Rotational Motion & Moment of Inertia

Transcription

1 Rotational Motion & Moment of nertia Physics 161 ntroduction n this experiment we will study motion of objects is a circular path as well as the effect of a constant torque on a symmetrical body. n Part of this lab, you will use World n Motion to examine two-dimensional circular motion of an object. You will investigate the relationships among acceleration, velocity and position of the object. n Part, you will determine the angular acceleration of a disk when a constant torque is applied to the disk. From this you will measure its moment of inertia, which will be compared with a theoretical value. n Part, you will observe the relationship between torque, moment of inertia and angular acceleration for a rotating rod with two masses on either end. You will vary the mass connected to the rod by two pulleys. You will also change the moment of inertia of the rod system by changing the distance of the masses from the center of mass of the rod. Reference Young and Freedman, University Physics, 1th Edition: Chapter 3, sect. 4; Chapter 9, sect.1-4 Theory Part f an object is moving around a circle at any given instant the distance of the object to the origin must be constant. Therefore r x y (1) Where x and y are the coordinates from the center of the object and r is the radius of the circular path. Uniform circular motion can be described as the motion of an object in a circle at a constant speed. The speed at any given instant can also be calculated as: v () v x v y Where v x is the velocity in x direction, v y is the velocity in y direction and v is the tangential velocity of the object. There is another quantity that is called angular velocity and is given by: v / r (3) The acceleration of the object is called centripetal acceleration which points towards the center of the circle. This acceleration can be calculated using the following equation: v a r (4) r You will use these equations to analyze the video of an object moving around a circle.

2 Part - The moment of inertia is a measure of the distribution of mass in a body and indicates how difficult it is to angularly accelerate that body. For both parts of the experiment, a falling mass will accelerate an object rotating in the horizontal plane. n Part, the object is a disk. n Part, you will find the moment of inertia of a rod with two masses attached to it. The basic equation for rotational motion is: (5) where is the rotational acceleration in units of rad/s, is the applied torque in N m, and finally is the moment of inertia or rotational inertia in units of kg m. For a uniform disk pivoted about the center of mass, the theoretical moment of inertia is 1 MR disk (6) where M is the mass of the disk and R is the radius of the disk. n Part, we measure α and use its value to calculate, which we will compare with the theoretical value of. n Part, the moment of inertia is the sum of the moments of inertia of the two masses and the rod. For the masses that slip onto the rod, we will assume point masses. Thus, the moment of inertia for one of the two masses is: mass mr (7) where r is the distance of the center of mass from the axis of rotation located at the center of the rod. Because the masses can be moved along the rod, r will be adjusted to change their moment of inertia. The moment of the inertia of the rod with mass M and a length L is: 1 M L 1 rod rod (8) The moment of inertia for a rod with length L and two masses on each end at a distance r is simply the sum of the components as defined by Equations 7 and 8: 1 1 mr M L (9) rod The first term is multiplied by two because there are two point masses. Given the moment of inertia of the entire system,, and the net torque, applied by the mass M hanging from the pulley, the angular acceleration,, can be found using equation 5 (just as F = ma is used for translational motion).

3 The torque, (the cross product between lever arm and applied force), causes rotation with an angular acceleration,, in an object with a moment of inertia,. The rotating object will be attached to a pulley placed at the center of the rotary motion sensor which has a string wrapped around it. The string has a mass tied to one end and is laid over an additional pulley which allows the falling motion of the mass to be converted into a torque on the rotary motion sensor. For the purposes of this lab, the applied torque will be due to a mass accelerated by gravity acting on the pulley of the rotary motion sensor. We will use the expression: MgR MR (10) where M is the mass hanging on the pulley and R is the radius of the Rotary Motion pulley. The full derivation of the above equation is given in the Appendix. The experimental moment of inertia can be found using the following equation: experiment / (11) n Part, by adjusting the masses on the rod, we can observe how an increased moment of inertia (where either the mass is distributed farther from the center of mass or the total mass is increased) will result in a decreased angular acceleration for the same torque. t is the same type of relationship as the one you observed for Newton's Second Law. Procedure Part : Object moving around a circle. The video to be analyzed is called UCM.avi and can be found in World in Motion 4.0, Video Analysis under z drive, videos. The video shows a turntable with two dimes placed on top of it, and was taken with a camera on a tripod pointing down at the turn table. Both x and y are real world horizontal. As the turntable rotates, the dimes will travel in circular paths. 1. First you need to establish a new origin. To do this, select Video>New Origin and follow the instructions given. The point you mark should be the tip of the shaft that comes out of the middle of the turn table.. Hit Play to return to the beginning of the motion. Place your cursor over the center of mass of the dime closer to the origin in each frame. Mark the center of the dime in each frame by clicking the mouse, then hit Step to proceed to the next frame. Advance the video in this fashion, stepping and clicking for a data point until the dime makes one complete rotation (note that the video covers more than one rotation). Click on Dataset 1 and notice that the video will return to its original frame. Repeat the above steps for the second dime and click on Dataset at the end. Click on Save and choose Copy Data to Clipboard.

