General Physics I Lab. M7 Conservation of Angular Momentum

Transcription

1 Purpose In this experiment, you will investigate the conservation law of angular momentum in a collision between a ball falling along an inclined ramp and a ball catcher fixed on a freely rotating disk. Equipment and components Science Workshop 750 interface, ramp, rotational apparatus with accessories, ball catcher, spirit level, electronic balance, calipers, plumb bob, meter ruler, scotch tape, carbon paper and white paper, G-clamp. Background The angular momentum L of a particle with mass m and velocity v is defined as L = R (mv) (1) where R is a vector from the origin to the particle. The actual expression for the magnitude of L depends on the choice of coordinate system. In this experiment, the object in question is a steel ball falling downward along an inclined ramp. It is to be captured by a ball catcher sitting on a freely rotating disk, whose moment of inertia, I disk, has already been determined in Experiment M6. To compare the angular momentum change before and after the capture, it is most convenient to choose the origin on the axis of rotation of the rotating disk and in the same plane as the velocity vector of the steel ball. This is shown in Fig. 1 below. Figure 1 Coordinate system used to determine the angular momentum In this case, the magnitude of the angular momentum L of the steel ball about the axis of rotation just before the capture is given by L before capture = mvr (2) After the steel ball is captured by the ball catcher, the angular momentum of the whole system is given by L after capture = (I disk with ball catcher + I ball ) ω (3) where ω is the angular velocity of the rotating system, I disk with ball catcher and I ball are the moment of inertia of the disk with ball catcher and the ball respectively. M7-1/8

2 Using the parallel axis theorem, the moment of inertia of the ball about the axis of rotation is given by I ball = (2/5) ma 2 + mr 2 (4) The values of v, m, a, ω and distance r can all be measured in the experiment. Procedure Part I: Angular momentum before capture In this part of experiment, you need to measure the initial linear velocity v of the ball by falling it from the ramp to the ground and then measuring the horizontal travelling distance, x = v t, of the ball before it hits the ground. The time t of flight for the ball is related to the vertical distance y travelled by y = ½gt 2. From the measured values of x and y, you can find the value of v. To determine the angular momentum L, you also need to measure the mass and radius of the steel ball and its distance from the axis of rotation. 1. Measure and record the mass, m and radius, a of the steel ball in Table Fix the ramp at the edge of a table using a G-clamp as shown in Fig. 2. Clear an area at least one meter from the table for the landing of the steel ball. Figure 2 Measurement of the initial velocity v of the falling steel ball 3. Choose and mark the initial height of the ball on the inclined ramp with a label. 4. Place the ball at the initial height marked on the ramp and release it to locate where the ball first hits the floor. 5. Tape a piece of white paper at the spot where the ball hits the floor. Tape a piece of carbon paper (carbon-side DOWN) above the white paper. 6. Release the ball from the ramp at the same marked height five times. 7. Measure and record the vertical distance y, from the ball as it leaves the ramp to the floor (as shown in Fig. 2) in Table 1. Calculate the time of flight t using the measured value and record it in Table 1. M7-2/8

3 8. Use a plumb bob to find the spot on the floor that is directly below the release point of the ball. 9. Measure the horizontal distances of the five dots (as shown in Fig. 2) and record these distances in Table 1. x 10. Calculate the initial velocity v = of the ball for each trial and record in Table 1. t 11. Find the average value of the initial velocity, v average and record in Table 1. Part II: Angular momentum after capture In this part of experiment, a ball catcher mounted on a rotating disk is used to catch the falling ball from the inclined ramp. The smart pulley measures the motion of the rotating disk after the ball is captured. A computer program is used to record and display the initial angular velocity ω of the rotating system. Equipment setup 1. The rotating system is assembled with the rotating disk and a ball catcher, as shown in Fig. 3. Figure 3 Setup of the rotating system 2. Place a spirit level on top of the base, and then adjust the leveling supports until the top of the base is leveled. 3. Adjust the height of the ramp so that the released ball can hit right into the catcher and the ball is caught with dropping. 4. Position the rotating platform and the ramp such that the edge of the catcher is perpendicular to the line-of-flight of the ball and the edge of the ramp as close as possible to the catcher. The layout of the entire experimental setup right before data recording is shown in Fig. 4. M7-3/8

4 (a) Top view (b) Side view Figure 4 Layout of the entire experimental setup: (a) top view and (b) side view Computer setup 1. Connect the smart pulley s stereo phone plug into Digital Channel 1 of the interface. 2. Open the program M7. The program will open with a graph display of Disk angular velocity (rad/s) versus Time (s). Data recording 1. Take the moment of inertia of the rotation disk, I disk = (6.96 ± 0.07) x 10-3 kg m 2 and the moment of inertia of the ball catcher, I ball catcher = (0.95 ± 0.10) x 10-3 kg m 2. Calculate and record the moment of inertia of the rotation disk with ball catcher, I disk with ball catcher in Table Place the ball on the marked initial position of the ramp (same as that in Part I). 3. Click the Start button to begin data recording. Release the ball from rest on the ramp. 4. Remove the ramp once the ball is caught. Warning: do not let the catcher collide with the ramp. 5. Click the Stop button to end data recording after about 10 data points have been collected or after the rotating disk has stopped. 6. Record the radial distance r of the steel ball away from the rotation axis in Table 2. And calculate the moment of inertia I of the system, record the value in Table Run #1 will appear in the Data list. Record the maximum value of the measured angular velocity in Table Print the angular-velocity-time graph and paste it in the lab report. 9. Repeat the steps 2 7 at least one more time. 10. Repeat the above measurements with 3 different values of r, while keeping the initial height of the ball unchanged. Data analysis (Record in Table 2) 1. Calculate the average of the maximum angular velocity for each value of r. 2. Calculate the angular momentum L before capture given by Eq. (2) for different values of r. 3. Calculate the angular momentum L after capture given by Eq. (3) for different values of r. M7-4/8

5 Name Date Lab session (Day & time) Lab partner Lab Report A. Answer the following questions BEFORE the lab session (6 pts each) 1. A competitive diver leaves the diving board and falls toward the water with her body straight and rotating slowly. She pulls her arms and legs into a tight tuck position. Her angular speed (a) increases, (b) decreases, (c) stays the same, or (d) is impossible to determine. Why? 2. In calculating the angular momentum L before capture given by Eq. (2), does the friction between the falling steel ball and the inclined ramp affect your results? 3. What is the purpose of leveling the base of the rotating system? M7-5/8

6 Name LA B. Results and data analysis (61 pts) Part I: Angular momentum before capture (30 pts) Table 1 Mass of the steel ball, m = Radius of the steel ball, a = Vertical distance, y = Time of flight, t = Horizontal distance, x Initial velocity, v Initial velocity, v average = Part II: Angular momentum after capture (31 pts) Table 2 Moment of inertia of the rotation disk with ball catcher, I disk with ball catcher = Data Calculations Trials Radial distance, r Maximum angular velocity Average value of the maximum angular velocity Angular momentum, L before capture Angular momentum, L after capture M7-6/8

7 Paste the angular-velocity-time graph here C. Answer the following questions after the experiment (7 pts each) TA signature: 4. In part II of experiment, why is the maximum value of the measured angular velocity used to calculate the angular momentum L after capture given by Eq. (3)? M7-7/8

8 5. Calculate the Percent difference between the angular momentum of the system before and after the capture. Discuss the main sources of errors. 6. Calculate the kinetic energy of the system just before and after the capture (using one set of data in Table 2). Is the energy conserved? Suggest a situation in linear collision which is analogous to the capture in this experiment. M7-8/8