University of Northern British Columbia. Physics Program. Physics 100 Laboratory Manual

Transcription

1 University of Northern British Columbia Physics Program Physics 100 Laboratory Manual Winter 2014

2 Contents 1 Newton s Second Law of Motion 5 2 Atwood s Machine 11 3 Measurement of the Acceleration of Gravity 17 4 Kinematics: Constant Acceleration 25 5 Hooke s Law and Simple Harmonic Motion 35 6 The Simple Pendulum 43 7 Rotation and Moment of Inertia 49 8 Standing Waves on a String 57 A Dimensional Analysis 63 B Error Analysis 65 B.1 Measurements and Experimental Errors B.1.1 Systematic Errors: B.1.2 Instrumental Uncertainties: B.1.3 Statistical Errors: B.2 Combining Statistical and Instrumental Errors B.3 Propagation of Errors B.3.1 Linear functions: B.3.2 Addition and subtraction: B.3.3 Multiplication and division: B.3.4 Exponents: B.3.5 Other frequently used functions: B.4 Percent Difference: iii

3 Introduction to Physics Labs Lab Report Outline: Physics 1?? Your Name section L? Student # date Lab Partner s Name Lab #, Lab Title The best laboratory report is the shortest intelligible report containing ALL the necessary information while maintaining university level spelling and grammar. Use full sentences and divide the lab into the sections described below. Object - One or two sentences describing the aim of the experiment. Be specific in regards to what you hope to prove. (In ink) Theory - The theory on which the experiment is based. This usually includes an equation which is to be verified. Be sure to clearly define the variables used and explain the significance of the equation. (In ink) Apparatus - A brief description of the important parts and how they are related. In all but a few cases, a diagram is essential, artistic talent is not necessary but it should be neat (use a ruler) and clearly labeled. (All writing in ink, the diagram itself may be in pencil.) Procedure - Describes the steps of the experiment so that it can be replicated. This section is not required. Data - Show all your measured values and calculated results in tables. Show a detailed sample calculation for each different calculation required in the lab. Graphs, which must be plotted by hand on metric graph paper and should be on the page following the related data table. (If experimental errors or uncertainties are to be taken into account, the calculations should also be shown in this section.) Include a comparison of experimental and theoretical results. (Except graphs, this section must be in ink.) Discussion and Conclusion - What have you proven in this lab? Support all statements with the relevant data. Was your objective achieved? Did your experiment agree with the theory? With the equation? To what accuracy? Explain why or why not. Identify and discuss possible sources of error. (In ink)

4 2 Introduction to Physics Labs Lab Rules: Please read the following rules carefully as each student is expected to be aware of and to abide by them. They have been implemented to ensure fair treatment of all the students. Missed Labs: A student may miss a lab without penalty if due to illness(a doctor s note is required) or emergency. As well, if an absence is expected due to a UNBC related time conflict (academic or varsity), arrangements to make up the lab can be made with the Senior Laboratory Instructor in advance. Students will not be allowed to attend sections, other than the one they are registered in, without the express permission of the Senior Laboratory Instructor. Failure to attend a lab for any other reason will result in a zero (0) being given for that lab. Late Labs: Labs will be due 24 hours from the beginning of the lab period, after which labs will be considered late. Late labs will be accepted up until 1 week from the beginning of the lab. Students will be allowed 1 late lab without penalty after which all subsequent late labs will be given a mark of zero (0). Any labs not received within one week will no longer be accepted for grading and given a mark of zero (0). Completing Labs: To complete the lab report in the lab time students are strongly advised to read and understand the lab (the text can be used as a reference) and write-up as much of it as possible before coming to the lab. We urge you to stay the entire lab time - it is provided for your benefit, if you run into problems writing up the lab, the instructor is there to help you. Conduct: Students are expected to treat each other and the instructor with proper respect at all times and horseplay will not be tolerated. This is for your comfort and safety, as well as your fellow students. Part of your lab mark will be dependent on your lab conduct.

5 Plagiarism: Plagiarism is strictly forbidden! Copying sections from the lab manual or another person s report will result in severe penalties. Some students find it helpful to read a section, close the lab manual and then think about what they have read before beginning to write out what they understand in their own words. Use Ink: The lab report, except for diagrams and graphs, must be written in ink, this includes filling out tables. If your make a mistake simply cross it out with a single, straight line and write the correction below. Professional scientists use this method to ensure that they have an accurate record of their work and so that they do not discard data which could prove to be useful after all. For this reason also, you should not have "rough" and "good" copies of your work. Neatly record everything as you go and hand it all in. If extensive mistakes are made when filling out a table, mark a line through it and rewrite the entire table. Lab Presentation: Labs must be presented in a paper duotang. Marks will be given for presentation, therefore neatness and spelling and grammar are important. It makes your lab easier to understand and your report will be evaluated accordingly. It is very likely that, if the person marking your report has to search for information or results, or is unable to read what you have written, your mark will be less than it could be! Contact Information: If your lab instructor is unavailable and you require extra help, or if you need to speak about the labs for any reason, contact the Senior Laboratory Instructor. Dr. George Jones gjones@unbc.ca Supplies needed: Clear plastic 30 cm ruler Metric graph paper (1mm divisions) Paper duotang Lined paper Calculator Pen, pencil and eraser 4

6 Experiment 1 Newton s Second Law of Motion Introduction Newton s second law of motion describes how the net force acting on an object and its mass affect its acceleration. This experiment will be performed with the aid of a computer interface system which will allow you to use the computer to collect, display and analyze the data. Theory Newton s second law of motion states that the acceleration of an object or system is directly proportional to the net force acting on it and inversely proportional to its total mass. For a motion in one dimension, Newton s second law takes the form: F = ma (1.1) This motion is an example of a uniformly accelerated motion for which the following kinematic relations should hold true: The instantaneous velocity v of the system is given by v = at + v 0 (1.2) where t denotes time and v 0 the initial velocity, and its displacement x is given by where x 0 is the initial position. x = 1 2 at2 + v 0 t + x 0 (1.3) Consider the system shown in Figure 1. We shall neglect for now the rotational motion of the pulley, the frictional force acting on the system, and the mass of the string connecting the two masses m and M. Let us apply Newton s second law to the motion of m: mg T = ma (1.4)

7 6 EXPERIMENT 1 Now apply Newton s second law to the motion of M (note that Mg and N cancel each other): Substituting (1.5) into (1.4) leads to: which can be rearranged to: T = Ma (1.5) mg Ma = ma (1.6) a = m m + M g (1.7) which gives the value of the acceleration of either m or M. One way to interpret this equation is to say that the acceleration of the (m + M) system is equal to the ratio between the net external force acting on it (mg) and its total mass (m + M).

