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PHYSICS LAB MANUAL FUNDAMENTALS OF PHYSICS SERIES
(PHYS 101-102-201)
2008 Edition
Roberto Ramos, Joseph Trout and Som Tyagi
Department of Physics
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This custom laboratory manual includes materials submitted by the Authors for publication by Pearson Addison-Wesley. The material has not been edited by Pearson Addison-Wesley and the Authors are solely responsible for content. Copyright © 2008 by Roberto Ramos and Somdev Tyagi. All rights reserved. This publication is protected by Copyright and permission should be obtained from the Authors prior to any prohibited reproduction, storage in a retrieval system, or transmission in anyt form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, 1900 E. Lake Ave., Glenview, IL 60025. For information regarding permissions, call (847) 486-2635. Manufactured in the United States of America.
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PHYSICS 101-102-201 Laboratory Manual
Table of Contents Page
1. Introduction . . . . . . . . 4 2. General Primer and Instructions . . . . . 5 3. List of Experiments
3.1 Physics 101 . . . . . . . . 9 Lab-01 Speed of Light . . . . . . 11 Lab-02 One- and Two-dimensional Motion . . . 21 Lab-03 Elastic Forces and Hooke's Law . . . 35 Lab-04 Conservation Laws – Collisions. . . . 49
3.2 Physics 102 . . . . . . . . 63 Lab-01 Coulomb's Law . . . . . . 65 Lab-02 Millikan Oil Drop Experiment . . . . 75 Lab-03 Motion of Charges in E and B Fields . . . 89 Lab-04 Capacitors and RC Circuits . . . . 101
3.3 Physics 201 . . . . . . . . 113 Lab-01 Interference and Diffraction Using Visible Light . 115 Lab-02 Photoelectric Effect . . . . . 129 Lab-03 Bohr's Model and Emission Spectra of Hydrogen and Helium . . . . . . . 139 Lab-04 High Temperature Superconductivity . . 153
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Introduction & Acknowledgements
This original collection of Physics Laboratory Experiments for students in the calculus-
based Fundamentals of Physics stream at Drexel University has been written with the objective of instilling the fundamental discipline of experimental physics. To prepare students for the 21st Century, much effort has been made to create a new physics laboratory curriculum infused with aspects of modern and contemporary physics. For instance, freshman and sophomore level students are exposed at an early stage in their scientific training to experiments involving high-temperature superconductivity, fiber optics and spectroscopy. At the same time, students are exposed to famous, classic experiments such as the Millikan oil drop and Coulomb’s law using a torsion balance.
Another unique feature of these experiments is a “Just for Fun” section which engages students to perform short but thought-provoking hands-on demonstration experiments “just for the fun of it”. Together with a pedagogical Pre-lab Exercise, this section emphasizes conceptual development and critical thinking beyond the meticulous analysis of data in the experiment.
Developing these labs was a monumental effort. Under the now defunct TDEC special engineering curriculum, no pre-existing Physics Labs were taught to Drexel’s large population of engineering students. This changed in 2006 when the physics curriculum was redesigned to have a lab component. Starting from scratch and within a short time frame, we have developed this collection of experiments which has since been rigorously tested at Drexel University and has received enthusiastic and positive feedback from students and faculty. The authors wish to acknowledge the assistance of graduate student Sam Kennerly in helping draft and test some of these experiments and Seth Meiselman for editing the draft. We acknowledge the assistance of Lisa Ferrara and Michael Downes in providing technical support in developing some of these experiments. We also acknowledge Physics Department Head Prof. Michel Vallieres for providing departmental support for this project. Roberto Ramos, Ph.D. November 17, 2008 Som Tyagi, Ph.D. Joseph J. Trout, Ph.D. Department of Physics Drexel University Philadelphia, PA
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General Primer and Instructions 1. Introduction Each lab report is organized roughly as follows:
1. Objective: This provides a brief description of what is expected to be done in the lab. 2. Theory: This provides the theoretical background needed to understand and perform the
experiment. 3. Experimental Details: A discussion of how the experiment will be carried out.
3.1 Apparatus: provides a list of instruments needed to carry out the experiment. 3.2 Experimental Procedure: A step by step description of experimental
procedure to obtain the data. 4. Pre-Lab: A graded exercise to be performed before coming to class, to help introduce
the student to the essential concepts relevant to the experiment. Please submit the completed Pre-Lab Section to your Lab Instructor at the beginning of class.
5. Experimental Data: This represents an accurate recording of data from the experiment. Please submit this completed section to your Lab Instructor at the conclusion of your class.
6. Discussion of Results and Conclusion. These consist of a discussion and analysis of your experimental results aimed at answering the objective of the experiment. This must be submitted together with your data.
7. Just for the Fun of It. A unique feature of this laboratory course, this consists of a quick hands-on demonstration experiment that engages and encourages the student to examine and analyze an unusual but interesting phenomena “just for the fun of it”.
2. Measurement and Error (Uncertainty)
2.1 Precision and Accuracy There are two kinds of numbers that are used in science - numbers that are defined and those that are measured. The essential difference is this: Defined numbers are exact while the measured numbers are always subject to error (uncertainty) or a combination of errors. If you count ten oranges in a fruit bowl or sixteen students in a classroom, then these numbers (ten and sixteen) are exact. But if you measure the weight of the ten oranges then the result would have some inherent uncertainty (error) in it. The errors in a measured number, of course, can never be stated exactly - if that were the case the experimental number would be known exactly by subtracting the error - and are estimated. The experimental errors are introduced primarily due to the fact that no instrument can ever be made infinitely precise. These errors are thus inherent to the measurement process and should not be confused with “human error” or errors introduced in a measurement due to carelessness. Uncertainty in a measurement is a function of the precision of the scale employed. For example, suppose you had to tell time using a clock (A) as shown in the diagram below. You could easily tell that the time is a little bit after 1:55pm (you know it is pm because it is daytime). You could even make a reasonable guess that it’s not quite 1:57pm and it is somewhere between 1:55 and 1:56pm. If you were to read the time from another clock (B), where the dial is marked off in one-minute intervals(only the relevant region has been shown marked in minute intervals), you could say with greater certainty that the time is closer to
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55.5 min. past 1pm. There is less uncertainty in your claim made using clock B than using clock A. Clock B is said to be more precise than clock A. Precision and accuracy are not the same thing. For example, consider clocks A, B, and C. Suppose it is known using an atomic clock or some other standard that the time is 55 min. and 30 sec. past 1pm. Clock A is more accurate than clock C, but clock C is more precise than clock A. Clock B is both precise and accurate. Accuracy is a measure of how close your measurements are to an accepted standard, whereas precision is related to the smallest change that can be detected in a quantity that is being measured.
