PHYSICS 126
ERROR AND UNCERTAINTY
In Physics, like every other experimental science, one cannot make any measurement
without having some degree of uncertainty. In reporting the results of an experiment, it is
as essential to give the uncertainty, as it is to give the best-measured value. Thus it is
necessary to learn the techniques for estimating this uncertainty. Although there are
powerful formal tools for this, simple methods will suffice for us. To large extent, we
emphasize a "common sense" approach based on asking ourselves just how much any
measured quantity in our experiments could be in error.
A frequent misconception is that the experimental error is the difference between our
measurement and the accepted "official" value. What we mean by error is the estimate of
the range of values within which the true value of a quantity is likely to lie. This range is
determined from what we know about our lab instruments and methods. It is conventional
to choose the error range as that which would comprise 68% of the results if we were to
repeat the measurement a very large number of times. In fact, we seldom make the many
repeated measurements, so the error is usually an estimate of this range. But note that the
error range is established so as to include most of the likely outcomes, but not all. You
might think of the process as a wager: pick the range so that if you bet on the outcome
being within your error range, you will be right about 2/3 of the time. If you
underestimate the error, you will lose money in your betting; if you overestimate it, no
one will take your bet!
Error: If we denote a quantity that is determined in an experiment as X, we can call the
error ΔX. Thus if X represents the length of a book measured with a meter stick we might
say the length l = 25.1 ± 0.1 cm, where the central value for the length is 25.1 cm and the
error, Δl is 0.1 cm. Both central value and error of measurements must be quoted in your
lab writeups. Note that in this example, the central value is given with just three
significant figures. Do not write significant figures beyond the first digit of the error on
the quantity. Giving more precision to a value than this is misleading and irrelevant.
An error such as that quoted above for the book length is called the absolute error; it has
the same units as the quantity itself (cm in the example) .We will also encounter relative
error, defined as the ratio of the error to the central value of the quantity. Thus the
relative error on the book length is Δl/l = (0.1/25.1) = 0.004. The relative error is
dimensionless, and should be quoted with as many significant figures as are known for
the absolute error.
Random Error:
Random error occurs because of sma.l1 random variations in the
measurement process. Measuring the time of a pendulum's period with a stopwatch will
give different results in repeated trials due to small differences in your reaction time in
hitting the stop button as the pendulum reaches the end point of its swing. If this error is
random, the average period over the individual measurements would get closer to the
correct value as the number of trials is increased. The correct reported result would be the
average for our central value and the error (usually taken as the standard deviation of the
measurements). In practice, we seldom take the trouble to make a very large number of
measurements of a quantity in this lab; a simple approximation is to take a few (3- 5)
measurements and to estimate the range required to encompass about 2/3 of the results.
We would then quote one half of this total range for the error, since the error is given for
'plus' or 'minus' variations. In the case that we only have one measurement, even this
simple procedure won't work; in this cue you must guess the likely variation from the
character of your measuring equipment. For example in the book length measurement
with a meter stick marked off in millimeters, you might guess that the error would be
about the size of the smallest division on the meter stick (0.1 cm).
Systematic Error: Some sources of uncertainty are not random. For example, if the meter
stick that you used to measure the book was warped or stretched, you would never get a
good value with that instrument. More subtly, the length of your meter stick might vary
with temperature and thus be good at the temperature for which it was calibrated, but not
others. When using electronic instruments such 1.5 voltmeters and ammeters, you
obviously rely on the proper calibration of these devices. But if the student before you
dropped the meter, there could well be a. systematic error. Estimating possible errors due
to such systematic effects really depends on your understanding of your apparatus and the
skill you have developed for thinking about possible problems. For example if you
suspect a meter might be mis-calibrated, you could compare your instrument with a
'standard' meter -but of course you have to think of this possibility yourself and take the
trouble to do the comparison. In this course, you should at least consider such systematic
effects, but for the most part you will simply make the assumption that the systematic
errors are small. However, if you get a va.1ue for some quantity that seems rather far off
what you expect, you should think about such possible sources more carefully.
Propagation or Errors:
Often in the lab, you need to combine two or more
measured quantities, each of which has an error, to get a derived quantity. For example, if
you wanted to know the perimeter of a rectangular field and measured the length l and
width w with a tape measure, you would have then to calculate the perimeter,
p = 2 x (l + w), and would need to get the error on p from the errors you estimated on l
and w, Δl and Δw. Similarly, if you wanted to calculate the area of the field, A = lw, you
would need to know how to do this using Δl and Δw.
