Department of Physics
GENERAL PHYSICS II
(1. and 2. Education)
LABORATORY MANUAL
(2011-2012)
Department of Physics, Abant İzzet Baysal University, Bolu, 14280 TURKEY
1
Name Surname :
Number
:
Group
:
Date
Exp-1 Measuring
Flow Rate
Exp-2 Basic
Archimedes’ Principle
Buoyancy and
Specific Gravity
Exp-3
Studying the
Transfer of Heat
Exp-4
Specific Heat
Capacity of Metals
Exp-5
Equipotential and
Electric Field Lines
Exp-6
Parallel and Series
Combinations
of Capacitors
Exp-7
Sound Waves
2
Quiz
Report
CONTENTS
Experiment 1.
Measuring Flow Rate………………………..…4-11
Experiment 2.
Archimedes’ Principle: Buoyancy and
Specific Gravity ….……………………..…….12-17
Experiment 3.
Studying the Transfer of Heat ……………....18-23
Experiment 4.
Specific Heat Capacity of Metals…………….24-29
Experiment 5.
Equipotential and Electric Field Lines……...30-33
Experiment 6.
Parallel and Series Combinations
of Capacitors…………………………………..34-37
Experiment 7.
Sound Waves………………………………….38-43
Appendices…………………………………….44-57
3
Experiment 1 - Measuring Flow Rate
This experiment is taken from the University of Nebraska at Lincoln with permission of
Physics Laboratory Manager Shawn Langan.
1.1 Your Circulatory System - A Simple Model
"Blood is one of the most interesting organs in the body.
It has two unique properties: it is a liquid and it is always on the move."
From How the Body Works by John Lenihan.
1.
What are the functions of blood flow in your body? When would your body need
to increase the blood flow rate (flow rate = J (m3/s) = V / t)? When would some parts
of your body need more blood flow than other parts? Explain your ideas.
Here is an engineer’s sketch of
the blood circulation system of a human
body. The circulatory system can be
thought of as a closed-loop circulation
system with two pumps. One-way valves
keep the flow unidirectional through the
pumps. The pressures are indicated in
mm of Mercury (Hg) where 1
atmosphere of pressure is 760 mm of
Hg.
2.
What are the properties of this
system that determine the flow rate of
blood? By what mechanisms does your
body regulate its blood flow rate
(increase or decrease the flow rate or
change where more of the blood flows).
Explain your ideas.
1.2 Discussion
Your instructor will lead a brief discussion of important concepts related to fluid flow and
demonstrate the operation of today’s equipment.
4
1.3 Application #1 - Flow Rate as a Function of Pressure
You will conduct the following experiment to determine how the flow rate
changes as a function of the pressure.
Change pressure (P) with height (h),
Keep Tube Length (L) and Tube
inside Diameter (d) constant.
Follow the guidelines for using the equipment as demonstrated by your instructor.
Use the values of length and tube diameter specified by your lab instructor.
1.3.1 Data Collection:
Have a member of your team pour water into the top of the container so the
height of the water surface above the exit opening stays constant while you
are capturing the water flowing out of the container.
Capture the water flowing out of the container. For each trial, collect water long
enough so the data is meaningfully accurate. That is, ALWAYS collect for at least
1 minute. Once 1 minute is up, keep collecting until you have at least 10 ml or
until you’ve collected for 5 minutes, which ever comes first.
5
Choose at least 4 different convenient starting heights for the water surface, i.e. 6
cm, 8 cm, 10 cm, and 12 cm. Please do not directly write on the equipment.
Repeat each measurement at least twice and record the total volume of water
expelled for each interval. Calculate the resulting flow rates.
Record your data in a data table in your logbook.
1.3.2 Data Analysis:
Take an average of the two flow rate measurements for each height.
1.4 Application #2 - Flow Rate as a Function of Length of
the Tube
You will conduct the following
experiment to determine how the flow rate
changes as a function of the total length of
the tube.
Change Tube Length (L),
Keep height (h) and Tube inside
Diameter (d) constant.
Follow the guidelines for using the
equipment as demonstrated by your instructor.
1.4.1 Data Collection:
Have a member of your team pour water into the top of the container so the height
of the water surface above the exit opening stays constant while you are capturing
the water flowing out of the container. Use a constant water height of 5 cm. You
might mark this height on a piece of tape on the equipment - just be sure to
remove it when finished.
Capture the water flowing out of the container. For each trial, collect water long
enough so the data is meaningfully accurate. That is, ALWAYS collect for at least
1 minute. Once 1 minute is up, keep collecting until you have at least 10 ml or
until you’ve collected for 5 minutes, which ever comes first.
6
Choose at least 5 different tube lengths, making sure that the diameter of the
inside of the tube remains constant for all trials.
Repeat each measurement at least twice and record the total volume of water
expelled for each interval. Calculate the resulting flow rates.
Record your data in a data table in your logbook.
1.4.2 Data Analysis:
Take an average of the two flow rate measurements for each length.
1.5 Application #3 - Flow Rate as a Function of the Inside
Tube Diameter
You will conduct the following
experiment to determine how the flow rate
changes as a function of the inside diameter
of the tube.
Change Tube inside Diameter (d),
Keep height (h) and Tube Length (L)
constant.
Follow the guidelines for using the
equipment as demonstrated by your instructor.
1.6.1 Data Collection:
Have a member of your team pour water into the top of the container so the height
of the water surface above the exit opening stays constant while you are capturing
the water flowing out of the container. Use a constant water height of 5 cm. You
might mark this height on a piece of tape on the equipment - just be sure to
remove it when finished.
Capture the water flowing out of the container. For each trial, collect water long
enough so the data is meaningfully accurate. That is, ALWAYS collect for at
least 1 minute. Once 1 minute is up, keep collecting until you have at least 10 ml
or until you’ve collected for 5 minutes, which ever comes first.
7
Choose at least 3 different tube diameters, making sure that the length of the
inside of the tube remains constant for all trials.
Repeat each measurement twice and record the total volume of water expelled for
each interval. Calculate the resulting flow rates.
Record your data in a data table in your logbook.
1.6.2 Data Analysis:
Take an average of the two flow rate measurements for each tube.
1.7 Data Summary
Analyze your data using the following procedures.
1.7.1 Data Analysis - Application #1
Graph the average flow rate (y axis) versus height (x axis) on a linear graph
paper.
Use your knowledge of mathematical models to determine a functional
relationship between the flow rate and the water height.
Record the value of tube length and tube diameter on your graph as well.
3. (a) Write the mathematical model (including appropriate variables and units) relating
flow rate and water height.
(b) Rewrite the mathematical model (including appropriate variables and units)
relating flow rate and pressure. (Recall, pressure = density*g*h)
(c) Describe the physical meaning of this model in words.
1.7.2 Data Analysis - Application #2
Graph the average flow rate (y axis) versus tube length (x axis) on a linear graph
paper.
Use your knowledge of mathematical models to determine a functional
relationship between the flow rate and the length of the tube. (Hint! Try a Power
fit)
Record the value of water height and tube diameter on your graph.
4. (a) Write the mathematical model (including appropriate variables and units) relating
flow rate and the length of the tube.
(b) Describe the physical meaning of this model in words.
1.7.3 Data Analysis - Application #3
Graph the average flow rate (y axis) versus tube diameter (x axis) on a linear
graph paper.
8
Use Excel and your knowledge of mathematical models to determine a functional
relationship between the flow rate and the inside diameter of the tube. (Hint! Try
a Power fit)
Record the value of water height and tube length on your graph.
5. (a) Write the mathematical model (including appropriate variables and units) relating
flow rate and the inside diameter of the tube.
(b) Describe the physical meaning of this model in words.
1.8 Summary Questions
6. Poiseuille (1799-1 869) was interested in the physics of blood circulation. He found
that the flow rate of a fluid undergoing laminar flow in a cylindrical tube is:
Flow rate
.pressure.(radius ) 4
8.viscosity .length
(a) Compare your results for application #1 to this predicted mathematical model.
Do they appear to agree? Explain.
(b) Compare your results for application #2 to this predicted mathematical model.
Do they appear to agree? Explain.
(c) Compare your results for application #3 to this predicted mathematical model.
Do they appear to agree? Explain.
7.
(a) Based on how well the results agreed with Poiseuille’s flow rate model, do you
think the flow in these experiments was laminar (made up of overlapping
layers, layered, laminated) or did you find evidence of turbulence?
(b) If there was turbulence in the flow, what things might you be able to change if
you repeated this experiment to reduce the turbulence? Explain your ideas.
(c) Do you think most practical applications of fluid flow (like water in pipes or
air in heating and cooling ducts) represent laminar or non-laminar flow?