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A B C
2.2 Significant Figures In experimental physics, significant figures are the digits in a measurement that are known with certainty plus one digit that represents a careful estimate.(Note: this careful estimate is allowed only for analog scales. When the scale is digital, such as a digital timer or counter, such estimates are not meaningful and you quote whatever the digital scale shows). When you quote your results you must keep in mind that you are making your experimental findings public and you are not only stating a number but are also conveying to the reader how well your results can be relied upon. For example, if the length of an object is reported as 13.5 cm, then you are implying that the measurements were made with a scale marked off in cm. ( 0.5 cm is the estimate part) Similarly if the length is reported as 13.45 cm it would be proper to conclude that the scale was marked off in mm. To report the length as 13.45 cm using a cm-scale would be scientifically irresponsible. On a digital scale if the time measurement is quoted as 12.3 sec implies that the digital scale’s least count (the smallest division on the scale) is 0.1 sec. There are some simple rules to remember when dealing with significant figures. Rule 1: When there are no zeros in the stated number, all digits are significant. For example, 1.23 has three significant figures and 4567 has four significant figures. Rule 2 [a]: All zeros between significant figures are significant: 2.034 has four significant figures and 203 has three significant figures.
Rule 2 [b]: Zeros left of the first non-zero digit are not significant: 0.00123 has three significant figures. Confusion about the number of significant figures can be avoided by expressing the number in scientific notation (a number between 1 and 10 multiplied by a power of ten). When expressed in this manner all digits are significant. For example: 1.2 *103 has 2 significant digits, 1.20*103 has three significant digits, etc.
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When a number does not contain a decimal point and ends in zeros, assignment of significant figures can be ambiguous. For example, the number of significant digits in 1760 is not clear. If it is a measured length, then the last digit (0) is not significant. However, if it represents the number of yards in a mile, than all four digits are significant.
3. Error Estimation Remember, you never know exactly the error (uncertainty) in your measurements, but there are several ways to estimate the uncertainty in your measured quantities. We will employ a very simple method to get such an estimate. [a] Basic quantities Suppose you measured the length of an object to be 12.35 cm with a scale marked off in mm. The last digit is an estimate. The length of your object was between 12.3 and 12.4 cm. The least count of the scale (the smallest division on the scale) is 1mm or 0.1cm. The estimated percentage error in the measurement of your reported length is then taken to be (0.1/12.35)*100 = 0.8%. [b] Derived or Compound quantities. When you measure a derived or compound quantity such as kinetic energy, KE = 1/2mv2, one way to estimate the error, called the log error or maximum percentage error is as follows. Suppose the quantity you measure is X =AkBl/Cm, then logX = k logA +l logB – m logC logX = X/X = kA/A +lB/B – mC/C When estimating the maximum fractional error all fractional error are considered additive. Thus, the maximum fractional error is taken to be X/X = kA/A +lB/B + mC/C and the maximum percentage error in the measurement of X is (X/X)*100. Let us illustrate this with an example. Suppose you were to measure the KE of a toy car of mass 204 gm measured with a scale of least count of 1 gm. Further, suppose that the velocity was determined by measuring the time taken by the toy car to travel between two sensors 50.8cm apart. The time was measured to be 10.1 sec. (KE = ½ mv2= ½(204*10-3)[50.8*10-
2/10.1]2 = 2.58*10-5J) Since KE = ½ mv2, logKE = - log2 +log m + 2log v =- log2 +log m + 2log x – 2logt Thus the maximum fractional error isKE/KE =m/m +2(x/x+t/t)= 1/204 +2(0.1/50.8 +0.1/10.8) Or the maximum percentage error is = [1/204 +2(0.1/50.8 +0.1/10.8)]*100 =0.5 + 0.4 + 2 = 2.9 ~ 3% For certain complicated experiments that involve measuring a parameter many times, the estimation of errors may require statistical analysis of experimental data. We have not discussed such analysis here but will return to it as the need arises.
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Physics 101 Lab Manual
Ramos, Trout & Tyagi Experiment Page
Lab-01 Speed of Light . . . . . . 10 Lab-02 One- and Two-dimensional Motion . . 20 Lab-03 Elastic Forces and Hooke's Law . . . 40 Lab-04 Conservation Laws – Collisions . . . 54
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PHYS-101 LAB-01
Speed of Light
1. Objective The objective of this experiment is to measure the speed of light in an optical fiber by measuring the time a light signal takes to traverse a known length of fiber.
2. Theory
If a light signal travels a length L in a medium of refractive index, n in time t, then the constant speed of light in that medium can be determined simply as cn = L/t. The refractive index of a medium is n = c/cn, where c is the speed of light in vacuum.
LED DETO I
CH-1
CH-2
VOLTAGE PULSE
FIBER
Figure 1. Speed of Light Apparatus.