There are simple rules for calculating errors of such combined, or derived, quantities.
Suppose that you have made primary measurements of quantities A and B, and want to
get the best value and error for some derived quantity S. For addition or subtraction of
measured quantities:
If S = A + B, then ΔS = ΔA + ΔB.
If S = A- B, then ΔS = ΔA + ΔB (also).
(Actually, these rules are simplifications, since it is possible for random errors (equally
likely to be positive or negative) to partly cancel each other in the error ΔS. A more
refined rule that requires a little more calculation for either S = A ± B is:
This square root or the sum of squares is called 'addition in quadrature'.)
For multiplication or division of measured quantities:
If S = A x B or S = A/B, then the simple rule is ΔS/S = (ΔA/A) + (ΔB/B).
Note that for multiplication or division, it is the relative errors in A and B which are
added to that errors in A and B may partly compensate:
For the case that S = A2 (or An), the error is ΔS/ S = 2(ΔA/A) (or ΔS/S= n(ΔA/A)).
As an example, suppose you measure quantities A, B and C and estimate their errors ΔA,
ΔB and ΔC and calculate a quantity S = C + πA2/B. You should be able to derive from
the above rules that:
ΔS = (πA2/B)[2(ΔA/A) + (ΔB/B)] + ΔC
(Hint: 'π' is a known constant with no error, and it may be helpful to decompose S as S =
P + C, with P = πA2/B.) As before, the more refined calculation replaces the additions in
this formula with addition in quadrature.
Obtaining values from graphs: Often you will be asked to graph results obtained in the
lab and to find certain quantities from the slope of the graph. An example would occur
for a measurement of the distance l traveled by an air glider on a track in time t; from the
primary measurement of l and t, you might plot 2l (as y coordinate) vs. t2 (as x
coordinate). In constructing the graph, plot a point at the central x-y value for each of
your measurements, Small horizontal bars should be drawn whose length is the absolute
error in x (here t2), and vertical bars with length equal to the error in y (here 2l). The full
lengths of the bars are twice the errors since the measurement may be off in a positive or
negative direction. The slope of the graph will be the glider acceleration (remember,
l=(1/2)at2). You can determine this slope by drawing the straight line which passes as
close to possible to the center of your points; such a line is not however expected to pass
through all the points the best you can expect is that it passes through the error bars of
most (about 2/3) of the points, In getting the numerical value of the slope of your line,
you must of course take account of the scale (and units) of your x- and y-axes.
You can get the error on the slope (acceleration) from the graph as well. Besides the line
you drew to best represent your points, draw straight lines which pass near the upper or
lower ends of most of your error bars (again, on average), These maximum and minimum
slopes will bracket your best slope. Take the difference between maximum and minimum
slopes as twice the error on your slope (acceleration) value.
Fall 2007 PHY126 Experiment 1
FLUID FLOW
Purpose:
The purpose of this laboratory is to study some aspects of viscous fluid flow and to see
demonstrations of fluid flow and static’s, which can be interpreted by the Bernoulli
equation.
Part 1 - Viscous Fluid Flow:
According to Poiseuille's law, the viscous fluid flow Q (see text) through a tube is proportional to the pressure difference P across the tube, so that P = RQ. The resistance to
flow is R = 8ηl /πr4 where l is the tube length, r is the radius of the tube, and η is the
fluid viscosity. The applicability of this law to water flow through glass capillary tubes is
investigated here. The apparatus is very simple. It is sketched below.
Cup A
Overflow Tube
Cup B
h
Overflow Tube
Capillary
Calibrated Beaker
The capillary is connected to the bottom of cups A and B. Cup A is positioned a
height h above cup B provided the capillary is horizontal, and that the water in cup A and
B is up to the overflow tubes, then the pressure difference on the capillary is just A = ρgh
(ρ is the density of water). As water flows from A to B it will collect in the calibrated
beaker. By measuring the collection time, the flow rate Q can be computed. It is
important to keep adding water to cup A so that the water level stays at the top of the
overflow tube.
Procedure:
1. Connect the capillary tube to the cups. Add water to A and pinch the rubber connecting tubes until the air bubbles are removed - and water commences to flow.