8 NEWTON S SECOND LAW OF MOTION 7 Apparatus Computer Science Workshop Interface Dynamics Cart and Track System Mass Hanger and Mass Set (4x100g, 1x50g) Procedure 1. At the start of your lab session, the Linear Dynamics System, Science Workshop interface box, and Dell computer will all be connected and ready to go. Familiarize yourself with the system and the various components of the apparatus. 2. A linear motion timing experiment uses a photogate to measure the motion of an object with uniformly spaced flags. These are normally provided by the black stripes of a picket fence mounted on the object whose motion is being studied. Set up the apparatus as shown in Figure 1. Make sure the picket fence and track are free of dust and dirt. If needed, ask for assistance from your laboratory instructor. 3. Follow the directions below to set up the DataStudio software: Double click on the Data Studio icon on the desktop. Choose No when the program asks whether DataStudio should choose an options set to load. Choose Create Experiment The Experiment Setup window displaying the Science Workshop 750 Interface will appear.

9 8 EXPERIMENT 1 Click on the first digital channel on the left. Choose Photogate & Picket Fence on the menu that appears. Deselect Acceleration, but make sure Position and Velocity are checked in the Measurements tab. In the Constants tab, set the Band Spacing to 0.010m. Select Sampling Options at the top of the Experiment Setup window. In the Delayed Start tab, select Data Measurement and make sure that it says: Position, Ch 1 (m) Is Above 0 m. Click OK. You can now close the Experiment Setup window. 4. To set up how the data will be displayed, follow these steps: In the Displays column (at the left of the window) double click Graph. In the Choose a Data Source window, select Velocity, Ch1 (m/s). Maximize the graphical display. 5. Run 1: Place 150g (one 100 and one 50) on the cart and make sure there are no slotted masses on the hanger. (The mass of the hanger, m, is 50g.) The mass of the cart (including the picket fence) is 510g. Therefore M = kg + added masses. Record m and M in the data table. 6. Start data collection by clicking Start and immediately release the cart starting at one end of the track. Make sure to halt the cart after it passes through the photogate. Stop data collection by clicking Stop. 7. Look at the graph of Velocity vs Time on the screen. Use Scale to Fit to adjust the ranges so the points take up most of the screen. If the line is uneven, delete the data from the run and repeat data collection until the plot is a relativity smooth line. 8. Once a smooth line is recorded, choose Linear Fit from the drop-down Fit menu to analyze the velocity-vs-time plot. Compare the equation of the line (Y=mX+b) with equation (1.2) (velocity is on the y-axis, time is on the x-axis, etc.) and then extract and record the value of the acceleration a ex and the initial velocity v 0ex.

10 NEWTON S SECOND LAW OF MOTION 9 9. Calculate and record the total mass, M + m, in the data table. 10. Using equation (1.1), calculate and record the force acting on the system, F = mg. 11. Then using Equation (1.7) calculate and record value of acceleration, a th = m m+m g. 12. Consult section B.4 and then calculate and record the % difference between the theoretical and experimental values for acceleration, a th a ex a th 100%. 13. RUN 2: Move 50 g from the cart load to the mass hanger. (M = 0.610kg and m = 0.100kg.) Repeat steps 6 to RUN 3: Move an additional 50g from the cart load to the mass hanger. (M = 0.560kg and m = 0.150kg.) Repeat steps 6 to RUN 4: Add 150g to the cart load. (M = 0.710kg and m = 0.150kg.) Repeat steps 6 to RUN 5: Add another 150g to the cart load. (M = kg and m = kg.) Repeat steps 6to On one piece of graph paper, sketch each of the five lines generated in the above trials and label each of them with the run # and corresponding equation. Be sure to title the graph, and use regular increments on the axis and label them. Discussion and Conclusion Comment on whether your results agree with Newton s Second Law. Explain. Outline and discuss at least two possible sources of error which may have occurred during the collection of the data and how they could have been avoided.

11 10 EXPERIMENT 1 Data Table: Newton s Second Law Run # M (kg) m (kg) a ex (m/s 2 ) v 0ex (m/s) (m + M) (kg) F net (N) a th (m/s 2 ) % difference (%) Date: Instructor s Name: Instructor s Signature:

12 Experiment 2 Atwood s Machine Introduction Lab 2 continues with the study of Newton s second law of motion. In this study an Atwood s machine will be used which allows us to evaluate the situation where gravity is acting on both masses in the system rather than just one as in Lab 1. Theory Atwood s machine consists of two masses connected by a cord and suspended by a pulley as shown in the Force Diagram. Suppose for now that the pulley and cord have negligible mass and that there is no friction acting on the system. In response to the downward force of gravity, the heavier mass M 1 will fall, with an acceleration of magnitude a. If the cord is of fixed length, the lighter mass M 2 must rise with an acceleration of magnitude a as well. Applying Newton s second law to each mass gives: M 1 g T = M 1 a (2.1) M 2 g T = M 2 a (2.2) where we have chosen the positive direction to be downwards in each case. Notice also that the magnitude of the tension T is uniform throughout the cord. We can solve for a by subtracting equation (2.2) from equation (2.1): (M 1 M 2 )g = (M 1 + M 2 )a (2.3) If we include the effects of the frictional force, f, then equation (2.3) should be modified to: (M 1 M 2 )g f = (M 1 + M 2 )a (2.4)