3. Experimental Details The basis of this experiment is very simple. An LED (light emitting diode) is energized by a square voltage pulse. The time, t of travel of the resulting light pulse emitted by the LED, through the fiber of length L, is measured and used to compute the speed of light. But here is the caveat. Since t is only tens of nanoseconds, and the signals from Channel 1(CH-01) and Channel 2(CH-02) travel through different electrical paths considerable errors can be introduced in the measurement of t. Therefore, the actual experimental execution of the measurement scheme outlined above is a bit more interesting. Suppose you tap the square pulse signal (divert part of it) at the input end of the LED and use its detection as t = 0.0sec. It is instructive to know the various time delays that can be introduced before the signals of CH-01 and CH-02 are detected and displayed on the oscilloscope. We have to make sure that the time delay we are measuring is only the time delay caused by the light traveling from O to I through the fiber and not due to any other electrical delays. The time delay between CH-01 and CH-02 can be expressed as T =t +t’ where t is the time light takes to travel through the fiber and t’ is the total differential time delay between CH-01 and CH-02 due to different electrical paths the signals travel through the two channels. In most experiments such differential time delays are of little concern and are often neglected. However, in the present experiment t ~100 ns and t’ has to be properly compensated for. To do this we have to calibrate the time axis, as detailed below, by finding
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T when t ~ 0. This is done by using a short length (15cm) of an optical fiber and measuring T. For a 15 cm fiber t ~ 0.75 ns and thus T0 =t +t’ = 0.75ns +t’. Since the expected t ~ 100ns for a 20m fiber, T0 = 0.75ns +t’ can be taken as a reasonably accurate measure of the differential electrical delay between CH-01 and CH-02. Thus if we subtract T0 from our time delay measurements, we can extract the time the light signal takes to travel through a given length of optical fiber.
3.1 Apparatus
1) A Speed of Light (SOL) Apparatus with an associated power supply. 2) One each of a 15cm, 10m, and a 20m long fiber-optic cable. 3) A 100-MHz oscilloscope with two oscilloscope probes.
3.2 Experimental Procedure
4) Turn on the oscilloscope a. Under TRIGGER:
i. Press Source – select 1 ii. Press Mode – select Auto
iii. Press Slope/Coupling – 1. select the Up-Slope (↑) 2. set Coupling to AC
b. Under HORIZONTAL: i. Press Main/Delayed and set the following
1. Horizontal Mode : Main 2. Vernier : Off 3. Time Ref : Cntr
ii. Adjust Time/Div knob until divisions are presented in increments of 50ns (displayed on upper-right of screen)
c. Under VERTICAL: i. Adjust Volts/Div to 1 Volt/div for CH-01 and 0.5 Volt/div for CH-02. The
scale (Volt/div) will be displayed on the upper-left of screen. ii. Press the CH-01 button and set the following
1. CH-01 : On 2. Coupling : AC 3. Bw Lim : Off 4. Invert : Off 5. Vernier : Off 6. Probe : 1
iii. Press the CH-02 button and apply the same settings as above for CH-01 iv. Adjust Position knobs until both CH-01 and CH-02 are at 0.00mv
d. Attach oscilloscope probes to the lead-outs under the position knobs of CH-01 and CH-02
5) Leaving the oscilloscope for now, turn your attention the SOL Apparatus. We will now begin the calibration of the SOL Apparatus.
a. Set Calibration knob to the 12 o’clock position b. Using the 15cm fiber-optic cable, fit one end into the light blue D3 Transmitter
LED (the cable should fit snuggly). If the cable does not fit snugly (it easily slips
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out) you may gently tighten (no more than ¼ to ½ turn) the fiber optic cinch nut. CAUTION: Over tightening will result in damaging the cinch nut assembly.
c. Gently bend the fiber and insert the other end into the black D8 Receiver Detector DET
d. Attach probe from CH-01 to SOL Apparatus on the Transmitter side i. Red → Reference
ii. Black → Ground e. Attach probe from CH-02 to SOL Apparatus on the Receiver side
i. Red → Delay ii. Black → Ground
f. Plug power supply cable into SOL Apparatus (at the far left of the circuit). When power is applied to SOL Apparatus the yellow light at the top left should be lit.
6) Returning our attention back to the oscilloscope: a. Under HORIZONTAL:
i. Turn the Delay knob to the left or right until you see an obvious signal, there should be two – one for the trigger from CH-01 and one from the delayed signal from CH-02
ii. Adjust the Calibration knob on the SOL Apparatus until the peaks of both signals are aligned vertically
iii. Using the Horizontal Delay knob on the oscilloscope adjust the signals back to the center of the screen
The SOL Apparatus is now CALIBRATED!
7) Disconnect the power supply from the wall socket. Carefully remove the 15cm fiber-
optic cable from the SOL Apparatus and replace it with the 10m long. Once more, plug in the power supply.
8) If you do not see an obvious signal on your screen, use the Horizontal Delay knob to
locate it. The new signal should have a CH-02 output that is some distance from the original trigger signal.
a. Measurement of t: Under MEASURE, select Cursors: i. On source 1
1. select t1 – use the knob within the gray box containing the Measure buttons to move this cursor to the peak of signal 1 from CH-01
2. select t2 by pressing the button below t2, adjust this cursor to the peak of the second signal
ii. Record the Δt between these two cursors given on the lower left hand part of the screen: Example: 51ns
b. Under HORIZONTAL, adjust the Time/Div knob to 20 ns/div and repeat Steps 1
and 2 above. The expanded scale (from 50 to 20ns/div will improve the precision of your readings).
c. Every member of your team should individually go through steps [a] and [b]
above and record their readings in the data sheet that follows.
9) Remove the 10m fiber cable and replace it with the 20m cable. Determine t as in the step above. Record your readings in the Data sheet.
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Gently pull out the end of the 20m fiber that is connected to D8 Receiver Detector DET. Using a coupling sleeve, couple the free end of the 20m cable to a 10m cable. Slide the free end of the 10m cable into D8 Receiver Detector DET. Again, determine t as above.
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LAB-01 Speed of Light Name:_______________________ Sec./Group__________ Date:_____________ 4. Pre-Lab
1. What is the value of the speed of light in vacuum? 2. How does the speed of light depend on the refractive index of the medium of propagation
of light?
3. From your text find the values of the refractive indices of [a] Air [b] Water [c] Glass
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LAB-01 Speed of Light Name:_______________________ Sec./Group__________ Date:_____________
5. Data Fiber Length t1 (ns) t2 (ns) ∆t = t2 - t1 (ns) Avg. ∆t (ns) ∆L (m) 50ns/div 20ns/div 50ns/div 20ns/div 50ns/div 20ns/div 50ns/div 20ns/div
10 1
2
3
20 1
2
3
30 1
2
3
6.1 Analysis
1) Plot L vs. t for the three fiber lengths. From the slop of the straight line determine the velocity of light in the fiber, cn =L /t.
