The Damped Pendulum. Physics 211 Lab 3 3/18/2016

Transcription

1 PHYS11 Lab 3 Physics 11 Lab 3 3/18/16 Objective The objective of this lab is to record the angular position of the pendulum vs. time with and without damping. The data is then analyzed and compared to the prediction above using fitting routines in either Capstone or Origin. The relationship between angular frequency ω and the decay constant γ will be evaluated. Introduction and Background We consider the motion of an oscillatory tem influenced by damping forces. A physical pendulum consisting of a netic disk placed at some point along a solid bar is allowed to swing such that at any given time, t, the bar makes an angle, θ, with the vertical (as shown). Let s describe the motion of this pendulum. The net torque about the pivot is m gl sin m gl sin bar bar where Lbar is the distance from axis of rotation (pivot) to the center of mass of the bar and L to the center of mass of the net. mbarlbar ml gsin Mhgsin L bar Pivot θ CM bar CM Mg h L net solid bar where h is the distance from the pivot to the center of mass of the tem (bar and net) and the mass of the tem is M m mbar. Newton s second law gives So, I I d d d Mgh I Mghsin I sin UDel Physics 1 of 8 Spring 16

2 PHYS11 Lab 3 If the angle moved through is small then we can take sin ~, d Mgh (1.1) I Which is similar to what we get for a simple pendulum, so our solution must be t cos t d sin t d cos t Where we have defined the amplitude to be. Such that eqn (1.1) becomes, Mgh. t cos t cos I So we have: Mgh I Mgh I We introduce the parameter frequency for this undamped tem Mgh I To distinguish it from a damped pendulum tem So, the period for an undamped pendulum is, T. Mgh I It is interesting to note in the special case where we are considering a point like object attached to a massless bar, the moment of inertia can be written as I mh T I Mh h Mgh Mgh g which is the period for a simple pendulum. UDel Physics of 8 Spring 16

3 PHYS11 Lab 3 If the pendulum is placed such that the net moves near an electrically conducting plate (as shown to the right), eddy currents will be created in the surface of the plate such that an opposing netic field arises in the plate. The disk and the center of the eddy currents will attract each other Since they are in motion relative to each other and the plate is fixed, the net effect is to slow the pendulum arm (relative to the plate). The resulting damping force is expected to be proportional to the velocity of the net. This can be derived from Maxwell s equations, but we will leave that to an electronetism class. Since the damping force always opposes the motion of the pendulum, the sign of the new term in the equation of motion must be negative. Fd bv where b is the damping constant and v is the linear speed of the net. Since the motion is circular, the direction of this force is perpendicular to the position vector of the net (measured from the axis of rotation to the point where the force is applied). The resulting damping torque is then written as d Lbv L is the distance from the pivot point to the center of mass of the net 1. We can now write Newton s nd law for rotational motion with netic damping: d Mghsin d Mghsin Lbv I Remember the velocity of the net is always perpendicular to the line connecting the pivot to the net. As such, the speed of the net can be replaced with angular coordinates, d v L, Once again, for small angles we approximatesin. The nd order differential equation becomes the following: 1 While it is true that there are some effects for this being a circular, not linear motion, the effects are small if r L. UDel Physics 3 of 8 Spring 16

4 PHYS11 Lab 3 d d d bl d Mgh Mgh bl I I I. Introducing the constant parameters now bl Mgh,, the differential equation is I Isy s d d Using standard techniques, we determine the solution to be exp cos t t (1.) where and is the phase of the cosine at t. Recall the form of these constants depends on measureable quantities: m bar, m,h, and L. The constant b is the only true unknown of the tem. Note that we do not need to know these quantities for this lab as we are concerned with how the damping affects the frequency and amplitude. UDel Physics of 8 Spring 16

5 PHYS11 Lab 3 Method The setup includes a Pasco computer interface, rotary motion sensor, a pendulum bar with nets on either side acting as the bob, and an aluminum plate. These pictures should give you an idea for the setup design. Verify support structure is sturdy to minimize possible errors. Double check the clamps and posts in your setup. Securely fasten the pendulum to the rotary motion sensor using a thumb screw. The position of the pair of nets should be adjusted for selecting a desired moment of inertia. UDel Physics 5 of 8 Spring 16