13 12 EXPERIMENT 2 We also know that the acceleration, a, the distance through which the heavier mass M 1 falls, h, and the time taken for the mass M 1 to fall to the floor, t, are related by the equation for uniformly accelerated motion: Apparatus a = 2h t 2 (2.5) Pulley System (Stand, Clamp, Mass Hangers, String, and Pulley) Tape Measure and Stopwatch Mass Set (3x100g, 1x50g, 1x5g, 2x2g, 1x1g) Procedure 1. Start with empty mass hangers. Since the masses of the hangers are equal (each has a mass of 50 g) the system should be in equilibrium. With one hanger touching the floor measure the distance from the bottom of the other hanger to the floor; this is the distance h in equation (2.5). Based on section B.1.2 the error in h, δh, will be 0.5mm. Record these values in your lab report. 2. While holding the system still add 0.160kg to M 2 and 0.200kg to M 1. Including the masses of the hangers the total masses are 0.210kg and 0.250kg, respectively. Record these values along with the total mass, M 1 + M 2, in the data table.

14 ATWOOD S MACHINE Gently release the system from rest. With a stopwatch measure the time it takes M 1 to travel the distance h from rest. Make three independent measurements of this time and record them as t 1, t 2 and t 3 in the data table. 4. By consulting section B.1.3 and knowing that since you made three trials N =3, calculate and record the average time of fall, t = t 1+t 2 +t 3, the standard deviation of your time 3 measurements, σ s.d. = (t1 t) 2 +(t 2 t) 2 +(t 3 t) 2, and the error in t, δt = σ s.d Using equation (2.5), determine and record the acceleration, a = 2h t 2, of the system. 6. Consult section B.3.4 to calculate the error in a, δa = a ( 1 δh h )2 + ( 2 δt ) t 2 and record it. 7. Calculate the force acting on the system, F = (M 1 M 2 )g, and assume the corresponding error, δf, is zero since the error in the masses is negligible. 8. Repeat steps 3 through 7 five more times, each time moving 2g from M 2 to M 1 (thus keeping the total mass of the system, M 1 + M 2, constant). 9. Use the data collected to plot a graph of Force versus Acceleration. Use the calculated values of δa to draw horizontal error bars to scale. Determine the best line through your points (It must intersect at least two points and have an equal number of points below and above the line) and use the error bars to find the maximum (steepest) and minumum (shallowest) slopes. 10. Determine the value of the best slope, m best, using two points you DID NOT plot, (x 1, y 1 ) and (x 2, y 2 ) and the formula: m = rise = y 2 y 1 run x 2 x 1. Similarly find the values of m max and m min. 11. Find the error in the slope, δm, by calculating m max -m best and m best -m min and choosing the larger value. 12. Compare the equation of the line (Y=mX+b) with equation (2.4) (net force is on the y-axis, acceleration is on the x-axis, etc.). 13. Based on the comparison in the previous step extract the value of the effective frictional force, f, acting on your system as well as it s error, δf (see section B.3.1).

15 14 EXPERIMENT 2 Discussion and Conclusion Comment on whether your results agree with Newton s Second Law. Explain. Outline and discuss at least two possible sources of error which may have occurred during the collection of the data and how they could have been avoided.

16 ATWOOD S MACHINE 15 M 1 (kg) M 2 (kg) (M 1 + M 2 ) (kg) t 1 t 2 t 3 t σ s.d. δt a (m/s 2 ) δa (m/s 2 ) F (N) Data Table: Atwood s Machine Date: Instructor s Name: Instructor s Signature:

17 16 EXPERIMENT 2

18 Experiment 3 Measurement of the Acceleration of Gravity Introduction In this lab you will study free fall motion using the free fall adapter which plugs into photogate timers and computer interfaces, allowing for a precise measurement of acceleration. Theory Free fall is the motion of any object subjected to the gravitational force exerted by the earth. The magnitude of this gravitational force is given by Newton s formula: F = G M EM d 2 = G M EM (R + r) 2 (3.1) where G is the gravitational constant, M E the mass of the earth, M the mass of the falling object, R the radius of the earth, and r the elevation of the object above the surface of the earth. If we defined the quantity g as: M E g = G (3.2) (R + r) 2 On the surface of the earth r is negligible in comparison to the much larger value of R; consequently, and to a high degree of accuracy, g can be considered constant. Combining equations (3.1) and (3.2) we get: Applying Newton s second law of motion: F = Mg (3.3) F = Ma (3.4)

19 18 EXPERIMENT 3 it becomes clear that the quantity g is nothing but the acceleration of a free fall motion, which, since it is independent of the mass of the falling object, is a constant regardless of how heavy or light the object is. We call g the acceleration of gravity. As you already know, the equation of motion for a body starting from rest and undergoing constant acceleration can be expressed as: x = 1 2 at2 (3.5) where x is the distance travelled, a the acceleration, and t the time elapsed. So the motion for free fall with no initial velocity is simply given by: d = 1 2 gt2 (3.6) Apparatus Free Fall Adaptor (automatic ball release mechanism, receptor pad, controller box)

20 MEASUREMENT OF THE ACCELERATION OF GRAVITY 19 Computer Science Workshop Interface Tape Measure 2 Steel Balls (13 and 16 mm) Procedure 1. At the start of your lab session, the Free Fall Adaptor, Science Workshop interface box, and Dell computer will all be connected and ready to go. item Follow the directions below to set up the DataStudio software: Double click on the Data Studio icon on the desktop. Choose No when the program asks whether DataStudio should choose an options set to load. Choose Create Experiment The Experiment Setup window displaying the Science Workshop 750 Interface will appear. Click on the first digital channel on the left. Choose Free Fall Adapter on the menu that appears. Deselect Acceleration, but make sure time is checked in the Measurements tab. Select Sampling Options at the top of the Experiment Setup window. In the Delayed Start tab, select time and set it to 0. Deselect the start signal generation before start condition option. In the Automatic Stop tab, select data measurement but don t change the options. Click OK. You can now close the Experiment Setup window. 2. To set up the timing window, look in the Displays column (at the left of the window) and double click 3.14 DIGITS. 3. The Free Fall Adapter consists of an automatic ball release mechanism, a receptor pad, and a controller box (see Figure 1). The controller box plugs into a computer interface which is basically a digital clock started when the ball is released and stopped when the