2. You must time the water flow in order to relate the amount of water accumulated in
the beaker to flow rate Q. Do this as follows: Put the end of your finger over the hole
in cup A so that the water cannot flow through the capillary; empty the calibrated
beaker and get it set up; start the timer and remove your finger; add water to cup A as
needed to keep the level constant. Do this until a few minutes have elapsed and/or at
least 50 ml have collected in the beaker.
State University of New York at Stony Brook
3. Take measurements on two different capillaries with 1 mm and 2 mm diameters. For
each tube, vary h to have at least 4 different values over the range from 5 cm to 30 cm.
4. For each tube, compute R = Δp/Q for each value of h.
5. Now compute R for each tube from the equation given above.
6. Next take the two tubes and hook them first in series and then in parallel, as shown
below. Take new measurements for these combinations. Compute the effective resis
tance of the two combinations. You should find R(series) = R1+R2 and R(parallel) =
R1R2/(R,I+R2)
R1
Series
R1
R2
R2
Parallel
Questions:
1. Why is p = ρgh?
2. Within the estimated precision of your experiments, do you find: A) that R. is constant
for a given capillary?, B) that R(exp.) equals R(predicted)?, and C) that R(series) and
R(parallel) are given by the above formula? If not, give some possible systematic
uncertainties.
Part 2 - Bernoulli Effects, etc.:
In this part of the laboratory, you are asked to comment or make computations on the
exhibits.
Exhibit I: A container of water has holes iii the sides. Why does the water from hole
A not go as far as the water from hole C? Give a quantitative explanation.
A
B
C
Exhibit II: There is a Venturi tube apparatus for study. Give a qualitative description
and explanation of what you see.
Exhibit III: Hydraulic jacks are very useful for lifting heavy objects by application
of relatively small forces. Study the model jack setup in the laboratory. Push on both
sides so that you feel the jack action. What is the principle?
Fall 2007 PHY126 Experiment 2
THERMAL EXPANSION
In this experiment, the coefficients of linear expansion of copper and
aluminum will be measured. When a metal rod of length l has its
temperature changed by ΔT, its length changes by an amount Δl. For small
changes in temperature, the fractional change in length is proportional to the
temperature change. The proportionality constant, α, is the coefficient of
linear expansion as shown in the equation below:
Δl
= α ⋅ ΔT
l
By measuring l, Δl, and ΔT, we will determine the coefficient of linear
expansion, α.
A. Equipment
Steam generator, thermometer, battery, voltmeter, “heat tube” with
micrometer.
B. Method
A rod of either copper or aluminum rests inside a hollow metal tube (see
figure below). The tube and rod start at room temperature. A steam
generator can be used to fill the tube with steam and thereby raise the
temperature of the rod.
steam
Thermometer
Micrometer
Heat Tube
Voltmeter
Battery
steam
At each end of the rod is an electrical contact. If the circuit is complete, the
voltmeter will read the voltage of the battery. If contact is broken, the
voltmeter will read zero. One of the contacts to the rod is made through a
small micrometer. By measuring the micrometer settings at which electrical
contact is just made for a cool and hot rod, we can measure the change in
length of the rod for a known temperature change.
C. Procedure
1. Your apparatus will either contain a copper or aluminum rod.
Measure the length of the rod with a meter stick and the starting
temperature.
2. Turn the micrometer screw gently back and forth while observing the
voltmeter. Find the point at which contact is just barely made and record the
micrometer setting.
3. Repeat step #2 several times so that the random error can be
estimated.
4. Back the micrometer screw well away from the end of the rod to
allow for expansion while heating.
5. Pass steam through the “heat tube” and watch the temperature of the
rod as it rises. Wait until the temperature reading has been stable for several
minutes. Record the temperature.
6. Using the same procedure as above determine the new position of the
end of the rod (and its uncertainty).
7. Calculate the coefficient of linear expansion of the rod.
Q1. Does your value of α agree within error with that in the textbook ?
8. Change places with another group and repeat the measurements on the
other kind of metal rod.
Q2. Compare your result to the accepted value for α.
Fall 2007 PHY126 Experiment 3
THE MECHANICAL EQUIVALENT OF HEAT
Historically, the relationship between heat flow into a material and its
resulting temperature change was deduced prior to mankind’s understanding
of heat as a form of energy. A unit of heat (the calorie) was invented to
quantify heat flow. A calorie is defined as that amount of heat necessary to
raise one gram of water by one degree Celsius.