6 PHYS11 Lab 3 I. Set-up of equipment. Procedure 1. Mount the rotational sensor onto the higher horizontal rod which has been set on your table.. Connect the yellow plug of the rotational sensor to digital channel 1 of the Pasco interface, and the black plug to channel. 3. Attach the steel bar (the physical pendulum) to the rotation axis of the sensor using the screw.. Mount the aluminum plate onto the lower horizontal rod. Leave it close to the pendulum for now. 5. Add the pair of nets to the pendulum bar at a position that allows them to move across the aluminum plate. 6. Adjust the aluminum plate to be away from the pendulum. II. Set-up of computer and interface. 1. Turn on the Pasco 75 interface first. Verify that the indicator light is on.. Turn on the computer and login (if necessary). 3. Startup Pasco Capstone on the computer. In hardware setup, left click channel 1 and choose Rotational Motion Sensor. Click the properties button just underneath the setup window and set the resolution to High. Close the properties and setup windows.. Drag a Graph onto the empty page from the right column. 5. Click on <Select Measurement> on the vertical axis and select Angle (rad) option in menu. III. Recording the angle-time data of the physical pendulum without damping. (It is important that your physical pendulum does not change throughout your recordings. If the nets move, it will change the moment of inertia ) 1. If present, remove the aluminum sheet near the pendulum/net such that there will be no damping.. Choose an appropriate recording frequency (e.g., 5 Hz) on the mid-bottom of the software window. 3. Use the Recording Condition button near the bottom to set a Start Condition to be Time Based, set the time delay to be 3-5 seconds.. While the pendulum is hanging vertically downward at rest, click Record (the red circle) on the lower-left corner of the window. UDel Physics 6 of 8 Spring 16

7 PHYS11 Lab 3 5. Pull back the physical pendulum to a small angle (~ degrees) away from its equilibrium position and release it just before the recording timer reaches zero () to ensure oscillation at the start of the recording data run. 6. Wait until you have at least 5 seconds (possibly more much more) of recorded time and click Stop (the red square). 7. If the data run was successful, record the Run ## and specifics to identify it (i.e. Trial run #1- no netic damping. Trial run #3 do not use, etc.) 8. Repeat step III.3-6 for two more trials. 9. In Capstone or Origin you will include these three data sets in one graph, containing the data as points and fit one of these sets with the appropriate damped sinusoid equation. (This graph will be part of your lab report.) 1. Record the value of the angular frequency (This may require conversion from the fit notation to our own.) [ /w]. IV. Recording the angle-time data of the physical pendulum with damping caused by the motion of a netic pole near an aluminum plate. (It is important that your physical pendulum does not change throughout your recordings. If the nets move, it will change the moment of inertia ) 1. Place the aluminum plate close to the pendulum and secure. a. Adjust the height and horizontal direction of the rods such that the net is roughly in the middle of the aluminum plate. b. Adjust the plate and pendulum such that net stays near the center of the arc of the plate during its swing. c. Visually inspect the pendulum-plate setup to ensure a parallel swing (the pendulum should not swing towards or away from the plate appreciably throughout the oscillation.. Do steps III.3 5. (the recording condition should still be a delayed). 3. If the data run was successful, record the Run ## and specifics to identify it (i.e. Trial run #- netic damping approx. 1 inch away. Trial run #5 do not use, etc.). Repeat step IV.1 3 for two more trials. 5. In Capstone or Origin you will include these three data sets in one graph, containing the data as points and fit one of these sets with the appropriate damped sinusoid equation. (This graph will be part of your lab report.) 6. Record the value of damping parameter [ 1/t ] and the angular frequency [ ]. (These will require conversion from the fit notation to our own.) /w 7. Adjust the distance between the net and aluminum plate at least once more (twice if time permits). Repeat steps -7. UDel Physics 7 of 8 Spring 16

8 PHYS11 Lab 3 Expected tables and graphs in the lab report: Similarly, for Table 3 and Table (time permitting). Analysis Analyze the data and evaluate the relation between, and γ for the two (three, if time permits) damping conditions. Based on Newton s second law we derived the following where and are measured for the two damping which can be rewritten as conditions. While, here, is the implied damped-free oscillation from these measurements. It can be shown the resulting error on the implied, using error propagation of the fitted errors and, is the following: Make a plot of the quantity with error bars. The value of which is on the y-axis and the trial number on the x-axis for the data sets of different damping conditions. The plot should include a horizontal line indicating the direct measurement of from Table 1 (part III of the procedures). UDel Physics 8 of 8 Spring 16