21 20 EXPERIMENT 3 ball hits the pad. The digital reading of the timer is then simply the time taken by the ball to fall a distance d (see Figure 2). Set up the apparatus as shown in Figure 2. If needed, ask for assistance from your laboratory instructor. 4. Mount the 16mm diameter steel ball tightly in the Ball Release Mechanism and being as accurate as possible (so that δd can be assumed to be zero) set the distance from the bottom of the ball to the top of the pad d to 1.00m and record it in Data Table 1. Make several trials to be sure the ball will hit the center of the Receptor Pad when released. 5. Take a first measurement of t and record it in the table as t 1. Repeat the measurement twice more and record these values as t 2 and t By consulting section B.1.3 and knowing that since you made three trials N =3, calculate and record the average time of fall, t = t 1+t 2 +t 3, the standard deviation of your time 3 measurements, σ s.d. = (t1 t) 2 +(t 2 t) 2 +(t 3 t) 2, and the error in t, δt = σ s.d Square the value of t and consult section B.3.4 to find the corresponding error, δt 2 2 δt = t2. t Remember to record all values in the tables. 8. Repeat steps 4 through 7 for the 13 mm steel ball and record the values in Data Table Repeat steps 4 through 8 for the following values of d: 1.05, 1.10, 1.15, 1.20, 1.25, 1.30, and Record all your measurements in the Data Tables. 10. Plot a graph of Distance versus Time 2, for the 16mm ball bearing. Use the calculated values of δt 2 to draw horizontal error bars to scale. Determine the best line through your points (It must intersect at least two points and have an equal number of points below and above the line) and use the error bars to find the maximum (steepest) and minumum (shallowest) slopes. 11. Determine the value of the best slope, m best, using two points you DID NOT plot, (x 1, y 1 ) and (x 2, y 2 ) and the formula: m = rise = y 2 y 1 run x 2 x 1. Similarly find the values of m max and m min. 12. Find the error in the slope, δm, by calculating m max -m best and m best -m min and choosing the larger value.

22 MEASUREMENT OF THE ACCELERATION OF GRAVITY Compare the equation of the line (Y=mX+b) with equation (3.6) (distance is on the y-axis, time squared is on the x-axis, etc.). 14. Based on the comparison in the previous step extract the value of the gravitational acceleration, g 16, acting on your system as well as it s error, δg 16 (see section B.3.1). 15. Repeat steps 10 through 14 for the 13 mm steel ball Discussion and Conclusion Comment on whether your results support that the acceleration of gravity is independent of mass. Explain. Outline and discuss at least two possible sources of error which may have occurred during the collection of the data and how they could have been avoided.

23 22 EXPERIMENT 3 d (m) t 1 t 2 t 3 t σ sd δt t 2 (s 2 ) δt 2 (s 2 ) Data Table 1: Free Fall of a 16mm Steel Ball Date: Instructor s Name: Instructor s Signature:

24 MEASUREMENT OF THE ACCELERATION OF GRAVITY 23 d (m) t 1 t 2 t 3 t σ sd δt t 2 (s 2 ) δt 2 (s 2 ) Data Table 2: Free Fall of a 13mm Steel Ball Date: Instructor s Name: Instructor s Signature:

25 24 EXPERIMENT 3

26 Experiment 4 Kinematics: Constant Acceleration Introduction In this experiment a dynamics cart will be launched across the floor, and its change in velocity (acceleration) measured. The results will then be analyzed using two different methods of error analysis. Theory If the acceleration of the cart does remain constant throughout its journey, the calculated values of t th, the time required to travel the distance d, should be equal to the directly measured value, t ex. To calculate t th the distance equation is used, d = v o t th at2 th (4.1) Now if both the acceleration a and the initial velocity v o are known, then the time required to travel an intermediate distance t th can be calculated. We do this by applying the quadratic formula, if ax 2 + bx + c = 0 then x = b± b 2 4ac 2a, to the distance equation (4.1). The dynamics cart will be launched across the floor using the built in spring plunger. The cart will then accelerate (in everyday language we often use the word decelerate for an object that slows down. In Physics we normally use the word accelerate for objects that speed up or slow down as they are just two aspects of the same effect) due to various effects of friction (friction in the wheel bearings, rolling friction) and the slope of the floor.

27 26 EXPERIMENT 4 Eventually the cart will roll to a stop. The total distance covered is denoted D and the total elapsed time from the launch to the stop is denoted T. Then the average velocity of the cart on its journey is given by: v av = D T (4.2) If the acceleration of the cart remains constant throughout its journey across the floor, then the initial instantaneous velocity of the cart at the moment it is launched is given by: v o = 2v av = 2D T (4.3) The value of this constant acceleration is then given by: a = change in v change in t = v final v initial t final t initial Note: the acceleration is negative, ie. the cart is slowing down. = 0 v o T 0 = v o T = 2D T 2 (4.4) Apparatus Dynamics Cart (PASCO ME-9430) Stopwatch with lap timer Tape Measure and Electrical Tape

