Linn–Benton Community College



Ph 212 Lab2 – Rotational Dynamics and Conservation of Angular MomentumObjective: In this lab you will determine the height for a roll of toilet paper falling freely and the height for when it unrolls to the floor, keeping the time for both fall the same. You also analyze the conservation of angular momentum in a gentle collision.Introduction:To analyze the rotational motion of a rigid body, you apply the newton’s 2nd law to the motion: F=ma and τ=Iα about a fixed point.Where τ is the torque(s) acting on the object. τ=r×F (cross product)r is the vector position from a fixed point on the object (axis of rotation) to the point of the application of the force F. Magnitude of the torque can be found by: t = r F sin q where q is the angle between r and the line of action of the force.Angular momentum of a rotating rigid body is L=Iω where ω is the angular velocity vector. the direction of vector ω is determined by right hand rule (Curl you right hand fingers in direction of plane of rotation and your thumb is pointing in direction of angular velocity).In the absence of external torques the total angular momentum of a rotating system is conserved. That isLi=LfAn example of conservation of angular momentum is when a figure skater is spinning with her arms stretched out and the tucks her arms and as a result spins faster. That is because there is nearly no external torque acting on her, therefor her angular momentum must remain the same throughout her rotation. When she brings her arms in her moment of inertia decreases and hence she must spin faster in order to keep angular momentum unchanged.Materials: Two identical rolls of toilet papers, meter stick (ruler), and stopwatch (timer)Part I – Dynamics of a rotating roll of toilet paperIn this part you and your lab group perform the experiment first and them make the predictions.Obtain two identical rolls of toilet papers (you can use a single roll as well). Hold the first sheet of paper in your hand and from a fix height gently let the roll drop by unrolling itself to the floor. It might be better to press the first sheet on the edge of a table with the roll sliding off the edge vertically to the floor. Measure the time of fall to the floor as accurately as you can. Repeat the experiment 2-3 times and average the time. Record the times and the height of fall (call it h). Now you want to let the same roll fall freely from a height such that it would take exactly same amount of time as the unrolling one. Ensure that the roll is wrapped tight, falls straight down like a rock (no spin) and from rest. Adjust the height until the fall time is the same as part 2. Once you find the proper height repeat the experiment for 2-3 times to obtain an average height. You may do this near a wall for easier height observation. Record the height of the fall (call it H). Measure the inner radius r, outer radius R? and mass of the roll as accurately as you can. Predictions for part I: Show your work clearly and completely to determine the theoretical ratio of H to h in terms of the parameters involved. That is find H/h in terms of R (outer radius), r (the inner radius) and M (mass). Do NOT insert any of your measurement values from your experiment.Hint 1: Use free fall kinematics equation(s) to determine the tdrop from height H. Hint 2: The unrolling motion is also with constant acceleration. Therefore use kinematics equation(s) to find the tunroll from height h with acceleration a. Hint 3: Refer to problem 1 of in-class activity we did in last class to find a and a. Note: Hint 4: The roll of toilet paper is considered a hollow cylinder (not a disk and not a ring) and is rotating about its edge (not its center).Part II - Conservation of angular momentum:Do the following before performing the experiment:Two identical solid disk each of mass Mdisk and radius R are on top of each other and are rotating with angular speed of ?i rad/sec in clockwise direction. A book of mass Mbook with length a and width b is very gently dropped on disks; the book and the disk couple and combination rotates at speed ?f rad/sec in clockwise direction.Is the collision elastic or inelastic? Is angular momentum of the system conserved? How about kinetic energy?Use conservation of angular momentum to fine an expression for ?f. in terms of Mdisk, Mbook, R, a, b, ?i and ?f.504634525082500Calculate the KEi and KEf in terms of Mdisk, Mbook, R, l, w, ?i and ?f.Set up For Part IIA rotary sensor with two disks mounted on is rotating and labQUEST is graphing the angular velocity with time. The book is dropped gently on disks to make. The disks are rotated clockwise by hand and shortly after that I drop the book on it with center on center.Data and data reductions:For Part I, record your measurements for Heights time t, H, h, mass M of the roll, and its inner and outer radii. Properly organize them label each item.For Part II, measure and record the geometric values for the disk and book, a, b, Mbook, mass and radius of the disk. Also the initial and final values of the angular velocities. Organize the data and label them properly.Data analysis:Part I - Dynamics of a rotating roll of toilet paperFor this part calculate the following: Write the formula for moment Inertia of a hollow cylinder rotating about an axis along it length. Show all steps for using parallel axis theorem.Insert your data for mass and radii of toilet paper roll and calculate I.In the equation you derived for H/h in prediction part, insert your measured values of R and r and calculate the H/h pare your calculated value to the measured ratio of your heights H and h. Calculate percent error considering your measured heights as actual values. Discuss sources of errorsPart II – Conservation of Angular MomentumUsing your data for this part, calculate numerical values for ?Li , ?Lf , ?KEi , ?KEi , and ?f.. Verify if total angular momentum and kinetic energy of the system is conserved. To check the quality of your experiment compare your experimental value of ?f using your predicted formula in step 4 of part II to that of measured (from graph).Again calculate %error and discuss sources of error.You will now set up the experiment of C in your prediction 1 (prediction 1C), with known d, m and F. Once the rod is in equilibrium, record all information. Calculate force F using the rubber band force equation.Calculate mass of the rod using equilibrium of the rod.Calculating mass of the rod using rotation of the rod in vertical plane:Now set up the experiment shown in the figure above, where the rod is released from horizontal position. The rotation of the rod is measured by smart pulley with data displayed as angular position and angular velocity of the rod as a function of time, q(t) and w(t).Use excel again to make scatter plot of q(t) and w(t). Fit a power function to data points of q(t). Record the equation. Use the regression equation q(t) and find and expression for w(t) (recall ω=dθdt.) and from that determine the angular speed of the rod when it reaches the vertical position (at C, see the figure above).Read the maximum angular speed from your w(t) graph. How does this compare to your calculation in part C?Use your maximum angular speed from part D and the formula you obtained in your prediction2 part C to calculate the moment of inertia of the rod.Use the moment of inertia for a rod rotating at one end to find an expression for mass of the rod. Comparing them allYou have determined mass of the rod in three different ways. Let’s take the mass you found in part 1C in procedure section to be actual mass of the pare the mass you have calculated in part 2C and part 3F above the actual mass. Calculate % error for each.Lab ReportTitle of the lab, Your name and the names of lab partners.A summary (abstract) of the report (goal or objective)A statement of your group’s experimental goals.Show all predictions and derivation of formulas.A description, including a sketch, of the experimental situation. Include sufficient detail for other students to reproduce your experimental work. Getting the apparatus to work well can be a significant task, so outline how you did it.Original data, which may be in table and or graphical form. Include explanations for excluding any data in your analysis.Analysis leading to your conclusions. Include what would be the consequences of conservation of energy and conservation of angular momentum, and whether either one, none, or both were ruled out by your experiment.Summary of your conclusions. ................
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