The equivalence of heat energy and mechanical energy was deduced by
measuring the amount heat created when an object undergoes a known
amount of work due to a non-conservative force. We will use this technique
to measure the proportionality constant between the old heat unit (the
calorie) and the MKS energy unit (the Joule).
A. Equipment
Mechanical equivalent of heat apparatus, water, stir rod, thermometer.
B. Method
A schematic diagram of the apparatus is shown below. The inner brass cup
is partly filled with water. The outer brass cup is connected to a crank
handle and turned about the stationary inner cup. The work done by the
frictional force is converted into heat.
Thermometer
Water filled brass
cup (stationary)
Brass cup
(spins)
Since the inner cup is in equilibrium, the torque due to friction is balanced
by the torque due to the external force applied to the edge of the aluminum
disk. Since work is equal to torque × angle, we find:
W = τ × θ = FR 2πN = 2πR FN
(1)
where R is the radius of the disk, F is the force applied to keep the inner cup
stationary, and N is the number of turns of the outer cup.
We can analyze the amount of heat input to the system by measuring the
change in temperature. In general, Q = mcΔT for a single material which
does not undergo a phase change. In our case, several pieces of the
apparatus are heated at the same time. The total heat input is:
Q = mwcwΔT + mbcbΔT + mscsΔT + mthcthΔT
(2)
where w represents the water, b the brass pieces, s the stir rod, and th the
thermometer. Using the known specific heats of each substance (in
CALORIES/kg K), we can determine the heat flow in calories and compare
this result to the work done (in Joules).
C. Procedure
1. Disassemble the apparatus and measure the mass of each relevant
part. The mass of the water can be deduced by weighing the inner cup with
and without the water.
2. Reassemble the device with the inner cup 3/4 filled with cold water
(roughly 6-8˚ C below room temperature). Stir the water and measure the
starting temperature. Place a scrap of paper between the inner and outer cup
during reassembly so that the crank will turn smoothly.
3. The crank handle is attached to a counter so that the number of turns
can be measured. Record the starting value of the counter and begin to
crank.
4. The force necessary to keep the inner cup stationary is read from a
spring balance. Record its value. Try to keep the force steady by turning the
crank very smoothly and continuously.
5. While one partner cranks, the other should record the temperature of
the water. DO NOT Stop cranking the device during these “spot-check”
temperature measurements. If fatigue should set in, the lab partners should
switch jobs.
6. Continue to crank the apparatus until the final temperature is roughly
as far above room temp as the initial T was below it. After cranking is
finished, record the new reading on the counter.
7. Take several temperature measurements even after the cranking is
finished since the T value will continue to rise for a short time.
8. Calculate the the work in Joules from equation (1) and the heat in
calories from equation (2). Determine the ratio, W/Q and compare it to the
accepted value.
Q1. Why is it best to start the experiment below room temp and end above?
Q2. Why is it necessary to stir the water before each temperature measurement?
Q3. Why is there a residual upward drift in temperature after the cranking is
stopped?
Q4. It takes about 100,000 Joules to toast break. Compare this amount to the
mechanical work you did during the experiment.
Fall 2007 PHY126 Experiment 4
THE QUEST FOR ABSOLUTE ZERO
The kinetic theory of gases predicts that an ideal gas will obey the relation
pV = nRT
(1)
where p is the pressure in Pascals, V is the volume in m3, n is the number of
moles of gas, R is the gas constant (8.31J/mol-K), and T is the temperature
in K. When V and n are kept constant, we see that equation (1) shows a
linear relation between p and T. Using the pressure and temperature
characteristics of air, we will estimate the absolute zero of temperature on
the Celcius scale.
A. Equipment
Hot plate, beaker, aluminum gas cell, large graduated cylinder with
reservoir, pressure transducer, power supply, digital voltmeter, ice water.
B. Method
The pressure transducer is a very clever device using piezoelectric crystals.
It outputs an electric voltage proportional to the applied pressure.
Piezoelectric crystals produce an electric voltage across their faces when
squeezed or stretched. By orienting a number of them carefully inside the
transducer, one can measure the applied gas pressure from the produced
output voltage. The transducers are calibrated so that they will output 0
volts at 1 atm. The linearity of the transducers will be checked over a small
range by changing the height of a column of water pressing on the air in the
connecting tube. Next, the tube will be evacuated by attaching it to a
vacuum pump, and the voltage output will be compared with the expected
value. Finally, the pressure of the gas in the aluminum gas cell will be
measured after placing the aluminum gas cell into boiling water and ice
water, at temperatures of 100oC and 0oC respectively.