28 KINEMATICS: CONSTANT ACCELERATION 27 Procedure 1. Mark a fixed launch point for the cart. To launch the cart with reasonable repeatability, you need to cock the spring plunger. To cock the plunger, push the plunger in, and then push the plunger upward slightly to allow one of the notches in the plunger bar to catch on the edge of the small metal bar at the top of the hole. 2. Launch the cart a couple of times to roughly determine its range. Now mark a distance d that is a bit more than halfway between the start and finish. Measure the distance from the launch point to d and record it above the data table in your lab report (Data Table 1). 3. Using the stopwatch (make sure it is in cumulative mode ) and the meter stick, it is possible to measure t, T and D for each launch. You will probably want to practice this a few times before you start recording data. The stopwatch is started when the cart is launched, the t is recorded using the lap time feature, and the total time for the journey is recorded by stopping the stopwatch. The total distance the cart travels, D, is determined using the tape measure. Note: To reduce errors due to reaction times it is important that the person who launches the cart also do the timing. You should take turns. Also, trials in which the launcher/timer thinks that there was some distraction or other unusual error that interfered with the measurement should not be recorded, as this will decrease the accuracy of data. 4. Launch the cart and record t, T and D for six trials in a table similar to the one shown Data Table 1. Note: We will be switching back and forth between absolute and relative (%) errors to simplify the error calculation (yes, this is the easy way). This allows us to simply add the errors in quadrature, which is done by combining the errors using the following formula: Error = (Error 1 ) 2 + (Error 2 ) 2 For addition/subtraction add the absolute errors in quadrature. For multiplication/division add the relative (%) errors in quadrature. For powers multiply the relative (%) errors by the exponent. For coefficients the relative (%) errors are unaffected.

29 28 EXPERIMENT 4 Although some of the following looks intimidating, if you follow the steps given in the manual carefully by substituting in your values, it is time consuming but not difficult. 5. Calculate the relative errors of d, D, T and t ex and fill them into your table. A reasonable error estimate for hand timed intervals (due to human reaction time) is ±0.09 s and distances should be measured to within ± m. Show your calculations as in the following example: Sample calculation of relative errors: In a case where, d = ( ± )m D = ( ± )m T = (7.84 ± 0.09)s t ex = (3.12 ± 0.09)s the absolute errors are converted to relative (%) errors by: relative error = absolute error value of variable 100 % For example: d = m ± ( m 100%) = m ± 0.05% m D = m ± ( m 100%) = m ± 0.03% m T = 7.84s ± ( 0.09s 100%) = 7.84s ± 1.14% 7.84s t ex = 3.12s ± ( 0.09s 100%) = 3.12s ± 2.88% 3.12s

30 KINEMATICS: CONSTANT ACCELERATION Work out v o, a and their absolute errors. Show your calculations in the following way: Sample calculation of v o : v o = 2D T = 2(1.8000m ± 0.03%) (7.84s ± 1.14%) since this is division the relative errors must be added in quadrature: v o = 0.459m/s ± (0.03%) 2 + (1.14%) 2 = 0.459m/s ± 1.14% The relative (%) errors are converted to absolute errors by: absolute error = relative error 100 % value of variable For example: v o = 0.459m/s ± ( 1.14% 100% 0.459m/s) = (0.459 ± 0.005)m/s Sample calculation of a: a = 2D ± 0.03%) m ± 0.03% = 2(1.8000m = T 2 (7.84s ± 1.14%) s 2 ± 2.28% Note: When adding in quadrature the relative error of T is multiplied by 2 because T is squared in the equation for a. a = m/s 2 ± (0.03%) 2 + (2.28%) 2 = m/s 2 ± % a = m/s 2 ± ( 2.28% 100%.0585m/s2 ) = ( ± 0.001)m/s 2

31 30 EXPERIMENT 4 7. For trial 1 only, calculate the expected (theoretical) value for t. Follow the sample calculation exactly and show all the steps in your lab. Sample calculation of t th : In an equation of the form: 1/2at 2 th + v o t th d = 0 the quadratic equation can be used to solve for t th : which can be simplified to give: t th = v o ± v 2 o 4( 1 2 a)( d) 2( 1 2 a) (a) plug in the values: t th = v o ± v 2 o + 2ad a t th = (0.459m/s ± 1.14%) ± (0.459m/s ± 1.14%) 2 + 2( 0.059m/s 2 ± 2.28%)(1.100m ± 0.05%) ( 0.059m/s 2 ± 2.28%) (b) do the exponent and multiplication: t th = (0.459m/s ± 1.14%) ± (0.211m 2 /s 2 ± 2.28%) (0.130m 2 /s 2 ± 2.28%) ( 0.059m/s 2 ± 2.28%) (c) convert the errors inside the square root to absolute errors: t th = (0.459m/s ± 1.14%) ± (0.211m 2 /s 2 ± 0.005m 2 /s 2 ) (0.130m 2 /s 2 ± 0.003m 2 /s 2 ) ( 0.059m/s 2 ± 2.28%) (d) do the subtraction inside the square root: t th = (0.459m/s ± 1.14%) ± 0.081m 2 /s 2 ± 0.006m 2 /s 2 ( 0.059m/s 2 ± 2.28%)

32 KINEMATICS: CONSTANT ACCELERATION 31 (e) convert the errors back to relative errors: (f) do the square root: t th = (0.459m/s ± 1.14%) ± 0.081m 2 /s 2 ± 7.4% ( 0.059m/s 2 ± 2.28%) t th = (0.459m/s ± 1.14%) ± (0.285m/s ± 3.7%) ( 0.059m/s 2 ± 2.28%) (g) convert the errors in the numerator to absolute errors: t th = (0.459m/s ± 0.005m/s) ± (0.28m/s ± 0.01m/s) ( 0.059m/s 2 ± 2.28%) (h) do the addition/subtraction (note: you will have two possible values): t th = (0.18m/s ± 0.01m/s) ( 0.059m/s 2 ± 2.28%) OR t th = (i) convert the error in the numerator back to relative error: (0.74m/s ± 0.01m/s) ( 0.059m/s 2 ± 2.28%) t th = (j) do the division: (0.18m/s ± 5.56%) ( 0.059m/s 2 ± 2.28%) OR t th = (0.74m/s ± 1.35%) ( 0.059m/s 2 ± 2.28%) t th = 3.05s ± 6.01% OR t th = 12.5s ± 2.65% Whew! We re finished! Clearly the first t value is the one to choose since the second one is a time after the cart has stopped. Note that the resultant error is rather large (±6.01% or ±.2s). This is because of the subtraction that yielded a relatively small result although the error grew in each step. 8. Find the difference (and corresponding error) between the theoretical and measured t values. Call these values x and δx respectively. In this sample our measured value is t ex = (3.1 ± 0.2)s so the difference is: x = t ex t th = (3.12 ± 0.09)s (3.1s ± 0.2)s = (0.02 ± 0.2)s In this case, although there is a difference between the two t values, our error analysis shows that this difference is small compared to the error. For this trial, the hypothesis that the acceleration is constant is supported.