Explain why you think this.
8. During heavy exercise, the body increases its flow rate to 5-10 times greater than
when at rest. How does the body do this?
Is increasing blood pressure 5-10 times higher a viable option?
Is decreasing the length of your blood vessels a viable option?
The arterioles (small arteries) have sphincter-like muscles around them. How
much would the radius of a blood vessel have to increase to cause a 5 times
greater flow rate?
9
In fact the body does increase its blood pressure a little to increase the flow rate - but
most of the flow rate increase comes from vasodilatation (dilation of the blood
vessels).
9.
Arteries in the human body can be constricted when plaque (layer of bacteria
accumulation) builds up on the inside walls. How would this affect the blood flow
rate through this artery? If the body wants to keep a constant flow rate, then how
might this condition affect a person’s blood pressure? Explain.
10. The implications of Poiseuille’s law for some bioengineering applications can follow
from these conclusions about blood flow. What other kind of physics or engineering
applications can you imagine are relevant to the study of laminar (steady) flow?
Describe at least one example and explain why you think it is relevant. For example,
you might think about streamlines in airflows in wind tunnels (another example of
steady flow).
End of the lab Procedures
Empty all water into the sink in the hallway.
Return the metal/glass tubes to the provided storage bin at your station.
Remove any tape that you may have placed onto the apparatus.
10
Notes
11
Experiment 2 - Archimedes’ Principle:
Buoyancy and Specific Gravity
This experiment is taken from the book "Physics Laboratory Experiments" pages 287295 by Jerry D. Wilson (D.C Heath and Company - Third Edition)
2.1 Advance Study Assignment
Read the experiment and answer the following questions
1. Describe the physical reason for the buoyant force in terms of pressure.
2. Give the conditions on densities that determine if an object will sink or float in a
fluid.
3. Distinguish between specific gravity and density.
4. Describe how the specific gravity of an object less dense than water can be
determined using Archimedes’ principle. How about the specific gravity of a
liquid?
5. Why is it important to make certain that no air bubbles adhere to objects during
the submerged weighing procedures? How would the experimental results be
affected if bubbles were present?
2.2 Introduction
Some objects will float and others will sink in a given fluid - a liquid or a gas. The
fact that an object floats means it is "buoyed up" by a force greater than or equal to
weight. Archimedes (287-212 B.C.), a Greek scientist, deduced that the upward buoyant
force acting on a floating object is equal to the weight of the fluid it displaces. Thus, a
body will sink if its weight exceeds that of the fluid it displaces.
In this experiment, Archimedes’ principle will be studied in the application of
determining the densities and specific gravities of solid and liquid samples.
2.3 Equipment Needed
Triple-beam pan balance with swing platform (or single-beam double-pan balance
with swing platform and set of weights)
Overflow can (or graduated cylinder and eye dropper)
Two beakers
Metal cylinder or irregularly shaped metal object, or metal sinker
Waxed block of wood
Saltwater solution or alcohol
String
Hydrometer and cylinder
12
2.4 Theory
When a body is immersed in a fluid, it experiences an upward buoyant force that may
cause it to float. The reason for the buoyant force can be understood by considering the
pressure-depth relationship, p = gh, where p is the pressure, the mass density, g the
acceleration due to gravity, and h the depth below the surface of the fluid. In a fluid, the
pressure on the lower surface of an object is greater than that on the upper surface, so
there is pressure difference, or an upward force. (Consider a block or a cube for
simplicity.)
The magnitude of the buoyant force is described by Archimedes’ principle:
“When a body is placed in a fluid, it will be buoyed up by a force equal to the weight
of the volume of fluid it displaces.”
Archimedes’ principle applies to a body wholly or partially immersed in a fluid. The
magnitude of the buoyant force depends only on the weight of the fluid displaced by the
object and does not depend on the weight of the object.
Archimedes’ principle shows that an object
1. will float in a fluid if the density of the object 0 is less than the density of the
fluid f,
2. will sink if the object’s density is greater than that of the fluid’s, and
3. will float in equilibrium at any submerged depth where it is placed if its density is
equal to that of the fluid.
The weight of an object is w0
m0 g
0
gV0 where V0 is the volume of the object
and 0 m0 V0 . Similarly, the weight of the fluid displaced by the object, or the buoyant
force, is Fb w f m f g
f gV f . If the object is completely submersed in the fluid, then
V0
V f , and dividing one equation by the other,
Fb
w0
Hence, if
will float. If
Fb
0
0
f
f
, then Fb
, then Fb
f
or Fb
f
(2.1)
w0
0
0
w0 and the object will be buoyed to the surface and
w0 , and the object will sink. And, if
0
f
, then
w0 and the object is in equilibrium.
The specific gravity of a solid or liquid is defined as the ratio of the weight of a
given volume of the substance to an equal volume of water specific gravity
sp. gr .
weight of a substance (of given volume)
weight of an equal volume of water
13
(2.2)
Specific gravity is a density-type designation that uses water as a comparison
standard. Since it is a weight ratio, specific gravity has no units, and the numerical value
of a substance’s specific gravity is the same as the magnitude of its density in cgs units.
This can be seen as follows:
ws
ws
Vs
=
ww ww
Vw
sp. gr .
ms g
ms
Vs
mw g
Vs
mw
Vw
Vw
s
(2.3)
w
where the subscripts s and w refer to the substance and water, respectively, and by
definition Vs Vw . Since, for practical purposes, the density of water is 1 g/cm3 over the
temperature range in which water is liquid,
3
s
s ( g / cm )
(2.4)
sp. gr.
s
1 ( g / cm3 )
w
where s , is the numerical value of the density of a substance in g/cm3. For example, the
density of mercury is 13.6 g/cm3 and has a specific gravity of 13.6. A specific gravity of
13.6 indicates that mercury is 13.6 times more dense than water, s ( sp. gr .) w , or that
a sample of mercury will weigh 13.6 times more than an equal volume of water.
Archimedes’ principle can be used to determine the specific gravity (and density)
of a submerged object:
w0
w
= 0
ww
Fb
sp. gr .
since ww
(2.5)
Fb . For a heavy object that sinks, if its apparent weight w0' is measured while
it is submerged, we have ww
sp. gr .
Fb
w0
w0' . Hence,
w0
w0
=
ww
w0 w0'
(2.6)
or in terms of mass which is measured on a balance ( w mg )
sp. gr .
where
0
m0
m0
m0'
(
0
) (of submerged object)
is the magnitude of the density of the object in g/cm3.
To find the specific gravity and density of
a solid object less dense than water using
Archimedes’ principle, it is necessary to use
another object or sinker of sufficient weight and
density to submerge the light solid completely.
Using an arrangement as illustrated in the
figure, w1 w0 ws' is the measured weight
(mass) of the object and the sinker, with only the
sinker submerged, and w2 w0' ws' is the
14
(2.7)
measured
weight
both
are
submerged.
Then
'
w1 w2 ( w0 w ) ( w w ) w0 w , or put in terms of mass, m1 m2 m0 m0 and
the specific gravity can be found from equation (2.7).
The specific gravity of a liquid can also be found using Archimedes’ principle.
First, a heavy object is weighed in air ( w0 ) and then weighed when submerged in liquid (
'
s
w0' ). Then ( w0
'
0
when
'
s
'
0
w0' ) l is the weight of the volume of liquid the object displaces, by
Archimedes’ principle. Carrying out a similar procedure for the object in water,
( w0 w0' ) w , is the volume of water the object displaces. Then, by the definition of specific
gravity (Equation (2.2)),
( w0 w0' )l
(2.8)
sp. gr.
( l ) (of a liquid)
( w0 w0' ) w
where l , is the magnitude of the density of the liquid in g/cm3.
You may have been thinking that there are easier ways to determine the density or
specific gravity of a solid or liquid. This is true, but the purpose of the experiment is to
familiarize you with Archimedes’ principle. You may wish to check your experimental
results by determining the densities and specific gravities of the solid samples by some
other method. The specific gravity of the liquid sample will also be determined using a
hydrometer.
2.5 Experimental Procedure
A. Direct Proof of Archimedes’ Principle
1. Weigh the metal sample and record
its mass m0 , and the type of metal
in the Laboratory report. Also,
determine the mass of an empty
beaker mb , and record. Fill the
overflow can with water and place
it on the balance platform. Attach a
string to the sample and suspend it
from the balance arm as illustrated
in the figure.
2. The overflow from the can when
the sample is immersed is caught
in the beaker. Take a mass reading
mb' of the submerged object. Make
certain that no bubbles adhere to
the object. (It is instructive to place
the overflow can on a second balance, if available, and note that the "weight" of
the overflow can does not change as the sample is submerged.) Then, weigh the
beaker and water so as to determine the mass of the displaced water m w .