4 3. Before you leave the World in Motion program, estimate the uncertainty in the x and y components of the position of the dimes, by measuring the two extremes of one coin along the x axis. Assume that the uncertainties in the x and y components are equal. 4. Open Excel and Paste the data into the spreadsheet. 5. Your spreadsheet should have separate columns for t, x 1, y 1, x, and y. The time in seconds is in the first column. Subscript 1 and refer to the first and second dime respectively. 6. n Excel, calculate the radius of the circular path for the first dime using equation 1. Find r by taking the average of all the rows containing the r values and σ r by taking the standard deviation of the r column. Repeat this procedure for the second dime. 7. Now calculate the x and y components of the velocity for both dimes just as you did in the Projectile Motion lab. Calculate v 1 and v for any given time using equation. s v 1 constant? s v constant? s v 1 equal to v? Determine the uncertainty of the velocities v 1 and v by simply determining the standard deviation of them. Produce graphs of x- position, y-position and r vs. time (all in one graph) and v x, v y, and v vs. time (all in one graph) for the first dime. 8. Calculate the angular velocity ω 1 for the first dime and ω for the second dime using equation 3. s ω 1 equal to ω? Calculate the σ ω1 and σ ω using error propagation. How many σs are these two angular velocities apart? nclude the calculation in your sample calculation. 9. Calculate the centripetal acceleration using equation 4 for both dimes. Using error propagation, find the uncertainty for both accelerations. nclude the calculation in your sample calculation. Part : Moment of nertia for a Disk The experiment uses a mass hanging on a string over a pulley to exert a constant torque on the system resulting in a constant angular acceleration on the system. 1. Set up the apparatus as shown in Figure 1 (but without the ring on top), making sure that the rotary motions sensor is connected to the interface box.. Make sure that the pulley is set up to give positive values for angular position. This means that the rotary motion sensor will turn in a counterclockwise direction as the mass on the pulley drops. (The motion sensor may have an indication of which direction is positive taped to it.)

5 Figure 1 3. Adjust the measurement rate to 0 Hz on the Rotary Motion Sensor. 4. Choose angular velocity,, in radians/s under the Measurement tab only. 5. Measure and record the radius of the rotary motion sensor pulley around which the string is wound. Hang a mass of 50 grams from the pulley, press Start, and release it. 6. Press Stop just before the hanging mass reaches the ground. 7. Create a graph of angular velocity vs. time. 8. Using a linear trendline fit determine the angular acceleration. 9. Make the following table (specifying the appropriate units). Find the experimental moment of inertia for the disk (use Equation 10 to obtain ) and compare it to its theoretical value. R disk (m) R pulley (m) m hang (kg) m disk (kg) experiment / theory 0.5mdisk rdisk

6 Part : Moment of nertia of a Rod with Two Masses Attached The experiment uses a mass hanging on a string over a pulley to exert a constant torque on the system resulting in a constant angular acceleration on the system. 1. Set up the apparatus as shown in Figure, making sure to connect the rotary motions sensor to the interface box. Figure. Make sure that the pulley is set up to give positive values for angular position. This means that the rotary motion sensor will turn in a counterclockwise direction as the mass on the pulley drops. 3. Adjust the measurement rate to 0 Hz on the Rotary Motion Sensor. 4. Measure angular position,, in radians, angular velocity,, in radians/s and angular acceleration,, in radians/s². Create graphs of these quantities vs. time. 5. Measure and record the radius of the rotary motion sensor pulley around which the string is wound. Hang a mass of 50 grams from the pulley, press Start, and release it. 6. Press Stop just before the hanging mass reaches the ground. You will repeat this for three trials: 1) without attached masses, ) with masses located at the extremes of the rod (r 0 is the maximum length of the arm), and 3) with masses located at 0.8 r 0. Create a table with the following format:

7 Run r(m) r pulley (m) M hang (kg) masses mr theory mr experiment / rod r r r r r. 8r MPORTANT: The difference between M hang and m: the first is the hanging mass on the string which accelerates the rod; the second represents the point masses on each end of the rod. Another important distinction: r is the distance between the center of mass of the point mass and the axis of rotation (the center of the rod), r pulley is the radius of the pulley on the rotary motion sensor around which the string is wrapped. For Your Lab Report: For Part, include a sample calculation of r, v, ω and a with uncertainties. n your report, explain why acceleration is not zero, even though the speed was constant. Was v 1 =v? ω 1 = ω? Within how many sigmas? For Part, include a sample calculation of the torques MgR MR and moments of nertia theoretical and experimental. For Part, and compare theoretical to experimental using the % difference. Appendix The calculation of the torque applied to the rotary motion sensor is as follows. The rotary motion sensor has a radius, R, which is the distance from the axis of rotation at which the force, T, acts. T is the tension in the string attached to the pulley and is due in part to the weight on the string, Mg, where M is the mass of the hanging mass and g is the acceleration due to gravity. Figure 3 below shows a rough sketch of the set up. Figure 3

8 The application of Newton's Second Law to the hanging mass results: F y F y Ma Mg T Ma The downward direction is taken to be positive. We can substitute for the linear acceleration, a: a R Where and R are the angular acceleration and the radius of the rotary motion sensor, respectively. Continuing to solve the equation gives us: Mg T MR T Mg MR T is the tension in the string. t is a force, not a torque. To find the torque acting on the pulley, we must multiply by the distance from the axis of rotation at which the force acts: RT MgR MR For the set up of this experiment, take the following example where M=0.06 kg, R m and. 58 rad/sec². Substituting these values results in: MgR Nm MR From this example, we see that MgR >> MR so we can approximate MgR. 3 Nm