33 32 EXPERIMENT 4 9. Now calculate and record the values of v o, a, t th and x for the rest of the trials. 10. Compute the average time difference from your six trials. Also calculate the standard deviation of the time difference from your set of six trials. Sample calculation of x and σ x : If the x values are: (0.2 ± 0.4)s, (0.8 ± 0.3)s, ( 0.3 ± 0.4)s, (0.1 ± 0.4)s, (0.7 ± 0.3)s, ( 0.6 ± 0.3)s. Consider these values as x 1, x 2, x 3, x 4, x 5 and x 6 respectively. Calculate the mean value of x. We will call it x. x = 1 N N i=1 x i = 1 (0.2s + 0.8s + ( 0.3s) + 0.1s + 0.7s + ( 0.6s)) = 0.15s 6 where N is the number of trials. Estimate the standard deviation, σ x, in the set of trial values. In this calculation we don t use the calculated error values, instead we look at how much each x i value varies from x and combine this information to give us σ x. This is done using the following equation: σ x = 1 N N (x i x) 2 i=1 Plug in the values and solve: σ x = 1 6 [( )2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 ] = 1 6 (1.5s2 ) = 0.25s 2 = 0.50s So the value determined from our multiple trials is t ex t th = (0.15±0.50)s. From this we conclude that although the mean value is not zero, it differs from zero less than the standard deviation. Which means that zero is in the defined range (ie. 0.15s 0.50s = 0.35s, crosses zero). This supports the hypothesis that the acceleration of the cart remains constant. Note also that the standard deviation is roughly the same size as the errors in the individual trials.

34 KINEMATICS: CONSTANT ACCELERATION 33 Discussion and Conclusion Comment on whether your results support the hypothesis of constant acceleration. Explain. Outline and discuss at least two possible sources of error which may have occurred during the collection of the data and how they could have been avoided.

35 34 EXPERIMENT 4 Date: Data Table: Kinematics t ex δt ex (%) T δt (%) D (m) δd (%) v o (m/s) δv o n/a n/a n/a n/a n/a (m/s) a (m/s 2 ) δa n/a n/a n/a n/a n/a (m/s 2 ) t th δt th n/a n/a n/a n/a n/a x δx n/a n/a n/a n/a n/a Instructor s Name: Instructor s Signature:

36 Experiment 5 Hooke s Law and Simple Harmonic Motion Introduction Hooke s law describes the motion of a spring. This law is exactly valid only in the case of an ideal (i.e., perfectly elastic) spring. However, we can do experiments with real springs which exhibit ideal behavior over a limited range of extension. We will also study one of the most interesting types of motion in nature, the so-called simple harmonic motion. Such a motion takes place whenever the force acting on a system takes the form of a restoring force of the form given below. Examples include the oscillatory motion of a mass-spring system and the periodic motion of a pendulum. Theory Consider the mass-spring system in the following Force Diagram. The spring elastic force acting on the mass is a restoring force having the form: F = kx (5.1) where x = (x f x 0 ) is the amount of stretching undergone by the spring away from its unstretched position x 0. k is a constant characteristic of the spring called the force constant. The minus sign indicates that the tendency of the spring force is to oppose this stretching and push the system back toward the unstretched position (hence the term restoring force). Equation (5.1) is called Hooke s law, which we will apply to two cases. Static case: When the system is in equilibrium, equation (5.1) is simply a statement that the amount of extension x is proportional to the force applied to the spring which in our case is equal in magnitude to the weight of the suspended mass:

37 36 EXPERIMENT 5 F = Mg = k(x f x 0 ) = kx (5.2) Note: We have neglected the small mass of the spring compared to the suspended mass. Dynamic case: If the mass is pulled down from its equilibrium position by a small distance and released gently, then what follows is an oscillatory, periodic motion (or simple harmonic motion) wherein the block moves up and down around the equilibrium position. Such a motion is characterized by its period of oscillation T which is given by: T = 2π M k (5.3) Apparatus Hooke s Law Apparatus (Stand, Ruler, Mass Hanger) Stopwatch 2 Springs (small and large) Mass Set (1x100g, 5x20g, 1x10g)

38 HOOKE S LAW AND SIMPLE HARMONIC MOTION 37 Procedure Static Case: 1. Using the larger spring and with no mass on the hanger, measure the position of the pointer attached to the spring, x 0. Based on section B.1.2 the error in x 0, δx 0, will be 0.5 mm. Record these measurements the appropriate data table. 2. Add 10g to the mass holder whose mass is 10 g. Record the total mass acting on the spring, M = 0.020kg, in the appropriate data table. Calculate and record the force being exerted on the spring due to gravity, F = Mg. Assume the corresponding error, δf, is zero since the error in the masses is negligible. 3. Measure the pointer s new positon, x f. Since the measuring device being used is the same as that used to measure x 0, the error in x f, δx f, will be the same as δx 0, 0.5mm. 4. Calculate and record the total displacement, x = (x f x 0 ). By consulting section B.3.2, find the error in displacement, δx = (δx 0 ) 2 + (δx f ) Repeat steps 2 through 4 nine times, each time adding 10 g to the hanger and record your results in the appropriate data table. 6. Use the data collected to plot a graph of Force versus Displacement and test for linearity. Determine the best line through your points (It must intersect at least two points and have an equal number of points below and above the line) 7. Determine the value of the best slope, m best, using two points you DID NOT plot, (x 1, y 1 ) and (x 2, y 2 ) and the formula: m = rise = y 2 y 1 run x 2 x Compare the equation of the line (Y=mX+b) with equation (5.2) (force is on the y-axis, displacement is on the x-axis, etc.). 9. Based on the comparison in the previous step deduce the value of the spring constant and call it k Large. 10. Repeat steps 1 through 9 for the smaller spring and call the spring constant you deduce in this case, k Small.