15
3. According to Archimedes’ principle, the buoyant force Fb
m0 g
m0' g should
equal the weight of the displaced water ww m w g or ( m0 m0' ) g m w g .
Compute the buoyant force and compare it with the weight of the displaced water
by finding the percent difference.
B. Density of a Heavy Solid (
0
w
)
4. Determine the specific gravity and density of the metal sample. This can be
computed using the data from part A.
C. Density of a Light Solid (
0
w
)
5. Determine the specific gravity and density of the wooden block by the procedure
described Theory Section 2.4 and illustrated in the figure. First, measure the mass
of the wooden block alone (in air). Then set up as shown in the figure. Make
certain that no air bubbles adhere to the objects during the submerged weighing
procedures. [The first weighing procedure may have to be modified to fit the
apparatus (i.e., weighing the block and submerged sinker separately).] The block
is waxed so that it does not become waterlogged.
D. Density of a Liquid (
l
)
6. Determine the specific gravity and density of the liquid provided by the procedure
described in Theory Section. Again, make certain that no air bubbles adhere to the
object during the submerged weighing procedures.
7. Determine the specific gravity of the liquid using the hydrometer and cylinder.
Compare this value with that found in procedure 6 by computing the percent
difference.
2.6 Questions
1. Look up the density of the type of metal of the object used in parts A and B of the
procedure, and compare it with the experimental value. Comment on the purity of
the metal of the object. (Archimedes developed his principle while working on a
similar inquiry. His problem was to determine whether an apparent gold crown
had been made with some content of cheaper metal.)
2. In part B, the string will cause error. When does it lead to an experimental density
that is too high? Too low?
3. What would be the situation of an object immersed in a fluid if the object and the
fluid had the same density?
4. (a) Explain how a submarine is caused to submerge and surface without the use of
its propulsion propeller. (b) Oil floats on water. What can you say about the
density of oil?
16
5. A block of wood floats in a beaker of water. According to Archimedes’ principle,
the block experiences an upward buoyant force. If the beaker with the water and
floating block is weighed, would the measured weight be less than the sum of the
weights of the individual components? Explain.
6. A person can lift 45 kg (= 100 lb). Using the experimental value of the specific
gravity for the metal object in part B, how many cubic meters of the metal could
the person lift (a) in air; (b) in water? How many actual kg of metal is this (a) in
air; (b) in water?
7. Explain the principle and construction of a hydrometer. What is the purpose of the
common measurements of the specific gravities of an automobile’s radiator
coolant and battery electrolyte?
17
Experiment 3 – Studying the Transfer
of Heat
Activity 1: Exploration Activity
How reliable are you as a temperature-sensing device? Check this out by having
each person in your team perform the following experiment.
Take two cups to the sink in the hallway and fill one with warm water and one with
cool water and bring them back to the lab.
Quickly prepare three containers: one of warm water, one of cool water, and a
container of a mixture of those two waters.
Put a good temperature-sensing finger, i.e. your index finger, in each container, your
left finger in the warm water and your right one in the cool water, leave them in that
water for about 30 seconds.
Then, simultaneously place both of those fingers into the water mixture.
1.
Describe how the water mixture feels to your two fingers. Write an explanation for
these differences that sounds scientific and not wrong, from your point of view.
2.
Discuss the comparison between your human temperature sensing process and the
results from the thermometers. How would you explain these differences?
Invention Discussion
Your instructor will lead a brief discussion of important concepts related to this lesson.
Application #1 - Transfer of Heat
Equipment
Thermometers; Tripod support stand with thermometer holder; Hot pot; Container with
room-temperature water.
Comparing Heat Transfer in Water to Heat Transfer in Air
Fill the hot pot about 2/3 full of cool water and turn it on so that it will come
almost to a boil.
Do not put the thermometer in the water (that is, it should just be sitting in the air
on the table top).
While the water is heating, please discuss the prediction questions listed below.
18
Prediction Questions
Today you will be doing three experiments:
1) You will take the thermometer and place it into boiling water until it reads a
stable temperature. Then, you will remove it and place it immediately into a
room-temperature water bath and see how it cools.
2) You will repeat the experiment, only this time you will leave the sensor
suspended in air to cool after you remove it from the boiling water.
3) You will repeat the experiment a third time, only this time you will have a fan
blowing on the sensor as it is suspended in air to cool after you remove it from
the boiling water.
Discuss with your partners your thoughts on the following three questions:
(a)
(b)
(c)
What will happen to the value given by the temperature sensor when you
place it in the boiling water?
What will the difference(s) be for the data of the heated temperature sensor
when it is placed in the cool water bath, the data for the one left in still air, and
the data for the one placed into moving air?
Discuss what you think the temperature vs. time graphs will look like for the
three different experiments. Your graph should start (time = 0 s) when the
temperature sensor was just sitting on the table (before you placed it into the
hot water bath).
Do not continue unless the water has fully heated!
Experiment #1: Boiling water
Room-temperature water
Data Collection:
Leaving the thermometer sitting on the table and start recording the thermometer data
and time.
After fifteen seconds, take the thermometer and place it into the boiling water. The
tip should be immersed in the water, but do not let it touch the bottom of the hot pot.
After the temperature reaches its maximum value, remove the thermometer from the
boiling water. Place it immediately (as quickly as possible!) into a room-temperature
water bath. Let it cool until it reaches a steady temperature.
Stop recording data once the cooling process has finished.
Graph the temperature (ºC) vs. time (in seconds.) on a linear graph paper. (The total
experiment should take no more than 300 seconds.)
Record the data in your logbook.
19
Data Analysis:
Adjust the graph so that it shows the data from time = 0 until it has finished cooling
and so that the data fills the graph. Print a copy for each person in your group. Be
sure to label this graph in your logbook.
Label the following events on this graph:
Sitting at room temperature
Placed into hot water bath
Placed into room-temperature bath
Reached final temperature when heating
Reached final temperature when cooling
Create a summary data table like the one illustrated below that can be used for all
three experiments. Record the information for this experiment.
Exp. Cooling
#
method
3.
4.
Initial
Temp
(ºC)
Final
Heating
Temp
(ºC)
Total
Heating
Time
(s)
T/ t
Heating
(ºC/s)
Final
Cooling
Temp
(ºC)
Total
Cooling
Time
(s)
T/ t
Cooling
(ºC/s)
Compare the shape of the temperature vs. time curve when the thermometer was
heating in water to when it was cooling in water. What factors do you think affect
how quickly the change takes place?
What kind of heat transfer mechanism(s) accounts for the heat transfer from the
boiling water to the sensor? How do you know?
20
Experiment #2: Boiling water
Room-temperature air (not blowing)
Data Collection:
Leaving the thermometer sitting in the room-temperature water, start recording the
thermometer data and time.
After fifteen seconds, take the thermometer and place it into the near-boiling water.
The tip should be immersed in the water, but do not let it touch the bottom of the hot
pot.
After the temperature sensor reaches its maximum value, remove the thermometer
from the boiling water. Take it out of the water and leave it sitting in air. (Hang it
from the support stand so that the thermometer is touching nothing but air.) Try to
move it as little as possible as you do this!!
Stop recording data once the cooling process has finished
Graph the temperature (ºC) vs. time (in seconds.) on a linear graph paper.
Record the data in your logbook.
Data Analysis:
Adjust the graph so that it shows the data from time = 0 until it has finished cooling
and so that the data fills the graph. Each person must have his/her graph in the group.
Be sure to label this graph in your logbook.
Label the following events on this graph:
Sitting at room temperature
Placed into hot water bath
Placed into room-temperature air
Reached final temperature when heating
Reached final temperature when cooling
Record the information for this experiment in your summary table.
5.
6.
7.
8.
How does the heating curve compare to that from experiment #1? Does it take the
same amount of time to reach a maximum value? Should it?
Compare the shape of the temperature vs. time curve when the thermometer was
heating in water to when it was cooling in air. What factors do you think affect
how quickly the changes takes place?
Compare the shape of the temperature vs. time curve when the thermometer was
cooling in water to when it was cooling in air. What factors do you think affect
how quickly the changes takes place?
What kind of heat transfer mechanism(s) accounts for the heat transfer from the
heated thermometer to the air? How do you know?
21
Experiment #3: Boiling water
Room-temperature air with blowing fan
Data Collection:
Dry off the thermometer and leave it sitting in the room-temperature air.
Start recording data and after fifteen seconds, take the thermometer and place it into
the near-boiling water. The tip should be immersed in the water, but do not let it
touch the bottom of the hot pot.