C. Procedure
I. Measuring the linearity of the pressure transducer.
1. Attach a pressure transducer to the T-connector at the bottom of the
large graduated cylinder and open the clamp.
2. By moving the can up and down, you can change the level of the
water in the graduated cylinder. Measure the transducer’s voltage output at
4 different water levels.
3. Using the relation P = P0 + ρgh, calculate the pressure at the
transducer for each height of water used in step #2.
4. Graph the voltage of the transducer vs. the pressure of the water.
Q1. Is the graph linear? Calculate the slope of the graph.
Q2. What can you say about the linearity of the transducer?
II. Zero pressure
1. Record the output voltage of a dry transducer when it is open to
atmosphere.
2. Attach the transducer to the vacuum pump. Open the valve. Wait
several minutes for the transducer’s output voltage to stabilize to ensure an
accurate result. Record the final voltage in your notebook.
3. Your results from #1 and #2 determine the output voltage of the
transducer at 1 atm and 0 atm of pressure. Write a linear equation which
expresses the pressure at the transducer as a function of the output voltage
(i.e. Pressure = Const1 * Voltage + Const2). This equation represents the
calibration of your transducer.
III. Pressure thermometry
1. The pressure inside the aluminum gas cell will now be measured at
0oC and 100oC. Attach the pressure transducer to the gas cell and immerse
the gas cell in the boiling water. Caution: Boiling water is hot. Proceed
with caution! Avoid making any sudden moves that might knock over
the apparatus.
2. Record the voltage after waiting a few minutes for the gas to reach the
temperature of the water. Calculate, using your calibration equation, the
pressure at this temperature.
3. Next, put the gas cell into the bucket of ice water and wait for it to
come to equilibrium. Record the voltage. Calculate the pressure.
4. Graph the pressure versus temperature for the points at 100˚C and
0˚C.
5. Extend the line in your graph to pressure = 0. The temperature at this
point should be absolute zero.
Q3. At what temperature (in ˚C) would the pressure go to zero?
Q4. How does this compare to the expected value of absolute zero?
Fall 2007 PHY126 Experiment 5
SIMPLE HARMONIC MOTION
The phenomenon of simple harmonic motion will be studied for masses on
springs and suspended pendulums. Such motion occurs in any system for
which the force exerted on a mass is linearly proportional (by a negative
constant) to its displacement from equilibrium. The relationship may be
written in general as F = -mω2x, where m is the mass, ω is the angular
frequency, and x is the displacement from equilibrium.
A. Equipment
1 air track, 1 glider with “flag”, 1 photo-gate and timer, audio tape, small
masses, 1 simple pendulum.
B. Method
Throughout the experiment, a system (either a glider between springs, or a
pendulum) will be displaced from equilibrium. The period, τ, of the
resulting motion will be measured. The dependence of the period on the
amplitude of the motion will be assessed. For true simple harmonic motion,
there should be no dependence of the period on the amplitude.
C. Procedure
I. Measurement of the Spring Constant.
1. Your glider is held between two springs. Record its equilibrium
position.
2. Attach a piece of audio tape to the glider and lay it across the “air
pulley” with a small mass suspended on the end of the tape.
3. Measure the displacement of the glider from equilibrium for 4
different hanging masses.
4. Graph the weight of the hanging mass (y axis) vs. the measured
displacement.
Q1. Is your weight-displacement curve linear? Comment.
5. Determine the spring constant, k, from your graph.
II. Measurement of the Period of the Motion.
1. Remove the audio tape and place the photo-gate so that it is centered
over the flag on the glider when the glider is at equilibrium. Measure the
mass of the glider.
2. Set the Pasco Timer to “two-pulse” mode. This will measure the time
interval between two successive times at which the flag blocks the photogate.
3. Start the system oscillating. Measure the time between successive
blockages of the photo-gate using the Pasco timer. This time is 1/2 of the
system period.
Q2. Present an argument proving that the timer measures only 1/2 of the system
period.
4. Measure the 1/2 period several times using the “reset” button on the
timer.
5. Compare your results to the theoretical value τ = √(m/k).
6. Repeat the measurement for 2 other masses by taping extra mass to
your glider.
