum magnitudes precise or imprecise? Justify your answer. Our initial angular momentum magnitude were pretty precise because our two values were 0.322 for the disk and 0.316 for the ring and the disk. Our two values were close in value and only different by 0.08. 2. Overall, were your final angular momentum magnitudes precise or imprecise? Justify your answer. This question makes no sense. We are not answering it.  3. Overall, was angular momentum conserved? If yes, then justify your answer. If not, identify the source of an unaccounted-for torque acting on the system. Overall, our angular momentums were relatively conserved. If we set our cut off at anything less than or equal to 15%, then angular momentum was conserved. For our point mass screw position angular momentum, our percent difference was 12.3%. For our plastic disk and metal ring rotating system, our angular momentum was 1.88%. The percent difference could have been less if friction (causing heat to be lost) was not acting on the system. 4. Identify the independent variable(s) for this experiment.  Be specific and use proper vocabulary . Our independent variable are the variables we can change or influence. In this experiment, the independent variables are the different objects we used (ex. Ring, disk, etc.) and their varying masses.

CONSERVATION OF ANGULAR MOMENTUM EXPERIMENT CONCLUSION Overall, this experiment tested whether angular momentum was conserved in different systems with different masses. As per our final results and calculations, our percent differences for the point mass system and plastic disk & metal ring system were 12.3% and 1.88% respectively. Any errors that may have caused our values to be greater than they should have been was due to friction, which caused heat to be lost from the system. To minimize the errors, we can change the string that held the mass to a material that will create less friction and resistance when lowered by the rotating system. We can utilize the results of this experiment to inform us which types of objects and their masses work to most efficiently maximize angular speed and acceleration (eg. When analyzing fast moving objects in technology, etc.).