After the thermometer reaches its maximum value, turn on the fan so that it is
blowing toward the support stand. Take the thermometer out of the water and hang it
from the support stand so that the sensor is touching nothing but air and the fan is
blowing on it.
Stop recording data once the cooling process has finished.
Graph the temperature (ºC) vs. time (in seconds.) on a linear graph paper.
Record the data in your logbook.
Data Analysis:
Adjust the graph so that it shows the data from time = 0 until it has finished cooling
and so that the data fills the graph. Each person must have his/her graph in the group.
Be sure to label this graph in your logbook.
Label the following events on this graph:
Sitting at room temperature
Placed into hot water bath
Placed into room-temperature air
Reached final temperature when heating
Reached final temperature when cooling
Record the information for this experiment in your summary table.
Unplug the hot pot.
Take the hot pot out to the sink in the hallway and dump out all the water.
Be careful not to burn yourself since the water is VERY hot!!!
9.
10.
11.
12.
Does the moving air change the cooling of the sensor? Does this make sense?
Explain using some examples from real life.
Which appears to be a better conductor of heat, water or air? Explain your
conclusion.
How does convection seem to affect the transfer of heat? Explain your ideas based
on these experiments.
If you want to warm your cold hands quickly, what is more effective, entering a
warm room or putting them in some warm water? Why?
22
Notes
23
Experiment 4 – Specific Heat Capacity
of Metals
Objectives
To determine the specific heat capacity of an unknown metal
To gain experience with a constant-pressure calorimeter
Introduction
When heat is added to a substance, its temperature rises. Of course everyone already
knows this, but rarely do we ever think about how much heat must be added to raise the
temperature. Different substances require different amounts of heat to raise their
temperatures. Water, for instance, needs a lot of heat to raise its temperature. It takes
4.184 J of heat to raise the temperature of one gram by 1 C. Contrast this with gold,
which only needs 0.128 J of heat to raise one gram by one degree.
Before we go further it is very important that you understand the following terms:
1) Heat Capacity (C) – the amount of heat needed to raise the temperature of a
substance by one Celsius degree.
2) Specific Heat Capacity (c) – the amount of heat needed to raise the temperature
of one gram of a substance by one Celsius degree.
If we know c for a substance, and we know the mass (m) of the substance, then we can
easily calculate C:
C=c m
(1)
For example, as we saw above, the specific heat capacity of gold (cgold) is 0.128 J/(g C).
If we had 2.0 g of gold, its heat capacity (C) would be:
C = c m = 0.128 J/(g C) 2.0 g = 0.256 J/ C
You can see that the two terms are very similar, except that specific heat capacity (c) is
defined for a given amount of material (1 g) and heat capacity (C) is not. We say that
specific heat capacity is an intensive property of a substance (independent of the amount)
while heat capacity is an extensive property (depends on the amount).
If we know either C or c for a particular substance, it is very easy to calculate the amount
of heat (q) needed to raise the temperature of the substance. If we don't know the mass of
the substance, then we need to use the heat capacity (C):
q=C
T
24
(2)
where T = Tfinal – Tinitial. If we know the mass of the substance, then we can use the
specific heat capacity (c):
q=c m T
(3)
Notice that if the temperature of the substance is increasing then T > 0 and q > 0. A
positive value for q means that heat is being gained. We can also use equations 2 and 3 to
calculate the amount of heat that must be removed from a substance to lower its
temperature. In this case T < 0 and q < 0. A negative value for q means that heat is
being lost.
Determining Specific Heat Capacity
The principle goal of the experiments you will conduct in this lab is to determine the
specific heat capacity (c) of an unknown metal. To do this you will use a device called a
coffee-cup calorimeter. The coffee cup calorimeter, shown in Figure 1, is simply two
Styrofoam coffee cups nested one inside the other. The inner cup is covered with a
cardboard lid containing a thermometer used to measure the temperature of the liquid
inside the calorimeter. We assume that the calorimeter is well insulated and that no heat
is able to enter or exit.
A better name for this instrument is a constant-pressure calorimeter since it is an
unsealed container and, therefore, the calorimeter is always at atmospheric pressure. This
is in contrast to a sealed calorimeter, or bomb calorimeter, which has constant volume.
To carry out the experiments, you will set up the calorimeter and fill it with 50 mL of
water. You will then place the unknown metal in a loosely sealed test tube and heat the
test tube in a beaker of boiling water. (We will assume that the metal is at the same
temperature as the boiling water.) After measuring the temperature of both the
25
calorimeter and the boiling water, you will take the metal from the boiling water, drop it
into the calorimeter, and measure the final temperature of the calorimeter.
Now how do we use this information to calculate the specific heat capacity of the metal?
First, let's look at all the variables we'll need. The following table shows a summary of all
the important measured and calculated values:
Think about what's going on in the experiment. Some hot metal at Tmetal, initial is placed in
a cool calorimeter at Tcal, initial. The metal cools and the calorimeter heats up until they
eventually end up at the same final temperature, Tfinal. The key is in realizing that the heat
lost by the metal is equal to the heat gained by the calorimeter:
heat lost by metal = heat gained by calorimeter
(4)
If we plug in q for heat, and show the proper signs, this becomes:
– qmetal = + qcal
(5)
Using equations 2 and 3, and plugging in the proper variables, this then becomes:
– [ cmetal mmetal
Tmetal ] = Ccal
Tcal
(6)
Since we're trying to determine the specific heat capacity of the metal, let's solve for cmetal
to find the equation we'll actually use in the end:
(7)
This equation looks pretty simple to use, except for the Ccal term. How can we get this
value? This is answered in the following section
Calibrating the Calorimeter
We have to calibrate the calorimeter to determine Ccal. This is actually very easy. We'll
simply run the experiment described above. But instead of using an unknown metal, we'll
26
use a metal for which we know the specific heat capacity. In our case, that metal will be
copper (ccopper = 0.387 J/g C). We can then rearrange equation 6 to give:
(8)
Once we have Ccal from this calibration process, we have everything we need to calculate
cmetal for our unknown using equation 7. Note that Ccal is the same, whether the metal in
the calorimeter is copper or our unknown.
The Experiment
Required Metals
copper shot
sample of unknown metal
Procedure
A. Preparation of the Metal Samples
1. Fill a 1000-mL beaker about ½ full of distilled water and add a few boiling
chips. Use a Bunsen burner to bring the water to a boil.
2. Put approximately 30 g of dry copper metal into each of two large test tubes.
Be sure to record the exact mass of the metal in each test tube. Loosely cork
the test tubes and place them in the boiling water.
3. Obtain an unknown metal from your lab assistant. Be sure to record the ID
number of your unknown. As before, put an accurately weighed amount of
this metal into each of two additional test tubes. (Your instructor will tell you
how much metal to use.) Loosely cork the test tubes and place them in the
boiling water.
4. The test tubes should be placed in the boiling water for at least 20 minutes. Be
sure that the metal is submerged beneath the water level. Add water as
necessary to maintain the water level. Use a thermometer to record the
temperature of the boiling water to the nearest 0.1 C.
NOTE: If you notice any condensation on the inside of a test tube, remove that test tube from the
water bath. Pour the metal on a paper towel to remove any moisture, and then place the metal back
in the water bath in a new, dry test tube.
27
B. Preparation and Calibration of the Calorimeter
1. Construct a coffee-cup calorimeter by placing one Styrofoam cup inside
another. Obtain a cardboard top and a thermometer. Push the thermometer
through the hole in the cardboard.
2. Pipet 50 mL of distilled water into the calorimeter. Place the cardboard lid and
thermometer on the calorimeter. DO NOT POKE THE THERMOMETER
THROUGH THE BOTTOM OF THE CUP!!
3. Record the temperature of the water in the calorimeter. Remove the cardboard
top of the calorimeter and use metal tongs to quickly transfer the copper from
one of the test tubes into the calorimeter. Pour the copper into the calorimeter
gently enough to avoid splashing water out of the calorimeter. Also avoid
getting any of the hot water from the outside of the test tube into the
calorimeter.
4. Place the cardboard top on the calorimeter and gently swirl the calorimeter
while monitoring the temperature. Record the highest temperature reached;
this will be your value of Tfinal.
5. Dump the water out of your calorimeter and dry the inside. Place the wet
copper in one of the large drying dishes on the side counter.
6. Repeat the procedure for the second test tube of copper.
C. Determination of the Specific Heat Capacity of an Unknown Metal
1. Repeat part B, but now with your unknown instead of the copper.
Data Analysis
1. Use your data from part B, along with equation 8, to calculate Ccal for each of
your copper samples. Average the two values and use this as your final value of
Ccal.
2. Use your data from part C, along with equation 7, to calculate cmetal for each of
your unknown samples. Again, average your two values and report this as your
final value for cmetal.