2) Speed of light in vacuum, c = n·cn .(take n = 1.5)
6.2 Error Estimate
3) What is the uncertainty in your measurement of t? 4) Suppose the length of the optical fiber is known to within a centimeter. What is the
maximum estimated error in your measurement of the speed of light?
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7. JUST FOR THE FUN OF IT
This part of the Lab is, as the name suggests, for fun. You don’t have to write a report about this section. The idea is to introduce you to concepts that are very exciting but are normally not discussed in freshmen physics courses. The experiment will give you something to think about. If you would like to pursue these concepts further, we would be very happy to direct you to additional reading material and/or discuss the significance of observations you have made.
BOX
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Figure 2. Diffraction using a Laser pointer.
The laser light you have used for the SOL experiment can be viewed either as a wave or as a stream of photons (quanta of energy). When light interacts with objects that are comparable to its wavelength, it can behave in very counterintuitive ways. This experiment is designed to illustrate this. Take a sheet from the pad and put it on a flat surface. With a razor blade make a cut about 3cm long. Turn the box upside down and paste the slit paper on the box as shown. Place the laser pen about 2 to 4 cm behind the slit. Adjust the position of the pen and/or slit so that the slit is roughly in the middle of the laser spot. Let the laser light after passing through the slit fall on a flat vertical white or light-colored surface about a meter away from the laser. If there is no such surface within your convenient reach, use a sheet of white paper as a screen. What you will see is a series of dark and bright fringes spread over several centimeters. The fringes are caused by the interference between light waves originating from different parts of the slit. Bright fringes are formed when the waves are superimposed in phase and dark fringes result when the waves meet on the screen out of phase. You should also notice that the beam has spread only along the width but not the length (height) of the slit. Here, you are seeing Heisenberg’s uncertainty principle in action. This is what is going on. The photons that end up at the screen must have passed through the slit of width x. Therefore the position of a photon is known with a maximum uncertainty of x.
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According to Heisenberg’s uncertainty principle, in a simultaneous measurement of position and momentum, the uncertainties in position (x), and in momentum (p) must obey
/ 2x p h where h = 6.6*10-34J.s is a fundamental constant of nature and is known as the
Planck’s constant. Thus if you decrease x, p – the uncertainty in the x-component of the photon’s momentum - must increase. This is what makes the laser spot ‘spread’ in the direction of the slit’s width (but not along the length where the confinement length is much larger) You can easily observe the inverse dependence of x on p by twisting/bending the slit gently (please ask your lab instructor how to do it.)
For further reading, please refer to the chapter of your textbook discussing the Uncertainty Principle of Heisenberg.
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Now using high speed digital cameras we can view the motion without the use of an inclined plane. You will drop a ball from about 1.5 – 2.0 meters and record the motion using the digital camera. Software will be used to calculate the y-position of the falling object as the frames are stepped through. You will use an object that has negligible, but not zero, drag forces and will not be affected by air resistance. If the acceleration due to gravity is indeed a constant, plotting y-position versus time should yield a parabola. Fitting a second order polynomial to this parabola will provide you with the value of the acceleration due to gravity. You can also find the approximate y-component of velocity of the object. (Although you are dropping the object from rest, you will not begin recording data until the object is in motion.) If the acceleration due to gravity is indeed a constant, plotting y-component of velocity versus time should yield a straight line. If you fit a straight line to this data, the slope will yield the acceleration due to gravity.
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Figure 3. This is a position versus time plot for an object experiencing acceleration. The slope tangent to any point is the velocity at that time.
The velocity at Point i should be the slope of the line tangent to point i. In order to compute the slope, you will need two points. Your first thought might be to use points i and i-1, but looking at Figure 4a shows that this might not be a good idea. You would fine the same problem using points i and i+1, as shown in Figure 4b. A better choice might be points i-1 and i+1, see Figure 4c.
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Figure 4. This is a position versus time plot for an object experiencing acceleration. The slope tangent to any point is the velocity at that time. Approximations of the slope are made using different sets of points.
It looks like the best method maybe the one in Figure 4c.
If time permits, take a look at all three methods. Your spreadsheet does all the work!
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5. Once the Stop button is pressed, editing may begin. Save the video clip as a .mov movie file using the File drop down menu at the top of the screen. 3.2 Editing the Video Clip: 6. Data analysis will be easier to accomplish if the length of the video is cropped to include only the frames of interest. After the stop button is pressed, the editing screen will appear. The editing screen contains a set of controls much like the set of controls on any DVD player. There are buttons for play, fast forward, fast rewind, step forward, and step backward. When the play button is pressed, the video will begin to play, and the button is transformed into a stop button. These controls are shown in Figure 10. There is also a slide bar with a black diamond on the top. Sliding this black diamond along the slide bar advances/recedes the video clip. Use these controls to review the film clip to decide where to crop the video clip. There are also two white triangles. These triangles are to mark the start and end of the frames that are to be analyzed. 7. After reviewing the video clip, use the white triangles to mark the start and finish of the frames that will be analyzed. Make sure that the start is marked after the ball has cleared the hand of the person dropping the ball. Deleting approximately five frames after the ball is released is recommended. Once you have chosen the start and finish, clip on the Confirm Edit button. This will crop the video clip to the chosen length. ( See Figure 11 )
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28
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17. Calculate the y component of the velocity using the equation:
2∆
18. Plot the y component of the velocity versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. 19. Calculate the y component of the acceleration using the equation:
2∆
20. Plot the y component of the acceleration versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. 3.5 Analyzing the Data Collected (Projectile Motion). 21. Repeat the above procedure for projectile motion. Have one of the lab partners toss the ball at an angle and with an initial velocity. 22. Open the spreadsheet containing the data collected. 23. Plot the y position versus time data. Add a trend line using a second order polynomial fit. Show the equation of the line and the R-squared valued. 24. Calculate the y component of the velocity using the equation:
2∆
25. Plot the y component of the velocity versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. 26. Calculate the y component of the acceleration using the equation:
2∆
27. Plot the y component of the acceleration versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value. 28. Plot the x position versus time data. Add a trend line using a linear fit. Show the equation of the line and the R-squared valued. 29. Calculate the x component of the velocity using the equation:
2∆
30. Plot the x component of the velocity versus time. Add a trend line with a linear fit. Show the equation of the line and the R-squared value.
