Embed Size (px)
Citation preview
Physics Laboratory Manual
PHYC 10180 Physics for Ag. Science
2012-2013
Name.................................................................................
Partner’s Name ................................................................
Demonstrator ...................................................................
Group ...............................................................................
Laboratory Time ...............................................................
2
3
Contents
Introduction 4
Laboratory Schedule 5
Springs 7
Newton’s Second Law 15
An Investigation of Rotational Motion 23
Superconductivity, Resistance and Temperature 33
The Viscosity of Water and the Factors Influencing Flow Rate.
43
Archimedes’ Principle and Experimental
Measurements
47
JagFit 533
4
Introduction
Physics is an experimental science. The theory that is presented in lectures has its origins in, and is validated by, experimental measurement.
The practical aspect of Physics is an integral part of the subject. The laboratory practicals take place throughout the semester in parallel to the lectures. They serve a number of purposes:
• an opportunity, as a scientist, to test theories; • a means to enrich and deepen understanding of physical concepts
presented in lectures; • an opportunity to develop experimental techniques, in particular skills
of data analysis, the understanding of experimental uncertainty, and the development of graphical visualisation of data.
Some of the experiments in the manual may appear similar to those at school, but the emphasis and expectations are likely to be different. Do not treat this manual as a ‘cooking recipe’ where you follow a prescription. Instead, understand what it is you are doing, why you are asked to plot certain quantities, and how experimental uncertainties affect your results. It is more important to understand and show your understanding in the write-ups than it is to rush through each experiment ticking the boxes. This manual includes blanks for entering most of your observations. Do not feel obliged to fill in all the blank space, they are designed to provide plenty of space Additional space is included at the end of each experiment for other relevant information. All data, observations and conclusions should be entered in this manual. Graphs may be produced by hand or electronically (details of a simple computer package are provided) and should be secured to this manual.
There will be six 2-hour practical laboratories in this module evaluated by continual assessment. Note that each laboratory is worth 5% so each laboratory session makes a significant contribution to your final mark for the module. Consequently, attendance and application during the laboratories are of the utmost importance. At the end of each laboratory session, your demonstrator will collect your work and mark it. Remember,
If you do not turn up, you will get zero for that laboratory.
You are encouraged to prepare for your lab in advance and bring
completed pre-lab assignments to the lab
5
Laboratory Schedule Depending on your timetable, you will attend on either Wednesday 2-4pm, Wednesday 4-6pm, Friday 1-3pm or Friday 3-5pm. Please consult the notice boards in the School of Physics, Blackboard, or contact the lab manager, Thomas O’Reilly (Room 1.30) to see which of the experiments you will be performing each week. This information is also summarized below.
Semester Room Dates Week Lab 127 Lab 131 Lab 132
28 Jan - 1 Feb 2 Springs: 1,4,7,10
Springs: 2,5,8,11
4 - 11 Feb 3 Springs: 3,6,9
Newton: 1,4,7,10
Rotation: 2,5,8,11
11 - 15 Feb 4 Viscosity: 2,5,8,11
Newton: 3,6,9
Rotation: 1,4,7,10
18 – 22 Feb 5 Rotation: 3,6,9
4 - 8 Mar 7 Viscosity: 3,6,9
25 - 29 Mar 8 Viscosity: 4,7
Newton : 5,8
1 – 5 Apr 9 Viscosity: 1,10
Archimedes: 4,7 Newton: 2,11
Superconductor: 5,8
8 - 12 Apr 10 Archimedes: 5,8
Archimedes: 1,6,9,10
Superconductor: 2,4,7,11
15 - 19 Apr 11 Archimedes: 11,2
Archimedes: 3
Superconductor: 6,9,10,1
22 - 26 Apr 12 Superconductor: 3
6
7
Name:___________________________ Student No: ________________ Date:___________ Demonstrator:_________________________
Springs What should I expect in this experiment? You will investigate how springs stretch when different objects are attached to them and learn about graphing scientific data. Introduction: Plotting Scientific Data
In many scientific disciplines, and particularly in physics, you will often come across plots similar to those shown here. Note some common features:
• Horizontal and vertical axes; • Axes have labels and units; • Axes have a scale; • Points with a short horizontal
and/or vertical line through them; • A curve or line superimposed.