28
Notes
29
Experiment 5 - Equipotential and
Electric Field Lines
5.1 Purpose
To investigate and map the equipotential lines of two oppositely charged
conductors. Mapping the electric field lines using the equipotential lines. To study the
effect of a metal ring on the electric field.
5.2 Theory
If we put a test charge at a point in a certain part of space and observe a force of
electrical origin acting on it. Then we say that there is an electric field in that region. The
electric field has both magnitude and direction; like force, it is a vector. The electric field
is represented by symbol E. The source of the electric field is the other charges present in
the region. The electric field E at any point exerts a force F on a test charge q0 placed at
that point. The force is directly proportional to the field and charge. The electrical force F
acting on the test charge is given by
F = q0E
(5.1)
The SI unit for the force F is Newton (N) and the charge q0 is coulomb (C). Thus, the SI
unit for the electric field E is newton per coulomb (N/C). The direction and magnitude of
the electric field at various points near a positive charge +q is indicated in Fig.5-1a. In
order to determine the electric field produced by +q at a point, a unit positive test charge
(+1C) is assumed to be placed at that point and the force acting upon this test charge is
found. The direction of the field at three points are shown in the figure by arrows all of
which are radially from the source +q Field lines are used as an aid to make a picture of
the field near the source that produces the field. These lines are drawn to show the
direction and intensity (magnitude: strength) of the field at all points in the vicinity of the
source. Fig.5-1b shows such lines for the field around the charge +q.
Figure 5.1(a, b): Electric field and field lines of a charge +q
30
Field lines have no physical existence, but they serve to indicate the structure of
an electric field. They are continuous lines starting at a positive charge and ending at a
negative charge. Their density, that is their closeness, gives a visual impression of the
field intensity. Denser the field lines are stronger the field is. At any point, the electric
field can have but one direction. Only one field line can pass through each point of the
field and therefore field lines never intersect.
Work must be done on a charge q0 in moving it in the electric field, since a force is
required to do so. Calculation of the work to move a charge from point A to B is
simplified by introducing the concept called potential. The potential difference between
two points, say A and B is equal to the work WAB done to carry the charge from point A to
point B divided by the charge, that is
(5.2)
where VA and VB represent the potential at point A and at point B, respectively. The SI
unit for the potential is volt (V).
In order to graphically represent the potential at various points in an electric field
we use the equipotential lines. The potential is the same at all points of an equipotential
line, and therefore no work is required to move a charge along the equipotential line. It
follows that; the equipotential line through any point must be at right angles to the field
line at that point. Fig.5-2 shows some of the equipotential lines of the field produced by
two equal and opposite charges.
Figure 5.2: Field and equipotential lines of two equal and opposite charges.
When a charge is given to a conductor, the charge resides on the surface. The
electric field just outside the conductor is perpendicular to the surface at every point. It
follow that when a charge is moved from one point to another along the surface of a
conductor, no work is required. Hence, a conducting surface is always an equipotential
surface, when all charges are at rest.
31
5.3 Apparatus
DC voltage supply, multimeter, metal electrodes, leads, metal ring, connecting wires,
graph paper enclosed by plastic cover, millimetric graph paper, and ruler.
5.4 Experimental Procedure
Clean the tray, the electrodes, the leads and the metal ring before you start the
experiment.
1. Place the millimetric graph paper enclosed by plastic cover in the tray. Put weak
acid solution in the tray until it covers the plastic cover completely and reaches a
depth of about one millimeter. Specify the x- and y- axes on the graph paper.
Place the two metal electrodes symmetrically on the graph paper and construct the
circuit as shown in Fig.5-3. Note that the electrodes must always be at the same
position in the solution. Keep the DC supply off and connect one of the electrodes
to the positive and the other to the negative terminal of the supply.
Figure 5.3: Construction of equipotential lines in a field due to two oppositely charged
electrodes.
2. Have your circuit checked and approved by your instructor. Turn the DC supply
on. A voltage of about 5V is sufficient to do the experiment. Take six reference
points (A- BC-.....etc.) on the x- axis between the electrodes such that they are
symmetrically and uniformly distributed with respect to the origin. Prepare a table
similar to Table 5-1 to write down the coordinates of points that you determine for
the equipotential lines.
3. Place and fix one of the leads of the multimeter at point A. Put the other probe in
the solution and use it to locate points which have the same potential with that of
point A. Find 10 such points, 5 above and 5 below the x- axis, and write their
coordinates in your data table. Note that when the two leads are the same potential
the multimeter reads zero.
32
Table 5.1
4. On a millimetric graph paper draw the x- and y- axes, the electrodes and the
reference points. From your data table, put the equipotential points you
determined and make a map of the equipotential lines. Using the property that the
equipotential line through any point is at right angles to the direction of the field
at that point, draw the electric field lines between the electrodes.
5. Place the metal ring in the solution somewhere between the electrodes. Determine
the new equipotential and field lines. Using a different colored pencil draw the
equipotential and field lines on the same graph paper you used in step 5. What is
the effect of the metal ring on the field lines and equipotentials? Determine the
equipotentials and field lines inside the ring.
5.5 Questions
1. Must an equipotential line (or surface) coincide with a physical line, like the edge
of a metal object? Must it be a conducting line?
2. If the electric field E were not perpendicular but had a component parallel to the
equipotential, would work be required for a displacement of a charge along the
equipotential?
3. How can you create equipotentials in the form of evenly spaced parallel lines?
4. A spherical conductor carries positive charge as shown in Fig.5-4a. It is placed
near a very large metal plate. Draw the electric field and equipotential lines.
Figure 5.4: (a) A charged metal sphere placed near a large metal plate. (b) A metal with a
cavity inside.
5. In the Fig. 5-4b. Point A is inside a piece of hollow metal. The hollow metal
carries a net charge +Q. Draw the equipotential line that passes though point A.
33
Experiment 6 - Parallel and Series
Combinations of Capacitors
6.1 Purpose
To learn how to connect capacitors in series and in parallel and to calculate the equivalent
capacitance of a circuit.
6.2 Theory
If there is more than one capacitor in a circuit, one can divide them into several
combinations (as it is done for resistors) and replace them with an equivalent capacitor.
This single capacitor has the same capacitance as the actual combination of capacitors. In
this way one can simplify the circuit and can calculate the unknown quantities of the
circuit more easily.
If the potential difference applied across a combination of capacitors is equal to
the potential difference being applied across each capacitor, then these capacitors are said
to be connected in parallel. This type of connection is shown in the figure below:
Figure 6.1: Parallel connection of capacitors
34
If the potential difference applied across the combination of capacitors equals to
the sum of the resulting potential differences across each capacitor then these capacitors
are said to be connected in series. This form of connection is shown in the figure below.
Figure 6.2: Series connection of capacitors
If the potential on each capacitor is written as:
35
6.3 Apparatus
DC power supply, multimeter, various capacitors, connection wires.
6.4 Experimental Procedure
1. Connect the circuits shown below.
Figure 6.3
2. Find the equivalent capacitance of each circuit by using the formulas shown in the
theory section. (Make sure that the capacitors are uncharged at the beginning)
Have your instructor check your circuit.
36
Figure 6.4
3. Connect the circuit shown below and charge the capacitor.
4. Before discharging the capacitor, construct the circuit below and calculate the potential
V.
Figure 6.5
37
Experiment 7 – Sound Waves
Exploration - Activities 1 & 2A-D
Activity 1: Exploring your hearing
Turn your back to another member of your team. Ask that person to snap her/his
fingers behind your head so you cannot see him/her doing it. The snapper should snap
his/her fingers first on one side of the person's head and then on the other, and then mix
them up and ask the person to tell you from which side the sound is coming. Try this
activity at least eight times; and complete this table in your lab book:
Side Guessed
1.
Actual Side of Origin
By guessing alone you would expect the correct answer only half of the time, so for
eight correct answers in a row the probability is only (1/2)8 or 1 time in 256 tries!
Is your correct percentage better than that for guessing? How do you explain your
phenomenal success using physics principles?
Snap your fingers, first near the back of the person's head, and then farther away
from the person's head. Mix up the location and ask the person to tell you from which
distance the sound is coming, i.e. near or far. Try this activity at least eight times; and
complete this table in your lab book:
Distance Guessed
2.
Actual Distance of Origin
Is your correct percentage better than that for guessing (1/256)? How do you
explain your phenomenal success using physics principles?
Invention Discussion: What is sound?
3.
(a)
Your instructor will make 3 different sounds. Describe these sounds and how
they're similar and how they're different.
(b)
How can you distinguish between different sounds? Discuss this in your
group and with the class and record definitions for the following concepts:
Pitch, Intensity, Frequency, Wavelength, and Wave speed
38
Equipment Comment
The coils are wrapped in paper to keep them from becoming twisted.