III. Maximum velocity and position.
1. Measure and record the width of the flag atop your glider.
2. Set the Pasco Timer to pulse width mode. In this mode the timer will
measure the time interval during which the light is blocked.
3. Start the oscillator by pulling it back a measured distance from
equilibrium and releasing.
4. Record the time interval for which the gate is blocked and calculate
from this the glider’s velocity as it passes through equilibrium.
5. Check conservation of energy by calculating both 1/2kA2 (maximum
potential energy) and 1/2 mvmax2 (maximum kinetic energy)
Q3. Do the energies measured in part 5 agree within error?
6. Repeat the measurement for several amplitudes.
IV. Simple Pendulum.
1. Measure and record the length of the simple pendulum.
2. Place the timer in two pulse mode. Operate the timer manually and
measure the time required for 20 swings. Repeat several times.
3. Compare the measured period to the theoretical value. Repeat for a
different length of the pendulum.
4. Measure the period as a function of amplitude of the swing.
Q4. It it independent? Why or why not?
Fall 2007 PHY126 Experiment 6
STANDING WAVES
In this experiment, standing waves will be observed in a vibrating string.
The wavelengths of the waves and the tension in the string will be measured.
From these measurements and the frequency of the wave, the mass per unit
length of the string will be determined.
The velocity of transverse traveling waves in a stretched string is given by
v= Tμ
where T is the tension in the string and μ is the mass per unit length of the
string. The wavelength of the wave is then
λ=
v 1 T
=
μ
f f
where f is the frequency of the wave.
In your apparatus, a 60 Hz traveling wave is generated by a vibrating reed
attached to one end of the string. The tension in the string is generated by a
weight hung from the other end of the string. (See Fig. 1) The traveling
wave is reflected back along the string at the pulley, and this reflected wave
is reflected again at the vibrator. When the second reflected wave is in
phase with the original wave, a standing wave pattern will be observed in the
string. The distance between nodes is one-half the wavelength of the wave.
1. Place a 100 g mass on the end of the string. Adjust the position of the
vibrating reed until a stable, standing wave pattern is observed. Measure the
wavelength λ of the wave, and estimate the uncertainty.
2. Try to make standing wave patterns for several different positions of
the reed (each different by 1/2 wavelength).
3. Repeat the above procedures using at least three other masses
Plot λ2 as a function of T, the tension of the string, and from the slope of
this curve (and your knowlege of the frequency!), determine μ, the mass per
unit length of the string. Estimate the uncertainty in this result. Compare
with the mass per unit length determined by weighing a known length of
string.
4. Repeat the procedure using a piece of copper wire in place of the
string.
Q1. What do you expect to change?
Fall 2007 PHY126 Experiment 7
THE VELOCITY OF SOUND
In this experiment we will determine the velocity of sound in air by studying
the propagation of 40 kHz sound waves between two piezoelectric crystal
transducers, one acting as a transmitter, the other as a receiver. An
oscilloscope will be used to measure the time difference between the sending
of a short sound pulse and its subsequent receipt at a microphone a known
distance away.
A. Equipment
1 Dual Trace Oscilloscope, 1 Function Generator, 2 Piezoelectric Crystal
Transducers, 1 Meter Stick, 1 DC Power Supply, 1 Frequency Counter.
B. Method
Sound waves are created by local variations in pressure above and below the
ambient. These waves travel through the air at swift but finite speed. We
will measure the speed of sound waves in air by measuring the time required
for a short sound pulse or “chirp” to travel from its source to a receiver.
This time interval is short and is measured with the help of an instrument
called an oscilloscope. The experimental setup is shown in the figure below:
Function
Generator
Oscilloscope
Ch 1 Ch 2
The function generator will be set to produce a square pulse. Each edge of
the square pulse will chirp the piezoelectric transmitter crystal. The function
generator is wired to both the crystal and to the oscilloscope. The sound is
received at a second crystal. A small amplifier is mounted to the receiver to
make its signal large enough to be easily visible on the ‘scope.
The oscilloscope is essentially a voltmeter in which a spot on a screen can be
displaced up or down or left or right by an amount proportional to an applied
voltage. The spot can also be made to move from left to right at a uniform
rate (known velocity) to show the time variation of a voltage that deflects the
spot up and down. The latter of these two modes is used in our experiment.