Why do we make such plots? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
8
______________________________________________________________________ ______________________________________________________________________
______________________________________________________________________
What features of the graphs do you think are important and why? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
9
Apparatus In this experiment you will use 2 different springs, a retort stand, a ruler, a mass hanger and a number of different masses. Preparation Set up the experiment as indicated in the photographs. Use one of the two springs (call this one spring 1). Determine the initial length of the spring, be careful to pick two reference points that define the length of the spring before you attach any mass or the mass hanger to the spring. Record the value of the length of the spring below.
Think about how precisely you can determine the length of the spring. Repeat the procedure for spring 2. Calculate the initial length of spring 1. ______________________________________________________________________
Calculate the initial length of spring 2. ______________________________________________________________________
Procedure Attach spring 1 again and then attach the mass hanger and 20g onto the spring. Measure the new length of the spring, using the same two reference points that you used to determine the initial spring length. Length of springs with mass hanger attached + 20g
Spring 1 Spring 2 Position of the top of spring
Position of the bottom of spring New length of spring
10
Add another 10 gram disk to the mass hanger, determine the new length of the spring. Complete the table below. Measurements for spring 1
Object Added Total mass added to the mass hanger
(g)
New reference position (cm)
New spring length (cm)
Disk 1 30 Disk 2 40 Disk 3 Disk 4 Disk 5
Add more mass disks, one-by-one, to the mass hanger and record the new lengths in the table above. In the column labelled ‘total mass added to the mass hanger’, calculate the total mass added due to the disks. Carry out the same procedure for spring 2 and complete the table below. Measurements for spring 2
Object Added Total mass added to the mass hanger
(g)
New reference position (cm)
New spring length (cm)
Disk 1 30 Disk 2
Graphical Analysis Add the data you have gathered for both springs to the graph below. You should plot the length of the spring in centimetres on the vertical axis and the total mass added to the hanger in grams on the horizontal axis. Start the y-axis at -10cm and have the x-axis run from -40 to 100g. Choose a scale that is simple to read and expands the data so it is spread across the page. Label your axes and include other features of the graph that you consider important.
11
Graph representing the length of the two springs for different masses added to the hanger.
12
In everyday language we may use the word ‘slope’ to describe a property of a hill. For example, we may say that a hill has a steep or gentle slope. Generally, the slope tells you how much the value on the vertical axis changes for a given change on the x-axis. Algebraically a straight line can be described by y = mx + c where x and y refer to any data on the x and y axes respectively, m is the slope of the line (change in y/ change in x), and c is the intercept (where the line crosses the y-axis at x=0). Suppose the data should be consistent with straight lines. Superimpose the two best straight lines, one for each spring, that you can draw on the data points. Examine the graphs you have drawn and describe the ‘steepness’ of the slopes for spring 1 and spring 2.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Work out the slopes of the two lines.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________ What are the two intercepts? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________ How can you use the slopes of the two lines to compare the stiffness of the springs? ______________________________________________________________________
______________________________________________________________________
13
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
In this experiment the points at which the straight lines cross the y-axis where x=0g (intercepts) correspond to physical quantities that you have determined in the experiment. How well do the graphical and measured values compare with each other? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Look at the graphs on page 7, the horizontal and vertical lines through the data points represent the experimental uncertainty, or precision of measurement in many cases. In this experiment the uncertainty on the masses is insignificant, whilst those on the length measurements are related to how accurately the lengths of the springs can be determined. How large a vertical line would you consider reasonable for your length measurements? ______________________________________________________________________
What factors might explain any disagreement between the graphically determined intercepts and the measurements of the corresponding physical quantity. ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
14
15
Name:___________________________ Student No: ________________ Date:___________ Demonstrator:_________________________
Newton’s Second Law. What should I expect in this experiment? This experiment has two parts. In the first part you will apply a fixed force, vary the mass and note how the acceleration changes. In the second part you will measure the acceleration due to the force of gravity. Introduction
Newton’s second law states amF = , a force causes an acceleration and the size of the acceleration is directly proportional to the size of the force. Furthermore, the constant of proportionality is mass. Apparatus The apparatus used is shown here and consists of a cart that can travel along a low friction track. The cart has a mass of 0.5kg which can be adjusted by the addition of steel blocks each of mass 0.5kg. String, a pulley and additional masses allow forces to be applied to the carts. Take care to ensure that the track is completely level before starting the experiments
16
Investigation 1: Check that force is proportional to acceleration and show the constant of proportionality to be mass. Calculate the acceleration due to gravity.