Be careful when using the coils! If it is stretched out and you just "let go" then it
can become hopelessly twisted and ruined. Always hold on to both ends.
When you are finished using it, carefully collapse it back into a neat pile and wrap
it in newspaper. Your instructor can assist with this as necessary.
Activity 2: Visualizing Sound
Usually you cannot see sound. You cannot see how a disturbance from a
vibration propagates through a medium. In this activity you are going to use coils to
visualize and model how disturbances from vibrations can travel. These traveling
disturbances are called waves. Get a coil as assigned by your instructor.
A. Preliminary:
With a person holding each end, stretch the coil out on the floor about 4 m.
4.
(a)
(b)
5.
Oscillate one end of a coil back and forth perpendicular to the length of the
coil. What happens? In what direction are the individual loops in the coil
oscillating? Describe the disturbance that travels down the coil. This is called
a transverse wave. Transverse waves can be seen in the strings of musical
instruments and in your vocal cords.
Oscillate one end of a slinky back and forth parallel to the length of the slinky.
What happens? In what direction are the individual loops in the slinky
oscillating? Describe the disturbance that travels down the slinky. This is
called a longitudinal wave. Sound waves are longitudinal waves.
There are 3 main characteristics of waves: frequency, velocity (speed), and
wavelength. Relate these terms to what you’ve seen in the coil.
You are now going to examine these characteristics further by completing Activities B, C
&D
B. Looking at wave frequency ( f )
6.
Oscillate one end of the coil making a transverse wave.
(a)
(b)
7.
Count how many times the hand is making a complete back-and-forth cycle in
10 seconds. What is the frequency of the vibration source – of the hand?
Count how many times the other end of the coil is oscillating back and forth in
10 seconds. What is the frequency of the disturbance at the other end of the
coil?
How does the frequency of a wave compare to the frequency of the source?
39
C. Looking at wave velocity ( v )
8.
(a)
Oscillate one end of a 3-m stretched out coil like you did in part B. Time how
long it takes for a disturbance pulse to travel down the coil. What was the
speed of the wave?
(b)
Repeat with the coil stretched out to 6 meters. Did the wave speed change?
(c)
Does the wave speed depend on the stretch of coil? Describe the relationship.
9.
In general, do you think that wave speed depends on the specifics of the medium
the wave travels in?
D. Looking at wavelength (
)
With the coil stretched at 3 meters, oscillate one end of the coil at a relatively
slow frequency. Measure and record the approximate wavelength of the wave using the
spacing of the floor tiles as a guide.
Repeat, with same coil stretch but with a frequency approximately twice as much.
Measure and record the approximate wavelength of the wave.
10.
Does the wavelength depend on the frequency? What is the relationship between
wavelength and frequency? Is it directly proportional? Inversely proportional?
Repeat, with the original frequency but with the coil stretched so that the speed is
faster. Measure and record the approximate wavelength of the wave.
11.
Does the wavelength depend on the wave speed? What is the relationship
between wavelength and speed? Is it directly proportional? Inversely proportional?
Be sure to wrap the coil in paper for storage when you are finished.
Application #1 - Measuring the Speed of Sound
You know that sound travels much slower than light. For example, you have seen
loud events that occur far away before you hear the sound from that event, e.g. thunder
from a distant lightning flash. Sound moves so slowly compared to the speed of a
computer that you can determine the speed of sound using a computer by measuring the
time it takes for sound to travel a short distance.
Equipment
File: Sound Recorder; Snapping human fingers, meter stick, sound reflecting
tube, and sound sensor (microphone).
This experiment may be sensitive to background noise. If there is too
much noise coming from all the groups working at once, then you may
have to do this one experiment group at a time.
40
Set the sound sensor (microphone) near the open end of the cardboard sound
reflecting tube. Open the computer file called Sound Recorder (Start  Programs 
Accessories  Entertainment  Sound Recorder).
You are now ready to start collecting data. The software has been set up to take a
short amount of data, starting with a "loud" noise (like a finger snap). The sound of the
snapping fingers starts the computer software and the original snap and the echo of the
snap reflected from the far end of the sound reflecting tube will be recorded. You can use
this data to measure the time it took for the sound to travel to the end and back.
Repeat the experiment until you have a good data set showing the original finger
snap and its echo. Print a copy of your snap records and paste it into your logbook.
Identify the two events on the graph.
12.
Using the length of the tube and the time between the original snap and its echo,
calculate the speed of sound in this lab. Consider the fact that the sound travels past the
sensor, down the tube and back again! Show your calculations.
13. Revisit Activity 1 (page 1). What is the time delay between sound hitting the right
ear and then the left when coming from the right side? Show your work.
Application #2 - Studying Wave Properties of Sound –
Pressure vs. Time
Equipment
File: Sound Recorder; Your voice;
Sound sensor (microphone)
Tuning fork;
Mallet (Tr = tokmak);
Open the file Sound Recorder. Experiment with this new sensor by humming into
the sound sensor's input. Answer the following questions based on your experiments.
Describe your evidence for each answer. Make sketches when helpful.
14. (a)
(b)
How does the displayed data change if you hum loudly versus softly? Try to
hum one constant note.
How does the displayed data change for a low-pitched hum versus a highpitched hum? Try to hum with a constant loudness. (Hint: you may want to
look at the time for each cycle)
Studying a Human Voice
Hum a single note with a constant loudness into the sound sensor's microphone
and record data. Adjust the axes to show 10 to 20 wavelengths of this sound. Print that
graph and put it in your logbook, being sure to identify what this graph represents. On
this graph, mark exactly 5 wavelengths.
41
15.
Calculate the frequency of your hum. Explain how you are able to do this from
your graphed data. Hint! Measure the time for five cycles and use that to determine the
time for one cycle.
Studying a Tuning Fork
Be sure to treat the tuning forks with care and to only "ring" them in the
manner demonstrated by your instructor.
"Ring" the tuning fork and hold it near the sound sensor's microphone and record
data. Adjust the axes to show 10 to 20 wavelengths of this sound. Print that graph and
put it in your logbook, being sure to identify what this graph represents. Mark exactly 5
wavelengths.
16. (a)
Calculate the frequency of the tuning fork (show your calculations).
(b)
How does your result for the tuning fork compare to the number stamped on
the tuning fork (at the base)?
(c)
How does the frequency of your hum compare to the frequency of the tuning
fork? How does that relate to how they sound to you?
17. Knowing the speed of sound in the room (from your earlier measurements), what is
the wavelength of the sound waves from the tuning fork? From your hum? Show
your work.
Discuss the shapes of the sound waves that you recorded (Amplitude versus time)
for your hum and the tuning forks. Relate the characteristics of these sounds to the
properties of waves and to what you hear.
18.
Application #3 - Studying the Frequency of Different Sounds –
Intensity vs. Frequency
Equipment
File: FFT display file, your voice, Tuning fork, Mallet, Sound sensor
(microphone).
Open the file FFT display file. This software is able to display the intensity of
sound as a function of the different frequencies contained within the sounds recorded by
the sound sensor. Note, time is not one of the axes of this display! It is Intensity vs.
Frequency. This display of the frequencies is often referred to as the "frequency
spectrum" of the sounds. Experiment with this new display by humming into the sound
sensor's input. Answer the following questions based on your experiments. Describe
your evidence for each answer. Include sketches to make your descriptions more
clear.
42
19. (a)
(b)
(c)
(d)
(e)
20.
Describe the frequency spectrum that results from a ringing tuning fork. What
frequency was found to be the loudest?
Describe the frequency spectrum that results when you hum a single lowpitched note. What frequency was found to be the loudest? How does the
spectrum compare to that of the tuning fork?
Describe the frequency spectrum that results when you hum a single highpitched note. What frequency was found to be the loudest? How does the
spectrum compare to that of the tuning fork?
Describe the frequency spectrum that results when you say a vowel sound,
such as "iiiiii." What frequency was found to be the loudest? How does the
spectrum compare to that of the tuning fork?
Describe the frequency spectrum that results when you say a consonant sound,
such as "mmmmmm." What frequency was found to be the loudest? How
does the spectrum compare to that of the tuning fork?
Consider the frequency spectra of different sounds made by human voices you
observed in the previous question. How might the FFT spectra help you to explain
hearing loss for aging people who lose the high frequency sounds first? Which
spoken sounds are they likely to lose first? Explain your thoughts.
End of Lab Procedures
Quit the software by selecting "Quit" from the "File" menu.
Return all coils to their proper storage location on the equipment room. Be sure they are
stored in a safe manner to keep the coils from being ruined.