32
33
LAB-02 One- and Two-Dimensional Motion Name:_______________________ Sec./Group__________ Date:_____________ 4. Pre-Lab
1. Consider a block of mass M on a frictionless plane inclined at an angle θ with respect to the horizontal. What is the acceleration of M along the plane?
2. A particle is launched in a gravitational field g with an initial velocity vo at an angle of θ with respect to the horizontal.
a. Write down the equations for displacements along x and along y, as a function of time t.
b. By combining these equations to eliminate time t, show that the resulting equation
of the form y = f(x) shows that the trajectory is parabolic.
θ
g
34
35
LAB-02 One- and Two-Dimensional Motion Name:_______________________ Sec./Group__________ Date:_____________
5. Data
Free Fall: gtheoretical = __________________ m/s2
gexperimental = __________________ m/s2 (as measured from the position versus time data.) %difference = ________________ gexperimental = __________________ m/s2 (as measured from the velocity versus time data.) %difference = ________________ Projectile Motion: gtheoretical = __________________ m/s2
gexperimental = __________________ m/s2 (as measured from the position versus time data.) %difference = ________________ gexperimental = __________________ m/s2 (as measured from the velocity versus time data.) %difference = _________________
36
6. Analysis 6.1 Free Fall. 1. Describe the relationship between the acceleration and time, the velocity and time, and the y-position and time. 2. What is the value for the acceleration due to gravity found using the y position versus time plot? 3. What is the error between the acceleration due to gravity found using the y position versus time plot and the accepted value of 9.8 m/s2? 4. What is the value for the acceleration due to gravity found using the y component of velocity versus time plot? 5. What is the error between the value for the acceleration due to gravity found using the y component of velocity versus time plot and the accepted value of 9.8 m/s2? 6. Which method gave you a better approximation for the acceleration due to gravity? Explain why that method gave the better approximation.
37
6.2 Projectile Motion. 7. Describe the relationship between the acceleration and time, the velocity and time, and the position and time for both the horizontal and vertical component? 8. What is the value for the acceleration due to gravity found using the y position versus time plot? 9. What is the error between the acceleration due to gravity found using the y position versus time plot and the accepted value of 9.8 m/s2? 10. What is the value for the acceleration due to gravity found using the y component of velocity versus time plot? 11. What is the error between the value for the acceleration due to gravity found using the y component of velocity versus time plot and the accepted value of 9.8 m/s2? 12. Which method gave you a better approximation for the acceleration due to gravity? Explain why that method gave the better approximation.
38
7. JUST FOR THE FUN OF IT Falling without Air Friction 1. In the lab, you will find clear cylindrical tubing that contains a feather and a coin. By flipping the tubing vertically, determine which falls faster: the feather or the coin. 2. Next, using a vacuum pump and a hose, evacuate the air from the cylindrical tube. Repeat the same experiment in (1). When the tube is evacuated, which falls faster: the feather or the coin? Explain your observation. Does the acceleration during free fall depend on the mass of the falling object?
39
40
PHYS-101 LAB-03
ELASTIC FORCES and HOOKE’S LAW 1. Objective
The objective of this lab is to show that the response of a spring when an external agent changes its equilibrium length by x can be described by Hooke’s Law, F. = -kx. Here F is the restoring force or the force exerted by the spring on the external agent and k is a proportionality constant characteristic of the “stiffness” of the spring and is often referred to as the spring constant. We will also study the response of springs when they are connected in series and in parallel. Furthermore, it will be demonstrated that systems described by Hooke’s Law give rise to oscillatory, simple harmonic motion. Most systems that exhibit oscillatory behavior can, to a very good approximation for small perturbations, be described by Hooke’s Law. The list of such systems is very big and varied – a pendulum, your car when you hit a bump, a marble in a bowl, atoms in a molecule or a crystal, your ear drum or any other drum, junk floating in the Schuylkill river, the quartz crystal in your watch, and electrons in the antenna of your cell phone. A mass attached to a spring is a good model system for such motion.
2. Theory 2.1 Single spring
From the free-body diagram in Fig. 1, F = -mg = - kx (symbols in bold type are vectors), where x is the displacement from the natural equilibrium length of the vertical spring. Because F = mg = kx, k can be determined as the slope, k = gm/x, from the experimentally obtained m vs. x plot.
2.2 Springs connected in parallel
Springs connected in parallel are shown in Fig.4. Using the free body diagram (FBD) in Fig. 2a, k1x + k2x = mg = kpx
kp = k1 + k2
where kp is the effective spring constant of the two parallel springs.
2.3 Springs connected in series
X
m
mg
F
FIG. 1
k1x k2x
mg
FIG. 2a
k1x
mg
k2x
mgk2x =
FIG. 2b
41
Springs connected in series are shown in Fig.5. Using the FBD for the spring with spring constant k2 in Fig. 2b: k2x2 = mg or x2 = mg/ k2. Similarly, for the other spring: kix1 = k2x2= mg or x1 = mg/ k1. The total displacement for the series combination is x = x1+x2. Thus, we can calculate the effective spring constant ks for the series combination ks (x1 + x2) = mg. Substituting for x1, x2 from equations above the following expression results ks (mg/ k1+ mg/ k2) = mg.
2.4 Simple Harmonic Motion
Systems whose responses can be described by F = - kx undergo oscillatory sinusoidal motion, also known as simple harmonic motion (SHM). This is seen most easily by expressing F = md2x/dt2 = - kx or d2x/dt2 + [k/m]x =0. Although you will study this differential equation in some detail later elsewhere, it should suffice here to note that the solution to this equation can be expressed as x = A sin(t+), where A is known as the amplitude of the SHM , is the initial phase and = [k/m]1/2is the angular frequency. The time period of the SHM is T = 2/ [Note: To see that x = A sin(t+) indeed satisfies the differential equation, take the second time derivative of x = A sin(t+) and substitute it in the differential equation.]