The apparatus should be set up as in the picture. Attach one end of the string to the cart, pass it over the pulley, and add a 0.012kg mass to the hook.
The weight of the hanging mass is a force, F, that acts on the cart. The whole system (both the hanging mass mh and the cart mcart) are accelerated. So long as you don’t change the hanging mass, F will remain constant. You can then change the mass of the system, M, by adding mass to the cart, noting the change in acceleration, and testing the relationship F=ma. If F=ma and you apply a constant force F, what do you expect will happen to the acceleration as you increase the mass? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________ There remains the problem of working out the acceleration of the system. According to kinematics, distance travelled is related to the acceleration by the formula
2
21 atuts += , where s is distance, u the initial velocity, t is time, and a acceleration.
If the cart starts from rest write down an expression for ‘a’ . Place different masses on the cart. Using ruler and stopwatch measure s and t and hence the acceleration a. Fill in the table below.
17
mh (kg)
mcart (kg)
M= mh+mcart
(kg) a (m/s2)
M.a (N) 1/a (m-1s2)
0.012 0.5
s (m)
t (s)
0.012 1.0
s (m)
t (s)
0.012 1.5
s (m)
t (s)
From the numbers in the table what do you conclude about Newton’s second law? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Analysis and determination of the acceleration due to gravity. If Newton is correct, ammF carth ⋅+= )( You have varied mcart and noted how the acceleration changed so let’s rearrange the formula so that it is in the familiar linear form y=mx+c. Then you can graph your data points and work out a slope and intercept which you can interpret in a physical fashion.
Fm
mFa
ammF hcartcarth +=⇒⋅+=
11)(
So if Newton is correct and you plot mcart on the x-axis and 1/a on the y-axis you should get a straight line. In terms of the algebraic quantities above, what should the slope of the graph be equal to? In terms of the algebraic quantities above, what should the intercept of the graph equal? Plot the graph and see if Newton is right. Do you get a straight line? What value do you get for the slope? What value do you get for the intercept?
18
Use JagFit to create your graph (see back of manual) and attach your printed graph below.
19
Now here comes the power of having plotted your results like this. Although we haven’t bothered working out the force we applied using the hanging weight, a comparison of the measured slope and intercept with the predicted values will let you work out F. From the measurement of the slope, the constant force applied can be calculated to be From the intercept of the graph you can calculate F in a different fashion. What value do you get? You’ve shown that F is proportional to a and the constant of proportionality is mass. If the force is the gravitational force, it will produce an acceleration due to gravity (usually written g instead of a) so once again a (gravitational) force F is proportional to acceleration, g. But what is the constant of proportionality? Are you surprised that it is the same mass m? (By the way, a good answer to this question gets you a Nobel prize.)
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________ So since that force was just caused by the mass mh falling under gravity, F= mhg, you can calculate the acceleration due to gravity to be Comment on how this compares to the accepted value for gravity of about 9.785 m/s2. How does the accuracy of your measurements affect this comparison? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
20
Investigation 2: Acceleration down an inclined plane.
The apparatus should be set up as in the picture. Raise one end of the track using the wooden block in place of the elevator. The carts can be released from rest at the top of the track.
A cart on a slope will experience the force of gravity which causes the cart to accelerate and roll down the incline. The acceleration due to gravity is as shown in the schematic. The component of this acceleration parallel to the inclined surface is, g.sinθ and this is the net acceleration of the cart (when friction is neglected). Note the acceleration depends on the angle of the incline.