Return the tuning fork(s) and mallet to the provided storage bin at your station.
If you are in the last lab section of the day, then shut down the computer.
43
Appendices
A - Guidelines For Creating A Graphical Model
One of the most important skills that you will be developing during this laboratory course
is the skill of analyzing and presenting data with graphs. Graphical models are an extremely
important tool of the physical sciences, the social sciences, business, and any other discipline that
is concerned with trends in data. Graphical models of data are helpful because they allow you to
actually picture how your data is behaving and to understand the relationship between your data.
As the saying goes, a picture is worth a thousand words.
Presented on the next page is a step-by-step guideline for making a graphical model of
experimental data. Read through these steps carefully. You can refer to the graphs below to see
an example of each of the steps put into practice.
Motion of a Toy Car
Sample Graph of Experimental Data as
discussed on the next page:
Displacement (m)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
4
8
12
16
20
24
28
32
28
32
Time (s)
Motion of a Toy Car
Sample Graph with Linear Graphical Model
as discussed on the next page:
Displacement (m)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
4
8
12
16
20
Time (s)
44
24
Guidelines for Creating a Graphical Model of Experimental Data
(1) Decide which two variables you are going to graph.
(2) Decide which variable will go along the vertical axis (often-called "y" axis) and which will
go along the horizontal axis (often called "x" axis). There are two possible ways to decide:
(a) If you are told to graph Something vs. Another Thing, then Something is always graphed
on the vertical axis and Another Thing is always graphed on the horizontal axis. It may be
helpful to keep in mind that such a graph reference always has the form y vs. x.
(b) If you are told to graph Something as a function of Another Thing, then Something is
graphed on the vertical axis and Another Thing is graphed on the horizontal axis. It may be
helpful to keep in mind y as a function of x.
(3) Once you have decided what to graph on an axis, then you will need to decide the range
(minimum value to maximum value) for the scale on that axis. Choose a scale (spacing) for
each axis so that your data will take up over half the graph paper, but will not go off the
edge of the graph paper. Your graphed data should cover at least half of a page.
The beginning of each scale (usually a value of zero) is the lower left-hand corner of the
graph. On a linear Cartesian graph, the scales must be marked off in equal increments.
Choose your increments wisely (such as intervals of 2's like in the sample graph on page 4).
You should include the origin (where the vertical and horizontal axes equal zero) on all
graphs unless there is a very good reason for not including it.
(4) Give your graph a descriptive title (like, Motion of a Toy Car) that is relevant to your data.
Write this title above the graph.
(5) Give each axis a descriptive label (like, Time) that is relevant to the data. Write this label,
including the appropriate units in parentheses, next to the corresponding axis.
(6) Plot your data carefully.
(7) Draw a single, best-fit, smooth curve representing all the plotted data points. Data
frequently follows a linear relationship (a special case of a smooth curve). In such cases,
your best-fit line should be drawn with a straight edge (like a ruler). Never play connectthe-dots with your graphed data!
This best-fit smooth curve (or line) is the graphical model of your data.
45
B1 - Guidelines For Creating A Linear Functional Model
A graph is a powerful tool because it is a picture that represents a complex situation.
Another way to express the information displayed in a graph is by determining a functional
(mathematical) expression that represents the best-fit line that follows the trend of your data. The
simplest of these trends is called a linear fit (that is, the data generally follows a straight line).
Linear functions are very important for describing much of the physical world around us.
Therefore, it is important that you have a good understanding of this group of functions. The
following description is an introduction to linear function models of data, but do not worry if you
do not understand it all at once. You will have many more chances to practice thinking about
these functions throughout the semester!
Introduction to Linear Functions
A function y(x) is linear if it has a constant rate of change. If a function is linear, then the
following four statements must be true:
(1) The data will appear linear on a Cartesian graph.
(2) The data can be represented mathematically with an expression of the following
form:
y=mx+b
(3) The change in y ( y) divided by the change in x ( x) is a constant.
(4) For every unit change in x (xn+1 = xn + 1), y changes by a constant amount
(yn+1 = yn + some constant).
These four statements are summarized in this diagram.
Linear
expression:
y
x
= constant
y=mx+b
Rate of
change of y
is constant
Change x by 1
Linear when
plotted on a
Cartesian graph
y changes by a
constant amount
46
The form of the mathematical equation that represents a linear fit is:
where:
y
x
m
b
y
mx b
is the variable on the vertical axis
is the variable on the horizontal axis
is the slope of the best-fit line
is the value of the variable y when x = 0
The expression y mx b is purely mathematical. Since this is a physics course and
not a math course, all such expressions used in this course must represent physical phenomena.
That is, your mathematical equation should be a functional model of your experimental data. In
order to create a linear functional model, you should complete the five steps outlined below.
Your functional models should always be put into the proper form—even if the model was
generated by Excel.
Guidelines for Creating a Linear Functional Model of Experimental Data
Note:
The examples given in parentheses in the following steps refer to the sample graph on
page 1 of Appendix A.
(1) Identify the name and units of the variable on the vertical axis of your graph. Assign a
single-letter variable based on this name. (For example, Displacement (m) d (m))
(2) Identify the name and units of the variable on the horizontal axis of your graph. Assign a
single-letter variable based on this name. (For example, Time (s) t (s))
(3) Identify the value of the variable on the vertical axis where the best-fit line crosses the
vertical axis. This value (and its corresponding units) is often called the "y intercept" and it
is represented as b in the mathematical expression of a linear function. (For example, b =
0.21 m)
(4) Calculate the slope of the best-fit line. To do this, you first select two positions on the bestfit line that you drew (one from the upper end of the line and the other from the lower end of
the line). These positions should not be actual data points (or you wouldn't need to draw a
line!). Determine (as accurately as possible by carefully reading the values from the
horizontal and vertical scales of your graph) the coordinates of these two points P 1 (x1,y1)
and P2 (x2,y2) from your graph and then calculate the slope using this formula:
Slope = m =
(For example, slope =
(5)
1.08m 0.27m
30s 2s
0.81m
28s
0.029
rise
run
y
y2
y1
x
x2
x1
m
)
s
Once you have completed steps 1 - 4, you are ready to put it all together into a functional
model of your experimental data. Just plug your answers into the expression (For example,
the functional model is: d (m) = (0.029 m/s) t (s) + 0.21 m). Note that the variables
represent exactly what was graphed with no substitutions. This makes the equation easy to
use later.
47
B2 - Examples of Different Functional Models
There are many different kinds of mathematical functional models that are useful
for physics. The following charts give examples of some common kinds of functional
models along with corresponding graphical models that show the general trends of such
functions. You can use these graphical models to help guide your selection of the form
of a functional model to represent your data sets throughout your physics labs.
Remember, when you use a graphical model or functional model to represent real data,
you must use appropriate variables (not x and y) and you must specify the appropriate
units.
Sample Functional Model
Graphical Trend
Linear Function
(Also known as Directly Proportional)
General form: y
mx
Excel Trendline type: "Linear"
Linear Function
General form: y
mx
b
Excel Trendline type: "Linear"
48
Inverse Function
(Also known as Inversely Proportional)
General form: y
m
1
x
mx
1
Excel Trendline type: "Power"
Mathematical Functional Model
Graphical Trend
Quadratic Function
General form: y
ax
2
bx
c
Excel Trendline type: "Polynomial
– order 2"
Squared Function
General form: y
ax
2
Excel Trendline type: "Power"
49
Inverse-Square Function
General form: y
a
1
2
x
ax
2
Excel Trendline type: "Power"
Exponential Decay Function
General form: y
ae
bx
Excel Trendline type: "Exponential"
50
C - Guidelines for Handling Numerical Values
Experimental data necessarily contain uncertainties. That is, you cannot experimentally
measure any quantity with perfect precision. Therefore, when you calculate results from data,
you need to preserve the highest honestly allowable precision. Such calculated results should
give both the value found and the degree of uncertainty in this resultant value. For example, a
student measuring the acceleration due to gravity may report her result as: g = 9.78 m/s2 0.04
m/s2.
Many methods for achieving the degree of uncertainty are found in many texts; a brief
review of some of the basic techniques is given below.
Significant Figures
Numerical results should be written paying proper attention to significant figures. A
significant figure in a number is a digit that affects the precision with which the number is given.
It is believed to be closer to the actual value than any other digit.
The following lengths all contain 3 significant figures (which are underlined):
6.10
103 cm
6.00 cm
0.596 cm
0.000610 cm.
In rounding off a result to the correct number of significant figures, the last digit
retained is increased by one if the first discarded digit is five or greater; otherwise it is not
changed.