3. Experimental Procedure 3.1 Apparatus
1. A set of springs. 2. A set of slotted weights. 3. A weight hanger. 4. A stand with support rod. 5. A meter stick, stop-watch, and a yoke.
3.2 Some general precautions.
As you conduct this experiment it is important to observe the following:
1. Never stretch the springs you are using to more than twice
their relaxed length. Overstretching a spring may cause an inelastic, permanent deformation.
2. After every additional mass, allow the spring to come to rest before measuring the spring extension.
3.3 Experimental Procedure
1/ks =1/ k1+ 1/ k2
k1
SR
P
S
H
SW
SR - support rodS - springP - pointerSW - slotted weightsH - hanger
FIG. 3
42
A. Measurement of k1.
Suspend the spring labeled ‘5N’ from the support rod as shown in Fig.3. Place the 100.0 cm end of the meter stick on the table (the meter readings decrease upward) and align it along the length of the spring. Read the position of the red pointer on the spring. Enter this as L0 in data Table-I. Suspend the hanger from the free end of the spring and note the pointer position – this is entered as L1 in Table-I. Add a 20-gm mass to the hanger and enter the pointer position as L2. Keep adding masses in increments of 20-gm until all entries in the Table are filled.
B. Measurement of k2.
Hang the spring labeled ‘3N’ from the support rod. Repeat the steps above and fill all the entries in Table-II.
C. Measurement of kp - the effective spring constant of springs connected in parallel.
k1k2
FIG. 4
FIG. 5
k2
k1
43
Suspend the “5N’ and the ‘3N’ spring in parallel from the support rod as shown in Fig.4. Add various masses to the hanger as in the experiments above and enter your readings in Table-III.
D. Measurement of ks - the effective spring constant of springs in series.
Suspend the “5N’ and the ‘3N’ springs in series from the support rod as shown in Fig.5. Add various masses to the hanger as above and enter your readings in Table-VI.
E. Simple Harmonic Motion.
Suspend the ‘3N’ spring from the support rod and the hanger from the free end of the spring. Add 40 g to the hanger. Holding the hanger from the bottom, pull it down to stretch the spring about 5-6 cm and let go. Using the stop watch, measure the time taken for 10 complete oscillations. Enter this reading in Table-V. Take two more readings as indicated in Table-V.
44
LAB-03 Hooke’s Law Name:_______________________ Sec./Group__________ Date:_____________ 4. Pre-Lab
1. Are the shock absorbers in your car connected in series or in parallel?
2. Consider the function x(t) = Asin(t+) , we discussed earlier.
[a] Find the first derivative of x(t).
dx/dt =
[b] Find the second derivative of x(t).
d2x/dt2 =
[c] By direct substitution show that x(t) = Asin(t+) satisfies the following equation,
d2x/dt2 + [k/m]x =0.
45
46
LAB-03 Hooke’s Law Name:_______________________ Sec./Group__________ Date:_____________ 5.1 Data k1
TABLE-I Spring constant, k1.
# Spring load in grams (suspended mass, M)
Pointer Position (in cm)
Extension L(in cm)
1 0 L0 = L0 - L0= 0
2 50 L1 = L1 - L0=
3 70 L2 = L2 - L0=
4 90 L3 = L3 - L0=
5 110 L4 = L4 - L0=
6 130 L5 = L5 - L0=
7 150 L6 = L6 - L0=
Slope of mass M (y-axis) vs. L(in cm) = ___________g/cm =____________ kg/m Spring constant, k1 = (slope in kg/m)(acceleration due to gravity in m/s2) = _____________N/m
Graph I: M vs. ΔL
Mass (g)
∆L (cm
)
47
LAB-03 Hooke’s Law Name:_______________________ Sec./Group__________ Date:_____________ 5.2 Data k2
TABLE-II Spring constant, k2.
# Spring load in grams (suspended mass, M)
Pointer Position (in cm)
Extension L(in cm)
1 0 L0 = L0 - L0= 0
2 50 L1 = L1 - L0=
3 70 L2 = L2 - L0=
4 90 L3 = L3 - L0=
5 110 L4 = L4 - L0=
6 130 L5 = L5 - L0=
7 150 L6 = L6 - L0=
Slope of mass M (y-axis) vs. L(in cm) = ___________g/cm =____________ kg/m Spring constant, k2 = (slope in kg/m)(acceleration due to gravity in m/s2) = _____________N/m
Graph II: M vs. ΔL
Mass (g)
∆L (cm
)
48
LAB-03 Hooke’s Law Name:_______________________ Sec./Group__________ Date:_____________ 5.3 Data kp
TABLE-III Spring constant, kp.
# Spring load in grams (suspended mass, M)
Pointer Position (in cm)
Extension L(in cm)
1 0 L0 = L0 - L0= 0
2 50 L1 = L1 - L0=
3 70 L2 = L2 - L0=
4 90 L3 = L3 - L0=
5 110 L4 = L4 - L0=
6 130 L5 = L5 - L0=
7 150 L6 = L6 - L0=
Slope of mass M (y-axis) vs. L(in cm) = ___________g/cm =____________ kg/m Spring constant, kp = (slope in kg/m)(acceleration due to gravity in m/s2) = _____________N/m
Graph III: M vs. ΔL
Mass (g)
∆L (cm
)
49
LAB-03 Hooke’s Law Name:_______________________ Sec./Group__________ Date:_____________ 5.4 Data ks
TABLE-IV Spring constant, ks.