Vary the angle of the incline (repeat for at least four different angles) and measure the
acceleration of the cart using 2
21 atuts +=
How will you measure the angle of the incline? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
21
Summarise your results in this table. Angle (radians) Sine Angle Distance (m) Time (t) Acceleration (ms-2)
Since the acceleration θsinga = , there is a linear dependence on the sine of the angle. Make a graph with sin θ on the x-axis and a on the y-axis. What value do you expect (theoretically) for the slope? What value do you expect (theoretically) for the intercept? What value do you obtain (experimentally) for the slope? What value do you measure for the intercept? Comment on your results. How could you improve your measurements of g? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
22
Use JagFit to create your graph (see back of manual) and attach your printed graph below.
23
____
Name:___________________________ Student No: ________________ Date:___________ Demonstrator:_________________________
An Investigation of Rotational Motion What should I expect in this experiment? This experiment introduces you to some key concepts concerning rotational motion. These are: torque (τ), angular acceleration (α), angular velocity (ω), angular displacement (θ) and moment of inertia (I). They are the rotational analogues of force (F), acceleration (a), velocity (v), displacement (s) and mass (m), respectively. Introduction: The equations of motion with constant acceleration are similar whether for linear or rotational motion:
Linear Rotational
12
12
ttssvaverage −
−=
12
12
ttaverage −
−=
θθω
Eq.1
221 vvvaverage
+= 2
21 ωωω
+=average
Eq.2
atvv += 12 tαωω += 12 Eq.3
s2 = v1t + at2
2 θ2 =ω1t +αt2
2
Eq.4
Furthermore, just as a force is proportional to acceleration through the relationship F=ma, a net torque changes the state of a body’s (rotational) motion by causing an angular acceleration.
ατ I= (Eq.5)
The body’s moment of inertia is a measure of resistance to this change in rotational motion, just as mass is a measure of a body’s resistance to change in linear motion. The equation ατ I= is the rotational equivalent of Newton’s 2nd law maF = . You will use two pieces of apparatus to investigate these equations. The first lets you apply and calculate torque, measure angular acceleration and determine an unknown moment of inertia, I of a pair of cylindrical weights located at the ends of a bar. The second apparatus lets you investigate how I depends on the distribution of mass about
24
the axis of rotation and lets you determine the value of I, already measured in the first part, by a second method. You can then compare the results you obtained from the two methods. Investigation 1: Measuring I from the torque and angular acceleration:
Place cylindrical masses on the bar at their furthest position from the axis of rotation. The bar is attached to an axle which is free to rotate. The masses attached to the line wound around this axle are allowed to fall, causing a torque about the axle
The value of the torque caused by the falling masses is Fr=τ where F is the weight of the mass attached to the string and r is the radius of the axle to which it is attached. Calculate the value of τ.
Mass, m attached to string (kg)
Force, F = m . g (N)
Radius, r of axle (m)
Torque, τ = F × r (Nm)
Next you have to calculate the angular velocity, ω, and the angular acceleration, α. Distinguish between these two underlined terms. ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Wind the string attached to the mass around the axle until the mass is close to the pulley. Release it and measure the time for the bar to perform the first complete rotation, the second complete rotation, the third, fourth and fifth rotation. Estimate your experimental uncertainties.
25
Number of
Rotations θ (rad) Time (s)
0 0 0 1 2 3 4 5
Since you have the angular distance travelled in a given time you can use Eq. 4 to find the angular acceleration α. Explain how you can do this and find a value for α. ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
α =
26
Using Eq.1, calculate the average velocity, averageω , during each rotation. If the acceleration is constant, this will be the instantaneous velocity at a time, tmid, exactly half-way between the start and the end of that rotation. Can you see this? Explain why it is so. ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Enter the values in the table below and make a plot of averageω on the y-axis against tmid on the x-axis.
Number of
Rotation averageω (rad / s) tmid (s)
1
2
3
4
5
Since this graph gives the instantaneous velocity at a given time, Eq. 3 can be used to find the angular acceleration, α. What value do you get for α?
Now you know τ and α, so work out I from Eq. 5.
27
Use JagFit (see back of manual) to create your graph of averageω on the y-axis against tmid on the x-axis and attach your printed graph below.
28
Investigation 2: How I depends on the distribution of mass in a body Take the metal bar on the bench and roll it between your hands. Now hold it in the middle and rotate it about its centre so that the ends are moving most. Which is easier? Which way does the bar have a higher value of I?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Attach the weights and the bar to the rotational apparatus known as a torsion axle. This consists of a vertical axle connected to a spring which opposes any departure from the angle of rotational equilibrium.