It is customary to carry one more figure in computations than is needed for the final
result. Avoid the labor (and bad form) associated with the use of too many figures. For example,
the statement that "the length is 3.20 meters" implies that the correct length lies between 3.195m
and 3.205 m. If the uncertainty you attach to this length is 0.005 meters, quoting the total length
as being 3.202465 meters conveys no additional useful information.
As illustrated in this example, zeros can be significant figures, and it is important to show
them when appropriate. Non-significant zeros are most conveniently taken into account by using
scientific notation. For example, if you wish to write 4900 s to three significant figures, you write
4.90 103 s.
Here is an example to help clarify this discussion. It is desired to determine the area of a
rectangle whose sides have been measured and found to be 65.27 cm and 73.83 cm. To
determine the area, the product of the lengths of the two sides is taken which yields:
Area = Height
Length = 65. 27 cm
51
73. 83 cm = 4818.8841 cm2
(As shown on your calculator)
Although there are eight digits in the answer produced by your calculator, not all of them are
significant. A working rule of thumb is to round off the product to the least number of significant
figures that appears in either of the terms from which it is formed. In the present example, both
terms have four significant figures. Therefore, the product should be rounded off to four
2
significant figures and recorded as 4819 cm .
The reason for this procedure lies in the fact that the last significant digit implicitly
specifies the precision of the measurement. The side of length 65.27 cm is known to lie between
65.265 cm and 65.275 cm. The length of the other side lies between 73.825 cm and 73.835 cm.
This means that the rectangle has an area greater than:
65.265
73.825 cm = 4818.188625 cm2
65.275
73.835 cm = 4819.579625 cm2
But smaller than:
These areas differ from each other in the fourth place, and so there is no significance to be
attached to the digits beyond the fourth place. Therefore only the first four places in the product
should be retained.
You should be mindful of significant figures during your physics labs, although not
overly worried about them. As long as you don't include ridiculously too many or too few
significant figures, your instructor will not mark you down. Keep in mind that your calculator
and the Excel software will gladly give you a pre-determined number of significant figures
(usually 8). It is your job to interpret these values and record an appropriate number of
significant figures in your logbook.
Precision
Precision is the extent to which a given set of measurements of the same data set
agrees with the average value. A precise measurement is one that is very repeatable.
This is a good way to compare two values obtained within the lab. You may estimate the
precision of your measurements using one of the following:
Percent Difference between Two Values =
value # 1 - value # 2
average of 2 values
Percent Difference between One Value and All Values =
52
100 %
certain value - average of all values
average of all values
100 %
Accuracy
Accuracy is the extent to which a given set of measurements of the same data set
agrees with the accepted value. An accurate measurement is one that results in a value
that is very near to the true value. You may estimate the accuracy of your measurement
using:
Percent Difference =
accepted value - experimental value
accepted value
53
100%
D - Error Analysis Guidelines
Propagation of Errors
Assume we have a function of two variables x and y, f(x, y). The variables x and y are
assumed to be independent in the sense that each can vary arbitrarily without affecting the other.
To find the change in the function f due to small changes in x and y, we calculate the total
differential of f. In our example of area, f = x y, so that
df = x dy + y dx.
Now, dx is to be associated with our error in x, and dy with our error in y. df will then be
associated with our error in f resulting from dx and dy. Now a difficulty arises. We do not know
whether dx and dy are positive or negative quantities since we have assumed our errors to be
random in nature. Thus, for example, the maximum uncertainty (overly pessimistic) is given
when both errors are taken to be in the same direction; hence
fmaximum = |x y| + |y x|
where the vertical bars denote absolute value, and where we have changed our notation to reflect
the fact that our errors are not true differentials, i.e., we have made the identifications
df
f;
dx
x; and dy
y.
On the other hand, the minimum uncertainty (overly optimistic) is given when both errors are
taken to be opposing so that the uncertainty would be
fminimum = |x y| - |y x|.
In order to circumvent this difficulty, we use the methods of statistics that tell us that a
valid estimate of the error in f (the ―probable‖ uncertainty) is obtained by squaring each of the
error terms, adding them together, and then taking the square root:
f
(x
y)
2
(y
2
x) .
Here is an example: Take f = area = x y, where x = (65.27 0.005) cm, y = (73.83
0.005) cm. Therefore, x = 0.005 cm and y = 0.005 cm. We then find that
2
[( 65 .27 )( 0.005 )] [( 73 .83 )( 0.005 )]
f
f = 0.49 cm2
0.5 cm2.
Therefore, we quote the area as being equal to 4818.9
2
cm2
2
0.5 cm .
For comparison, the maximum uncertainty for this same example would be given by:
fmaximum = (65.27) (0.005) + (73.83) (0.005)
54
2
0.7 cm .
Here are some additional formulas for finding the uncertainties in functions of
independent variables whose uncertainties are known.
Functions
Probable Uncertainty
f(x,y) = x + y
| f |
( x)
2
( y)
2
f(x,y) = x – y
| f |
( x)
2
( y)
2
f(x,y) = x/y
f(x,y) = x y
f(x) = xk
|
f
|
f
(
x 2
)
x
(
y 2
)
y
|
f
|
f
(
x 2
)
x
(
y 2
)
y
|
f
x
| |k
|
f
x
f(x) = sin x
| f| = |(cos x)( x )|
where x is in radians
f(x) = cos x
| f| = |(sin x)( x )|
where x is in radians
However, if f(x, y) is a complicated function of x and y it is generally easier to proceed by
calculating f0 = f(x,y), f1 = f(x+ x, y+ y) and f2 = f(x+ x, y- y), where x and y are the measured
values. (The important point is that x and y occur in f1 with the same sign, where- as in f2 they
occur with opposite signs.) The maximum uncertainty in f(x, y) is then given by the larger of the
two magnitudes |f0 - f1| and |f1 - f2|.
Repeated Measurements
Finally, we consider the situation when we measure an experimental quantity (call it x) n
times. In general, every measurement may yield a different value of x because conditions of
measurement are not precisely identical for the different measurements. It is clear that if one takes
the average value for these n measurements, the effect of random errors will be decreased.
Let the average of these measurements be denoted by x , and let xi be a particular
measurement, the ith measurement. Then
n
1
2
(x)
(xi x)
n(n 1) i 1
For example, suppose four determinations of g yield 9.80 m/s2, 9.79 m/s2, 9.82 m/s2, and 9.83
m/s2. Then
2
(9.80 9.79 9.82 9.83)m / s
2
g
9.810m / s
4
and
55
(g )
1
[(0.01)
2
4 3
2
2
0.91 10 m / s
(0.02)
2
(0.01)
2
2
(0.02) ]
We therefore write our result as g = 9.810 0.009 m/sec2, if we can assume that our systematic
errors are negligible in comparison to the random ones.
56

G - Hints for Saving Time in Lab

At the beginning of the semester, some students express concern that they are not able to
complete all of the activities in each of the labs. Whereas it is not the end of the world (or your
grade!) if you do not finish any given lab, experience has shown that there are a number of things
that can be done to improve your efficiency at completing your work during the set laboratory
period.
Try to keep the following pieces of advice in mind as you work through the experiments:









Come to the lab room ready to work! This means you should have read and thought about
the lesson you are about to complete.
Be sure to show up to lab on time so you are ready to start working right away. In addition,
you should be sure to leave lab at the end of the period so that you do not delay the next
students. Remember, all logbooks must be turned in before you leave!
Stop working on the experiment and start writing your concluding discussion at a set time
each week (say, 10 minutes before the end). It is more important that you reflect on your
work than for you to try to get every last question or experiment finished.
If you are using sensors to collect data, then test your equipment setup quickly first before
trying to collect real data. For example, if you are using a motion sensor to detect the
motion of a cart, then first move the cart slowly along the track with your hand. This way
you can tell if the motion sensor can see the cart for the full distance. If it can’t see the cart,
then try aligning the motion sensor, putting a flag on the back of the cart, or moving your
book bag and/or lab partner out of the way.
Write your data down in a coherent way the first time. You should never have to recopy
your data.
Don’t waste time reporting extra materials! You should only report what you need for your
logbook.
If you are collecting a large set of data and you know that the data will eventually have to be
graphed, then start out by setting up an appropriate data table and prepare a set of axes so
you can plot your data as it is collected. This can be very efficient and help you spot
problems during data collection instead of waiting until the end. If you are going to do this,
then be sure to choose appropriate values for your axes. One way to do this is to begin by
collecting data for the biggest and smallest values so you know what limits you will need.
If after thinking about the experiment, you are still confused about the directions in the lab
manual or about what you are trying to measure, then be sure to ask questions!
Keep in mind that after a few labs, all lab students become more efficient working in the lab
and usually are able to finish labs as the semester moves along.
57