# Spring load in grams (suspended mass, M)
Pointer Position (in cm)
Extension L(in cm)
1 0 L0 = L0 - L0= 0
2 50 L1 = L1 - L0=
3 70 L2 = L2 - L0=
4 90 L3 = L3 - L0=
5 110 L4 = L4 - L0=
6 130 L5 = L5 - L0=
7 150 L6 = L6 - L0=
Slope of mass M (y-axis) vs. L(in cm) = ___________g/cm =____________ kg/m Spring constant, ks = (slope in kg/m)(acceleration due to gravity in m/s2) = _____________N/m
Graph IV: M vs. ΔL
Mass (g)
∆L (cm
)
50
LAB-03 Hooke’s Law Name:_______________________ Sec./Group__________ Date:_____________ 5.5 Frequency
TABLE-V Oscillation Frequency
Notes: Tth = 2[m/k]1/2 ,wherem isthemassinkg,andk isthespringconstant(inN/m)youdeterminedexperimentally1. We have neglected the mass of the spring in all our equations. This is not completely justified. Would neglecting the spring’s mass make your experimentally-determined time period Tex > Tth or Tex < Tth?
#
Suspended Mass (in g)
Time for 10 Osc. (T in sec)
Time Period Tex=T/10(in sec)
Tth = 2[m/k]1/2
(in sec) % difference [(Tex - Tth)/ Tth]*100
1
90.0
2
130.0
3
170.0
51
6. Analysis
1. Plot a graph of mass added, M (y-axis) vs. spring extension, L for the data of Table-I. Your plots should be well labeled. For a plot to be well labeled, the label along an axis should indicate the parameter, with appropriate units, being plotted. The graph should indicate the source of the data being plotted. The plotting in this LAB will be done the same way as you did in LAB-02. Briefly, open Microsoft Excel on your computer. Enter the displacement values (L’s) as a function of mass load. Enter the L-values in the first column and the mass-values in the second column. Plot L-values as a function of load (mass) and extract the best straight line fit. You can do this by highlighting both columns and clicking “Insert…Chart…(select XY scatter)” on the Menu Toolbar. When a graph is generated, right-click on the data points, and select “Add trendline”, choosing the “Linear” option; click on Options and the Trendline window and check the option “Display equation on chart”. Print out a graph for each member of your group and attach them to your Lab write-ups.
2. The spring constant k1 = (slope in kg/m)*(acceleration due to gravity, g= 9.8m/s2).
3. Repeat steps 1-3, above for Table II, III, and IV.
4. Determine the percentage difference between the experimentally determined spring
constant kp and the theoretical value, kp(th) = k1+k2
kp(%) = [( kp – kp(th))/ kp(th)]*100 5. Determine the percentage difference between the experimentally determined spring
constant ks and the theoretical value, ks (th.) = k1k2/( k1+k2)
ks(%) = [( ks – ks(th))/ ks(th)]*100
52
LAB-03 Hooke’s Law Name:_______________________ Sec./Group__________ Date:_____________ 7. JUST FOR THE FUN OF IT
Consider a ball suspended from the lid of a can using a string as shown in A1. Another ball is attached to one end of a spring with the other end of the spring attached to the lid of a can, see B1. Think of the can as just a mass. The reason for using a can is simply that the lower rim of the can provides a convenient visual marker for any relative vertical movement of the ball.
A. For the ball and string case
Suppose you hold the can at shoulder height and drop it. Which of the following would you expect? Make a guess and don’t worry too much about being right (you are not being graded here for your answer) [a] the string would be taut (the tension in the string, T>0) and the ball would be as in A1 [b] the ball would appear as in A1, but the tension in the string would be zero. [c] the string would go limp and the ball would move up relative to the can as in A2
B. For the ball and spring case
Suppose you hold the can at shoulder height and drop it. Which of the following would you expect? Make a guess and don’t worry too much about being right. [a] the spring and ball would maintain their position and appear as in B1. The spring would neither get compressed or stretch from its pre-release position. [b] the ball would appear as in B2, the spring would stretch further from its pre-release position. [c] the spring would return to its equilibrium length as in B3.
C. The reality kicks in.
Perform the experiment as described in A and B above. Explain your observations.
53
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larger mass to provide a glancing blow (Figure 1). The film may now be edited with the VideoPoint Capture Software. 5. Once the Stop button is pressed, editing may begin. Save the video clip as a .mov movie file using the File drop down menu at the top of the screen. 6. Data analysis will be easier if the length of the video is cropped to include only the frames of interest. After the stop button, the editing screen will appear. The editing screen contains a set of controls much like the set of controls on any DVD player. There are buttons for play, fast forward, fast rewind, step forward, and step backward. When the play button is pressed, the video will begin to play, and the button is transformed into a stop button. These controls are shown in Figure 6. There is also a slide bar with a black diamond on the top. Sliding this black diamond along the slide bar advances/recedes the video clip. Use these controls to review the film clip to decide where to crop the video clip. There are also two white triangles. These triangles are to mark the start and end of the frames that are to be analyzed.
Figure 6. VideoPoint Capture Editing Screen.
7. After reviewing the video clip, use the triangles to mark the start and finish of the frames that will be analyzed. Make sure that the start is marked after the puck has cleared the hand of the person shooting the puck. Deleting approximately five frames after the puck is released is recommended. Once you have chosen the start and finish, clip on the Confirm Edit button. This will crop the video clip to the chosen length. 8. After checking that the film clip has the needed frames to complete the analysis, save the film clip. 9. Open the VideoPoint Physics Fundamentals Software. Using the File drop down menu, load the video clip movie file saved in step 8.
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59
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62
LAB-04 Conservation Laws (Collisions) Name:_______________________ Sec./Group__________ Date:_____________
4.1 Prelab
1.) Consider a 0.52 g puck that moves as shown in the graph above. Consider each point to be a picture of the puck and consider that 60 pictures were taken each second. Plot the x position versus time and the y position versus time and use a linear trend line to find the x component of the velocity and the y component of the velocity. Attach you graphs. a.) What are the x component of the velocity and the y component of the velocity? b.) Write the velocity of the puck in i and j notation and in magnitude and angle notation. c.) What is the momentum of the puck? (Write it both in i and j notation and as a magnitude and an angle.)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
Y Position vs X position
x(cm)
y(cm
)
LAB-04 Name:__
4.2 Prel
2.) Consiwith a m2initially mconsider a. a. c.