Note: The apparatus is very delicate. Please take care not to strain it.
When you rotate the bar (always taking care to compress the spring, but not to over-compress it) the spring causes a torque about the axis of rotation which acts to restore the bar to the equilibrium angle. Usually the bar overshoots, causing an oscillation to occur. This is exactly analogous to the way a mass on a linear spring undergoes simple harmonic motion. The period of the oscillations, T, is determined by the restoring torque in the spring, D, and the moment of inertia, I, of the object rotating, in this case the bar and cylindrical weights. They are related by:
DIT π2=
The value of D is written on each torsion axle. Note it here.
29
To investigate the influence of mass distribution vary the position of the cylinders along the torsion bar, measure the period of oscillation, T, and use the equation above to calculate the moment of interia, I for the combined system of cylinders plus rod. (To improve the precision with which you measure T take the average over 10 oscillations.)
r – Position of the cylinders along rod (m)
T – Period of oscillation (s)
I for the combined system of cylinder +
bar (kgm-2)
I for the cylinders (kgm-2)
0.05
0.10
0.15
0.20
0.25
Just as you can simply add two masses together to get the total mass (e.g. if the mass of a cylinder is 0.24kg and the mass of a bar is 0.2kg, then the mass of cylinder plus bar is 0.44kg) you can also add moments of inertia together. Given that the moment of inertia of the bar is 0.00414 kgm-2, will let you fill in the fourth column in the table above.
Plot the value of Icylinder against the position r.
Plot the value of Icylinder against the position r2.
What do you conclude? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
30
Use JagFit to create a graph of Icylinder against the position r and attach your printed graph below.
31
Use JagFit to create a graph of Icylinder against the position r2.and attach your printed graph below.
32
The last measurement in the table above, with r=0.25m, returns the bar + cylinders system to the orientation used in your first investigation. You have thus measured the moment of inertia of the bar + cylinder system in two independent ways. Write down the two answers that you got.
I from investigation 1:
I from investigation 2:
How do the two values compare? Which is more precise?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
33
Name:___________________________ Student No: ________________ Date:___________ Demonstrator:_________________________ Superconductivity, Resistance and Temperature. This experiment is in two parts. In the first part you will investigate the relationship between resistance and temperature for a normal material. In the second part you will observe the behaviour in a superconducting material, where below a certain critical temperature, the resistance goes to zero. Precautions You will be using liquid nitrogen to cool the materials. Liquid nitrogen is at an extremely low temperature and contact with any part of your skin or body must be avoided. Exercise extreme caution. The apparatus must remain behind the safety screen. When handling the flask, wear safety gloves and safety goggles. Do not place liquid nitrogen in any vessel other than those which have been designed for it.
34
Part 1: Resistance v Temperature in normal materials You are provided with a power supply, resistor, ammeter, thermometer and a method for cooling the resistor. Write a short report detailing your investigations into the behaviour of resistance with temperature. Take data at various temperatures, graph your results and draw conclusions. In particular, extrapolate your data to lower temperatures and suggest what might happen at absolute zero (0K or –273C).
Introduction: Outline what you are going to investigate and provide any theoretical background that will allow you interpret your results.
35
Apparatus and Procedure: Describe the apparatus. In particular, draw the electrical circuit that you are using to measure the resistance. Describe the procedure (what you adjust (and how) and what you measure (and how)).
36
Temperature (K) Resistance (Ohm)
Data: Tabulate your data here.
37
Affix your graph here
38
Conclusions: What have you found out? Extrapolate your graph towards absolute zero and comment on what the value of the resistance might be there.