39 38 EXPERIMENT 5 Dynamic Case: 11. Now you are to measure the period of oscillation of the simple harmonic motion of the mass-spring system and compare it to equation (5.3). First measure and record the mass of the larger spring, M spring. 12. You will begin by adding 100g to the 10g hanger. Record the total mass, M = 0.110kg in Data Table 2. Assume the corresponding error, δm, is zero since the error in the masses is negligible. 13. Gently set the system in motion by slightly lifting the mass and then releasing it. Try your best to obtain up-and-down oscillations with no twisting or swaying of the spring (this would complicate the motion and affect the measured values). Also, make sure that the oscillations occur inside the region where Hooke s law is applies (otherwise equation (5.3) would cease to be valid). Wait a few moments to ensure the stability of the oscillation then measure the time t for 10 complete oscillations (cycles). Assume an error in t of δt = 0.09 s. (0.09 s should be a reasonable error reflecting how accurately and how fast you are timing the start and end of the 10 cycles.) 14. Deduce the value of the period, T = t. Consult section B.3.1 to calculate the error in 10 T, δt = δt. Continue to record your data and results in Data Table Calculate and record T 2. By consulting section B.3.4 you can find its corresponding error, δt 2 = T 2 2 δt T. 16. Repeat steps 11 through 15 five times, each time adding 20g and record your results in Data Table Equation (5.3) can be rearranged algebraically to give: M = k 4π 2 T 2 (5.4) Use the data collected to plot a graph of Mass vs. Period 2 and test for linearity. Use the calculated values of δt 2 to draw horizontal error bars to scale. Determine the best line through your points (It must intersect at least two points and have an equal number of points below and above the line) and use the error bars to find the maximum (steepest) and minumum (shallowest) slopes.

40 HOOKE S LAW AND SIMPLE HARMONIC MOTION Determine the value of the best slope, m best, using two points you DID NOT plot, (x 1, y 1 ) and (x 2, y 2 ) and the formula: m = rise = y 2 y 1 run x 2 x 1. Similarly find the values of m max and m min. 19. Find the error in the slope, δm, by calculating m max -m best and m best -m min and choosing the larger value. 20. Compare the equation of the line (Y=mX+b) with equation (5.4) (mass is on the y-axis, period squared is on the x-axis, etc.). 21. Based on the comparison in the previous step extract the value of the spring constant, k = 4π 2 m. Also consult section B.3.1 to find its error, δk = 4π 2 δm. 22. In addition find and record the value of the best y-intercept, b, the maximum y-intercept, b max, and the minimum y-intercept, b min. Find δb by calculating b max -b best and b best -b min and choosing the larger value. 23. If the spring mass, M spring, is included, it can be shown that equation (5.4) should be replaced by the more accurate expression: Which can be rearranged to give: M + M s = k 4π 2 T 2 (5.5) M = k 4π 2 T 2 M s (5.6) Compare this equation (5.6) to the straight line equation and determine the experimental value of the mass of the spring, M springex = b, and its error, δm springex = δb. Discussion and Conclusion Comment on whether your results agree with Hooke s Law. Explain. Outline and discuss at least two possible sources of error which may have occurred during the collection of the data and how they could have been avoided.

41 40 EXPERIMENT 5 Data Table 1: Hooke s Law for the Larger Spring x o ± δx o = M F x f ± δx f x ± δx (kg) (N) (m) (m) Date: Instructor s Name: Instructor s Signature:

42 HOOKE S LAW AND SIMPLE HARMONIC MOTION 41 Data Table 2: Hooke s Law for the Smaller Spring x o ± δx o = M F x f ± δx f x ± δx (kg) (N) (m) (m) Data Table 3: Dynamic Case for the Larger Spring M spring = M t ± δt T ± δt T 2 ± δt 2 (kg) (s 2 ) Date: Instructor s Name: Instructor s Signature:

43 42 EXPERIMENT 5

44 Experiment 6 The Simple Pendulum Introduction Lab 5 looked at simple harmonic motion of a mass-spring system whereas in Lab 6 you will study the periodic, simple harmonic motion of a pendulum. We will use the method of dimensional analysis to construct a relationship between the period T of the pendulum and the length l of the string and experimentally test the constructed relationship using graphical analysis. Theory A simple pendulum consists of a mass M attached by a string to a fixed point. The mass will execute periodic motion along a circular arc when released from rest to one side of the vertical. The position of the mass may be given in terms of the length L of the string and the angle θ of the string as measured from the vertical (see diagram). The acceleration of the mass M is in the direction perpendicular to the line of the string.