Conservatio
__________
lab (Conti
ider the colli2 = 0.32 g pmoving, is shthat 60 pictu
.) What is th magnitud
.) What is th magnitud
.) What is th magnitud
on Laws (Co
___________
inues)
ision of a m1uck initiallyhown in the ures were tak
he initial mome and an ang
he final mome and an ang
he final mome and an ang
0 0.5
-1.75
-1.50
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
y(cm
)
llisions)
___ Sec./Gr
1 = 0.52 g puy at rest. Onl
graph. Consken each sec
mentum of thgle.)
mentum of thgle.)
mentum of thgle.)
5 1 1.5 2
Y Posit
roup______
uck moving ly the track osider each pocond.
he m1 puck?
e m1 puck?
e m2 puck?
2.5 3 3.5 4
tion vs X positi
x(cm)
_____ Da
as shown inof the m1 = oint to be a p
? (Write in b
(Write in bo
(Write in bo
4.5 5 5.5 6
ion
ate:_______
n the graph, w0.52 g puck
picture of the
both i and j f
oth i and j fo
oth i and j fo
6 6.5 7
_______
which collidk, which is e puck and
format and a
ormat and as
ormat and as
63
des
as a
a
a
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LAB-04 Conservation Laws (Collisions) Name:_______________________ Sec./Group__________ Date:_____________
5.1 Data ( First Collision – One Puck initially at rest.)
m1= _________________ kgm2= _________________ kg
v1xi= _________________ m/ s
v1yi= _________________ m/ s
v1x f= _________________ m/ s
v1y f= _________________ m/ s
v 2x f= _________________ m/ s
v 2y f= _________________ m/ s
p1xi= _________________ kg m/ s
p1y i= _________________ kg m/ s
p1x f= _________________ kg m/ s
p1y f= _________________ kg m/ s
p2x f= _________________ kg m/ s
p2y f= _________________ kg m/ s
pxi= _________________ kg m/ s
pyi= _________________ kg m / s
p x f= _________________ kg m/ s
p y f= _________________ kg m/ s
K i= _________________ JK f = _________________ J
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LAB-04 Conservation Laws (Collisions) Name:_______________________ Sec./Group__________ Date:_____________
5.2 Data ( Second Collision – Both Pucks initially in motion.)
m1= _________________ kgm2= _________________ kg
v1xi= _________________ m/ s
v1yi= _________________ m/ s
v2xi= _________________ m / s
v2y i= _________________ m/ s
v1x f= _________________ m/ s
v1y f= _________________ m/ s
v 2x f= _________________ m/ s
v 2y f= _________________ m/ s
p1xi= _________________ kg m/ s
p1y i= _________________ kg m/ s
p2xi= _________________ kg m/ s
p2y i= _________________ kg m/ s
p1x f= _________________ kg m/ s
p1y f= _________________ kg m/ s
p2x f= _________________ kg m/ s
p2y f= _________________ kg m/ s
pxi= _________________ kg m/ s
pyi= _________________ kg m / s
p x f= _________________ kg m/ s
p y f= _________________ kg m/ s
K i= _________________ JK f = _________________ J
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LAB-04 Conservation Laws (Collisions) Name:_______________________ Sec./Group__________ Date:_____________ 6.1 Analysis (First Collision – One Puck initially at rest.)
a.) Calculate the percent difference between the initial and final momentum in both the x direction and the y direction. (If a zero value ends up in the denominator, compute only the difference, not the percent difference.)
b.) Was momentum conserved during this collision? How do you know?
c.) Compute the percent difference between the initial and final kinetic energy of the system.
d.) Was kinetic energy conserved? What type of a collision was this collision (elastic or inelastic)?
e.) Briefly discuss your outcome. Are you surprised by your outcome? Did you expect this collision to be elastic or inelastic?
LAB-04 Conservation Laws (Collisions)
67
Name:_______________________ Sec./Group__________ Date:_____________ 6.2 Analysis (First Collision – Both Pucks initially in motion.)
a.) Calculate the percent difference between the initial and final momentum in both the x direction and the y direction. (If a zero value ends up in the denominator, compute only the difference, not the percent difference.)
b.) Was momentum conserved during this collision? How do you know?
c.) Compute the percent difference between the initial and final kinetic energy of the system.
d.) Was kinetic energy conserved? What type of a collision was this collision (elastic or inelastic)?
e.) Briefly discuss your outcome. Are you surprised by your outcome? Did you expect this collision to be elastic or inelastic?
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7. JUST FOR THE FUN OF IT Brownian Motion How do you know that there are atoms? Let’s suppose there are (such an assumption is called a hypothesis). We can not really see the atoms or molecules with an optical microscope – the resolution of a microscope is limited by the wavelength of the light that is used to illuminate the object being observed. The motion of atoms in a liquid (or a gas) must be random. If the motion were not random then a liquid or a gas would occasionally become ‘lumpy’ all by itself. (Can you guess why?). If you were to put a ball much larger than the molecules in, let’s say, water and the atoms/molecules were moving randomly then the ball should jiggle around. This jiggling is expected as a result of the imbalance of the momentum (which itself would fluctuate randomly with time) imparted to the ball by the random collisions between the ball and the water molecules. This random motion is called Brownian motion and played a crucial role in establishing the atomic nature of matter. The mathematical model of Brownian motion – known as the random walk or drunkard’s walk model – has many real-world applications, including stock market fluctuations. If water were not made up of discrete atoms/molecules but was a continuum of matter, the micro-beads should have stayed undisturbed at the position you originally placed it. In the lab you would be watching micron-sized beads suspended in a water-based solution. Notice that the beads jiggle around as they are being bombarded with randomly moving water molecules. The random motion of atoms/molecules in a fluid is temperature dependent – the speed of atoms between collisions and the frequency of collisions increase with increasing temperature. In the experiment you can raise the temperature of the fluid simply by increasing the intensity of the illuminating light. What would you expect to happen to the micro-beads’ jiggle as the temperature is increased? Make a prediction and see if it comes true.
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