39
Part 2: Resistance in a superconducting material Introduction The resistance of a metal decreases as the temperature of the metal decreases. It was discovered in 1911, by the Dutch physicist Kammerlingh Onnes, that the resistance of mercury absolutely disappeared at temperatures below 4 K. (He was awarded the Nobel Prize in Physics for this discovery in 1913). This phenomenon, which he named superconductivity, allows electrical currents to circulate through the superconducting conductor without any losses. The temperature at which a particular material undergoes the transition from a normal conducting state to a superconducting state is known as the critical temperature, Tc. In 1986, a new form of high temperature superconducting material, a ceramic material, YBa2Cu3O7-Δ, was discovered, the first substance to have a critical temperature higher than the boiling point of liquid nitrogen. The Nobel prize was awarded for this breakthrough. The aim of this experiment is to observe the transition of a high temperature superconductor from its superconducting state to a non-superconducting state. This transition is observed by its effect on the inductance of a coil surrounding the sample. Theory The superconducting sample is placed within an inductance coil. When a current passes through the coil, magnetic field lines are generated and these pass through the centre of the coil and any material present within. At room temperature, the magnetic field lines pass through the sample of material which is in its ordinary state. However, as the temperature is lowered past the critical temperature (Tc) of the ceramic sample the magnetic field lines are excluded, or expelled, from the material. The effect of this flux exclusion is to change the inductance of the coil. This change in the inductance of the coil can be measured directly through measuring the current I and the voltage V, as a function of temperature. When an A/C signal at a frequency, f, is passed through the inductor, the current and voltage are related by
LRIV ω+= 2 (Eq.1)
where ω = 2πf, R is the resistance of the coil and L is the inductance of the coil. Eq.1 is similar to Ohm’s law but it includes an extra term due to the energy required to set up or shut down the magnetic fields associated with the coil. The idea in this experiment is to vary the temperature and calculate L by measuring V and I. However, there are two unknowns in Eq.1: R and L. If however we first measure V and I at a given temperature for a D/C signal (where ω =0), then R can be determined. A second pass with the A/C signal allows L to be found.
40
Procedure The inductance coil with the sample of the high-Tc material within it, is buried within a styrofoam container with sand. To cool the coil and sample, liquid nitrogen is poured into the container. The temperature of the coil and sample is measured by means of a thermocouple, the output of which is displayed on a meter. When the liquid nitrogen is poured in, the coil and sample are cooled very rapidly and gradually over a period of some minutes their temperature will increase, ultimately returning to room temperature.
You have already recorded the resistance R, of the coil over the temperature range that will be used in this part of the experiment. The coil itself does not show superconducting properties in this temperature region; however the special material inside it should. In this part of the experiment you cool the coil and sample again with liquid nitrogen. This time, as the apparatus warms up, the frequency generator is used to pass an alternating sinusoidal signal at a frequency of 1 kHz through the induction coil. The voltage, V, across the induction coil and the current, I, passing through the circuit are measured. Enter the data in the table on the next page. To increase your precision on estimating Tc, it is useful to take data points at smaller temperature intervals in the region above and below the transition point The fourth column (Inductance) can be calculated using the graph you have already produced. For each temperature reading, estimate the resistance of the coil from your graph; then calculate the inductance using Eq.1.
41
Temperature (K) Voltage (V) Current (I) Inductance (H)
42
Graph the Inductance, L, as a function of Temperature T. The change from the superconducting state to the normal state should be clearly seen in this graph. Using the graph, estimate Tc for the superconductor. How precisely can you determine
this temperature?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
43
Name:___________________________ Student No: ________________ Date:___________ Demonstrator:_________________________
The Viscosity of Water and the Factors Influencing Flow Rate. Theory When a fluid moves, frictional forces, known as viscous forces, are brought into play which tend to reduce its velocity. In the case of a liquid flowing down a tube, provided that the pressure difference between the ends of the tube is not too great, a regular type of motion results called laminar flow with the velocity increasing linearly from zero at the wall to a maximum value at the axis. The viscous flow in a tube of small bore is described by Poiseuille's equation:
ηπlprQ8
4
= (Eq.1)
where Q is the rate of flow (volume of fluid per second), p is the pressure difference between the ends of the tube, r is the bore radius of the tube, l is the length of the tube and η is the coefficient of viscosity.
Apparatus The apparatus consists of a five litre jar of water connected to a set of four capillary tubes through which the water can flow into a collecting beaker. The pressure difference is measured through two vertical tubes connected to the input and output of the capillaries.