The oscilloscope sweep is triggered (started) by the function generator when
it chirps the transmitter crystal. The screen then shows two traces, one for
the transmitted pulse (always at the left edge), and one for the received pulse
(delayed). By knowing the velocity of the trace, we can measure the delay
of the received signal and hence the velocity of sound.
The oscilloscopes used here are expensive high-quality instruments and
should be handled carefully. Do not have the intensity control set too
high, and never allow a small very bright spot to remain stationary on
the cathode-ray tube face. The screen can easily be damaged if the
intensity is too high.
.
C. Procedure
1. Set the function generator to produce maximum amplitude square
waves at a frequency of 100 Hz.
2. The oscilloscope should be set to trigger from channel 1 in the AUTO
mode and the VERT MODE switch should be set to DUAL, so that two
traces are visible. Your TA will lend assistance if the oscilloscope is
adjusted improperly.
3. The receiver transducer has a small amplifier built onto its holder
because the received signal is quite weak. Adjust the 'scope setting until you
can clearly observe the elapsed time between the edge of the square wave
and the arrival of the sound pulse at the receiver.
4. Move the receiver back and forth and observe the motion of the
received pulse on the screen. If the pulse appears to move in proportion to
the distance between the transmitter and receiver, then your device is
adjusted properly.
Q1. Sketch both traces and explain clearly what is happening.
5. Record the setting of the “time base” of the oscilloscope. This setting
tells you the inverse of the velocity of the trace. For example, if your time
base setting were 10.0 milliseconds per centimeter, then a spacing of 3.6
centimeters between pulses would imply a 36 millisecond delay.
6. Record the location of both the transmitter and receiver. Record the
distance (in cm) between the transmitted pulse trace and the received pulse
trace on the oscilloscope screen.
7. Calculate the time delay of the received pulse using the oscilloscope
time base setting.
7. Repeat steps 5-7 for five additional spacings of the transmitter and
receiver.
8. Plot the distance of separation (vertical axis) vs. the time delay.
Calculate the velocity of sound from your graph.
Q2. Why did we not just use the distance between the transducers for this part?
(HINT: What is the exact position of each transducer?)
Q3. Compare your values with the theoretical value:
csound =
γ air RT
Mair
Assume that air is 78% N2, 21% O2, and 1% Ar. A table of atomic weights can
be found in Appendix D of your text and values of γ are listed pp. 495.
Fall 2007 PHY126 Experiment 8
LENS OPTICS
In this experiment we will investigate the image-forming properties of
lenses, using the thin-lens equation:
1 1 1
+ =
s s' f
where s and s’ are the object and image distances from the lens and f is the
focal length of the lens.
A. Equipment
1 Optical Bench, Several Convex and Concave lenses, 1 Light source,
1 Mirror, 1 Screen.
C. Procedure
I. Measuring Focal Length
1. The focal length of a lens is equal to the image distance of an object at
infinity. Image the overhead lights on a piece of paper and measure the
distance from the paper to the lens.
Q1. Prove using the thin lens equation the statement from step 1.
2. Set up on your optic bench the equipment shown below:
Lens
Mirror
Image
Object
3.Move the lens until the reflected image forms a clear focus at the same
position as the object. Determine how to get the focal length from this
configuration.
Q2. Is the location of the mirror important?
Q3. Draw a ray diagram to show how the image is formed.
4. Repeat for each of your lenses.
II. Converging lenses and Real Images.
1. For each of your convex lenses, measure the s and s’ (as well as the
image size) for at least six different object distances.
Q4. What is the smallest object distance for which a real image can be formed?
2.For each lens, make a plot of 1/s’ cs. 1/s. Determine the focal length of
each lens from the plot and compare with your previous results.
3. Use your measurements to test the relation m = -s’/s.
III. Two Lens Systems.
1. Mount a diverging lens on the optical bench in front of an illuminated
object.
Q5. Where is the image formed? Can you see it? Why?
2. Put a strong (short focal length) lens behind the concave lens. If the
converging lens is strong enough, it will be possible to form a real image.
3. The virtual image created by the diverging lens acts as the object for
the converging lens. Using the measured distance to the image of the
converging lens (s’ for that lens), and its known focal length, measure the
location of the virtual image from the diverging lens.
4. Using the known object location and JUST MEASURED image
location, calculate the focal length of the diverging lens.
5. Construct a simple two-lens telescope. Try a few different lens
combinations to verify the relationship between the focal lengths of the
lenses and the magnifying power.