45 44 EXPERIMENT 6 Using Newton s second law gives: for the magnitude of the acceleration a. F = Mg sin θ = Ma (6.1) Notice that we have taken the component of the weight along the direction perpendicular to the line of the string. For large angles, further analysis of the motion of the mass using equation (6.1) is complicated. However, if we restrict our investigation to small angles, we may make approximations which simplify matters. For small angles we have: where θ is in radians. sin θ = h/l s/l (6.2) Putting equation (6.2) into equation (6.1) gives us an expression for the magnitude of a: a (g/l)s (6.3) Notice that the acceleration is in the direction back towards the vertical. Whenever the acceleration is proportional and opposite to the displacement, the motion is called simple harmonic motion. This type of motion occurs for the simple pendulum when the displacement, s, is small. We can construct a relationship between the period T and the length L of the pendulum by using the method of dimensional analysis. In this method we ensure that the dimensions (units) are the same on both sides of the equation. We expect the period to depend on the acceleration due to gravity, g, because we know that no oscillations would occur if g was zero. We therefore expect a relationship of the form: T = cl α g β, (6.4) where c is a numerical constant having no dimensions, and the exponents α and β are to be found using dimensional analysis. It can be shown that, in order that equation (6.4) provide a good estimate for the period T of the simple pendulum, the magnitude of s/l should typically be no bigger than about 0.1. Apparatus Pendulum System (Stand, string and bob) Meter stick and Stopwatch

46 THE SIMPLE PENDULUM 45 Advance Preparation: It is essential that you read the Appendix on Dimensional Analysis and do steps 1. and 2. in the data section of your lab report before coming to the lab. 1. Using the method of dimensional analysis (see the Appendix), determine the values of the exponents α and β in equation (6.4). Record your equation (6.4) with these values of α and β. 2. Explain how you can test your equation using graphical analysis. What should be plotted on the y axis? What should be plotted on the x axis? Write your expression in the standard form y = mx + b. What is your expression for the slope m? Procedure 3. Spend a few moments familiarizing yourself with the apparatus and the method of adjusting the length of the string. 4. Start by adjusting the string to have a length of 20cm from the fixed point to the bottom of the mass and record this length, L, in the data table. Also record the error in the length, δl, which based on section B.1.2 will be 0.5mm. 5. Pull the mass to one side of the vertical. Make sure that the ratio s/l (see Diagram) is no bigger than 0.1. Release the mass from rest and start the stopwatch at the same instant. Allow the bob to oscillate through 10 cycles and stop the stopwatch at the instant that the mass reaches its greatest deflection from the vertical. Record your value for the total time, t 1, of the 10 cycles. 6. Repeat step 5 four times with the same length of string. Record the times for 10 cycles as t 2, t 3, t 4, and t By consulting section B.1.3 and knowing that since you made five trials N=5, calculate and record the average time of fall, t = t 1+t 2 +t 3 +t 4 +t 5, the standard deviation of your time 5 measurements, σ s.d. = (t1 t) 2 +(t 2 t) 2 +(t 3 t) 2 +(t 4 t) 2 +(t 5 t) 2, and the error in t, δt = σ s.d By dividing t by the number of oscillations in each trial it is possible to find the average period, T = t δt. Consult section B.3.1 to calculate the error in T, δt =

47 46 EXPERIMENT 6 9. Repeat steps 4 through 8 for lengths of 25, 30, 35, 40, 45, and 50 cm. Record all your values in the data table. 10. Calculate and tabulate your values for what you determined y, δy, x and δx to be in step Plot a graph of y versus x. Use the calculated values of δx and δy to draw horizontal and vertical error bars respectively. Determine the best line through your points (It must intersect at least two points and have an equal number of points below and above the line) and use the error bars to find the maximum (steepest) and minimum (shallowest) slopes. 12. Determine the value of the best slope, m best, using two points you DID NOT plot, (x 1, y 1 ) and (x 2, y 2 ) and the formula: m = rise = y 2 y 1 run x 2 x 1. Similarly find the values of m max and m min. 13. Find the error in the slope, δm, by calculating m max -m best and m best -m min and choosing the larger value. 14. Using the expression for slope you found in step 2 and your values for the slope and its error, determine the numerical value of c and its error. Find the percent difference between this value and the expected value of 2π. 15. Examine your graph, if you have been careful enough in collecting your data, you may well find that x intercept 0. After thinking about it, you will agree that the distance to the center of the mass is the correct length to use in equation (6.4). Therefore you should indeed expect that the x intercept will be nonzero. So equation (6.4) should be modified to read: T = c(l r) α g β, (6.5) where L is the length of the string and r is the radius of the weight. Measure and record the radius of the weight r, at the end of the string. 16. Find and record the value of the best x intercept, X intercept, the maximum x intercept, X max, and the minimum x intercept, X min. Find δx intercept by calculating X max -X best and X best -X min and choosing the larger value. Find the percent difference between the radius of the ball and the x-intercept.

48 THE SIMPLE PENDULUM 47 Discussion and Conclusion Comment on whether your results support the constructed relationship between period and length. Explain. Outline and discuss at least two possible sources of error which may have occurred during the collection of the data and how they could have been avoided.

49 48 EXPERIMENT 6 L (m) δl (m) t 1 t 2 t 3 t 4 t 5 t σ s.d. δt T δt Data Table: Simple Pendulum Date: Instructor s Name: Instructor s Signature:

50 Experiment 7 Rotation and Moment of Inertia Introduction This experiment examines rotational motion and the moment of inertia. We will experimentally test the theoretical relationship between the torque applied to a wheel, the acceleration of the wheel, and the moment of inertia of the wheel using the method of graphical analysis. Theory Consider a wheel and an axle attached by radial spokes. A thin cord is wound around the wheel and the hanging end is used to suspend a mass, M, as shown in the diagram. To investigate the motion of the system, we need to specify the forces acting on the mass and the torques acting on the wheel. The mass experiences a downward force, Mg, due to gravity, and an upward force, T, from the tension in the cord. The mass will accelerate downward with magnitude a according to Newton s second law: Mg T = Ma (7.1) Notice that the tension, T, must be less than the force, Mg, if the mass is to accelerate downward. Given that the mass accelerates downward, the wheel will rotate with angular acceleration, α. The tension in the cord acts downward at the edge of the wheel and thus produces a torque of magnitude, τ = RT, on the wheel, where R is the radius of the wheel. Assuming that the torque due to friction, τ f, also has constant magnitude, the magnitude of the angular acceleration, α, is given by