Procedure Set the pressure difference to a constant value by adjusting the height of the large container. The pressure difference is proportional to the difference in height between the levels of water in the two vertical tubes when the water is flowing. The height difference is then converted to Pascals using p = ρ g h where ρ is the density of water = 103 kg m-3, g is the acceleration due to gravity, and h is the height difference of the water in the two tubes in meters. For each tube, one at a time, calculate the flow rate, Q, by noting the time taken for the beaker to collect a certain volume of water. (Read the volume from the side of the beaker or measure its mass and divide by the density.) Fill in the table below.
44
Precautions • Ensure that the tubes are kept horizontal. • Keep a constant eye on the pressure differences before and after readings to ensure
that they do not vary. • Take all readings over as short an interval of time as possible, to avoid changes in
temperature, since viscosity also has a dependence on temperature. r, Radius of tube
(m) 0.597 x10-3 0.575 x10-3 0.377 x10-3 0.321 x10-3
ln (r)
Volume of water (m3)
Time taken (s)
Q, Flow rate (m3/s)
Height difference (m)
p, Pressure (kg m-1 s-2 )
l, tube length (m)
ln(Q/p)
Make a graph of ln(Q/p) against ln(r). By taking the natural logarithm of both sides of Eq.1, figure out what quantities the slope and intercept of the graph represent. ____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
Find a value for the viscosity, η , and comment on whether your data is consistent with the hypothesis that the viscosity increases as the fourth power of the tube radius. ____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
45
To verify that the flow rate depends linearly on the pressure, choose one of the pipes.
Make a set of measurements varying the pressure and recording the flow rate
Radius of tube (m) l, tube length (m)
Height difference
(m)
p, Pressure
(kg m-1 s-2 )
Volume of water
x10-5 (m3)
Time taken (s)
Q, Flow rate
x10-8 (m3/s)
Graph Q against p.
Does your data support the hypothesis that the flow rate varies linearly with pressure?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
From the slope of the graph, calculate a value for the viscosity. Does this agree with the value you previously determined? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
46
47
Name:___________________________ Student No: ________________ Date:___________ Demonstrator:_________________________
Archimedes’ Principle and Experimental
Measurements. Introduction All the technology we take for granted today, from electricity to motor cars, from television to X-rays, would not have been possible without a fundamental change, around the time of the Renaissance, to the way people questioned and reflected upon their world. Before this time, great theories existed about what made up our universe and the forces at play there. However, these theories were potentially flawed since they were never tested. As an example, it was accepted that heavy objects fall faster than light objects – a reasonable theory. However it wasn’t until Galileo1 performed an experiment and dropped two rocks from the top of the Leaning Tower of Pisa that the theory was shown to be false. Scientific knowledge has advanced since then precisely because of the cycle of theory and experiment. It is essential that every theory or hypothesis be tested in order to determine its veracity. Physics is an experimental science. The theory that you study in lectures is derived from, and tested by experiment. Therefore in order to prove (or disprove!) the theories you have studied, you will perform various experiments in the practical laboratories. First though, we have to think a little about what it means to say that your experiment confirms or rejects the theoretical hypothesis. Let’s suppose you are measuring the acceleration due to gravity and you know that at sea level theory and previous experiments have measured a constant value of g=9.81m/s2. Say your experiment gives a value of g=10 m/s2. Would you claim the theory is wrong? Would you assume you had done the experiment incorrectly? Or might the two differing values be compatible? What do you think? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
1 Actually, the story is probably apocryphal. However 11 years before Galileo was born, a similar experiment was published by Benedetti Giambattista in 1553.
48
Experimental Uncertainties When you make a scientific measurement there is some ‘true’ value that you are trying to estimate and your equipment has some intrinsic uncertainty. Thus you can only estimate the ‘true’ value up to the uncertainty inherent within your method or your equpiment. Conventionally you write down your measurement followed by the symbol ± , followed by the uncertainty. A surveying company might report their results as
5245 ± km, 1253± km, 1.02.254 ± km. You can interpret the second number as the ‘margin of error’ or the uncertainty on the measurement. If your uncertainties can be described using a Gaussian distribution, (which is true most of the time), then the true value lies within one or two units of uncertainty from the measured value. There is only a 5% chance that the true value is greater than two units of uncertainty away, and a 1% chance that it is greater than three units. Errors may be divided into two classes, systematic and random. A systematic error is one which is constant throughout a set of readings. A random error is one which varies and which is equally likely to be positive or negative. Random errors are always present in an experiment and in the absence of systematic errors cause successive readings to spread about the true value of the quantity. If in addition a systematic error is present, the spread is not about the true value but about some displaced value. Estimating the experimental uncertainty is at least as important as getting the central value, since it determines the range in which the truth lies. Practical Example: Now let’s put this to use by making some very simple measurements in the lab. We’re going to do about the simplest thing possible and measure the volume of a cylinder using three different techniques. You should compare these techniques and comment on your results. Method 1: Using a ruler The volume of a cylinder is given by πr2h where r is the radius of the cylinder and h its height. Measure and write down the height of the cylinder. Don’t forget to include the uncertainty and the units. Measure and write down the diameter of the cylinder.
h= ±
d= ±
49
Now calculate the radius. (Think about what happens to the uncertainty) Calculate the radius squared – with it’s uncertainty! One general way of determining the uncertainty in a quantity f calculated from others is to use the following method..
• From your measurements, calculate the final result. Call this f , your answer. • Now move the value of the source up by its uncertainty. Recalculate the final result. Call this +f . • The uncertainty on the final result is the difference in these values.
Finally work out the volume. Estimate how well you can determine the volume.
r= ±
V= ±
r2 = ±
50
Method 2: Using a micrometer screw This uses the same prescription. However your precision should be a lot better. Measure and write down the height of the cylinder. Don’t forget to include the uncertainty and the units. Measure and write down the diameter of the cylinder. Now calculate the radius. Calculate the radius squared – with it’s uncertainty! Finally work out the volume.
h= ±
d= ±
r= ±
V= ±
(Show your workings)
r2 = ±
51
Method 3: Using Archimedes’ Principle You’ve heard the story about the ‘Eureka’ moment when Archimedes dashed naked through the streets having realised that an object submerged in water will displace an equivalent volume of water. You will repeat his experiment (the displacement part at least) by immersing the cylinder in water and working out the volume of water displaced. You can find this volume by measuring the mass of water and noting that a volume of 0.001m3 of water has a mass2 of 1kg. Write down the mass of water displaced. Calculate the volume of water displaced. What is the volume of the cylinder? Discussion and Conclusions. Summarise your results, writing down the volume of the cylinder as found from each method. Comment on how well they agree, taking account of the uncertainties. _____________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
2 In fact this is how the metric units are related. A litre of liquid is that quantity that fits into a cube of side 0.1m and a litre of water has a mass of 1kg.
±
±
±
± ± ±
52
Can you think of any systematic uncertainties that should be considered? Can you estimate their size? ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
Requote your results including the systematic uncertainties. What do you think the volume of the cylinder is? and why? My best estimate of the volume is because ______________________________________________________________ ______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
± ± ± ±
± ±
± ±
53
Using JagFit
In the examples above we have somewhat causally referred to the ‘best fit’ through the data. What we mean by this, is the theoretical curve which comes closest to the data points having due regard for the experimental uncertainties. This is more or less what you tried to do by eye, but how could you tell that you indeed did have the best fit and what method did you use to work out statistical uncertainties on the slope and intercept? The theoretical curve which comes closest to the data points having due regard for the experimental uncertainties can be defined more rigorously3 and the mathematical definition in the footnote allows you to calculate explicitly what the best fit would be for a given data set and theoretical model. However, the mathematics is tricky and tedious, as is drawing plots by hand and for that reason.... We can use a computer to speed up the plotting of experimental data and to improve the precision of parameter estimation.
In the laboratories a plotting programme called Jagfit is installed on the computers. Jagfit is freely available for download from this address: http://www.southalabama.edu/physics/software/software.htm
Double-click on the JagFit icon to start the program. The working of JagFit is fairly intuitive. Enter your data in the columns on the left.
• Under Graph, select the columns to graph, and the name for the axes. • Under Error Method, you can include uncertainties on the points. • Under Tools, you can fit the data using a function as defined under
Fitting_Function. Normally you will just perform a linear fit.
3 If you want to know more about this equation, why it works, or how to solve it, ask your demonstrator or read about ‘least square fitting’ in a text book on data analysis or statistics.
