PHYSICS
LAB
NOTES
FOR
MECHANICS, HEAT
AND
SOUND
EXPERIMENTS
PHYSICS 6
Los Angeles Harbor College
J. C. FU
R. F. WHITING
© 1992
Eighteenth Edition
August 2005
TABLE OF CONTENTS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Measurement ................................................................... 1
Acceleration Due to Gravity ............................................. 7
The Addition of Vectors .................................................... 10
Projectile Motion .............................................................. 14
Newton’s Second Law...................................................... 17
Centripetal Force Thistle Tube Method .......................... 20
The Coefficient of Friction ................................................ 23
Conservation of Mechanical Energy ................................ 27
The Ballistic Pendulum .................................................... 30
Torque and Center of Mass ............................................. 34
Archimedes' Principle ....................................................... 37
The Coefficient of Linear Expansion ................................ 41
The Heat of Fusion of Ice................................................. 44
Standing Waves on Strings .............................................. 47
The Speed of Sound in Air ............................................... 50
The set of lab experiments that you will be doing this semester will, hopefully
elucidate for you some abstract concepts, enable you to test a few hypotheses or theories
using the scientific method, realize the capabilities or limitations of certain equipment and
procedures, and to think analytically.
Each lab period will begin with a presentation / discussion of the experiment
indicated in your lab notes. Come prepared, having read the references given. An attempt
has been made to have the labs run in parallel with lecture. Share work with your partner
so that each person will have an opportunity to have hands-on experience.
J. C. Fu, Ph.D.
R. F. Whiting, M.S.
Experiment 1: MEASUREMENT
PURPOSE:
To determine the precision of measurements made using different devices for
measuring length, mass and time, and to learn to report data with the appropriate number
of significant Figures.
INTRODUCTION:
The three basic units in the SI system of units are the meter, kilogram, and second.
For measurement of length we will use micrometers, vernier calipers, and tape
measures. When using the micrometer, be sure to take account of the zero reading of the
instrument. For all the various measuring devices, with the exception of the vernier caliper,
estimate to the nearest tenth of a division in order to get the most out of the instrument.
For measurement of mass, we will use the double pan balance. If the mass of your object
exceeds the capacity of the balance, use a counter mass on the other pan.
The electric stop clock will be used to measure time. The 60-Hertz oscillations
assure the accuracy of these devices.
It is good practice to tabulate (put in table form) your data whenever possible.
Now to say something about precision and accuracy. Precision is a measure of how
reproducible a measurement is, for example, if one measures an eraser 3 times, using the
same ruler, and gets the following readings: 5.2 cm, 5.1 cm, and 5.3 cm, the result can be
expressed as 5.2 ± 0.1 cm. The ± 0.1 indicates the precision of the result. The accuracy
of a measurement is an indication of how close the measurement is to the true or accepted
value.
SUPPLIES & EQUIPMENT:
Tape measure
Vernier caliper
Wire gauge
Metronome
100-gram slotted mass
Micrometer
Metal cylinder
Wire and nail samples
Rock samples
Electric stop clock
Metric ruler
Meter stick
Double pan balance
Electronic balance
PROCEDURE:
A. LENGTH
1. Determine the area of the classroom floor in square meters. Make three separate
measurements of both the length and width and report your result with the average
deviation.
-1-
2. Measure the diameter and length of a metal cylinder using the micrometer and a ruler.
Then measure it again by using the vernier caliper. Calculate the volume. Observe the
rules for the use of significant Figures.
3. Using a wire gauge, measure the diameter of a length of wire. Measure it again, this
time with a micrometer. Compare the results, and number of significant Figures that
can be recorded in each case.
B. MASS
1. Determine the mass of a 100-gram slotted mass on the balance and record your result
with the appropriate uncertainty. Repeat the measurement on the electronic balance.
2. Determine the mass of an irregularly-shaped object such as a rock.
C. TIME
1. With the electric stop clock, time 50 beats of the metronome set at 120. Calculate the
beats per second and the beats per minute. Repeat three times and calculate the
average value.
2. With the electric stop clock, time 50 beats of your own or your lab partner's pulse.
Calculate the beats per second and the pulse rate (beats / min.). Repeat three times
for the same person and determine an average value.
-2-
MICROMETER
5
Object to be measured
0
45
0 1 2 3 4
Each division is 1 mm
= 0. 001 m
40
Each division is 0.01 mm
= 0. 00001 m
An example of how to read the micrometer when making a measurement:
(
Read the line at the contact
If there is no line there, read
point.
line just before the contact
the
point.
)
0
4.5
mm
45
4
+ 0 . 4 75
mm
4.975
mm
0
45
Read to the nearest hundredth of a millimeter.
-3-
4 7.5 div. X 0.01 mm/div. = 0.475
mm
VERNIER CALIPER
2.1 on the main scale
0
1
2
3
0
Object to be measured
5
4
5
6
10
5 on the vernier scale
The zero line on the vernier scale lines up with the
main scale at 2.1 cm plus a fraction of a millimeter.
Read to the nearest tenth of a millimeter.
An example of how to read the vernier caliper when making a measurement:
1. The zero line on the lower or vernier scale points up to 2.1 cm (plus a fraction of a
millimeter) on upper or main scale. (Note where the long arrow is pointing on the
diagram above.)
2. The 5th line on the vernier scale happens to line up with a main scale line (any line),
therefore the last digit is 5. (Note where the short arrow is pointing on the diagram
above.)
3. Any line on the vernier scale that lines up with a main scale line is the last digit. Thus,
the length of the object is 2.15 cm
WIRE GAUGE
Place wire here
1 .00 inch
Front:
Gauge #
= 2.54 cm
-4-
Back:
Diameter
in inches
DATA: MEASUREMENT
I. LENGTH
a) Tape Measure*: Floor
Width Deviation = | Width - Average Width |
W
Width (m)
W
Deviation (m)
W
Average Width (m)
W
Average Deviation (m)
 _________________
___________________
Length Deviation = | Length - Average Length |
L
Length (m)
L
Deviation (m)
L
Average Length (m)
 _________________
___________________
* Measure to the nearest millimeter
L
Average Deviation (m)
(Less than 10 m  4 sig. figs.)
(Greater than 10 m  5 sig. figs.)
Area Calculations:
A = L X W = ____________ m2
Average area
Positive deviation of Area:
A+ = ( W + W ) ( L + L )
= ____________ m2
Negative deviation of Area:
A = ( W  W ) ( L  L )
= ____________ m2
A =
Average deviation of area
A  A
2
=  ___________ m2
Area of room = _____________  _____________ m2
Average area  Average deviation
A = A  A
b) Micrometer (diameter) (4 significant Figures); Ruler (length) (3 significant Figures)
Item
Length (m)
Diameter (m)
Metal Cylinder
-5-
Radius (m)
Volume (m3)
c) Vernier Caliper (3 significant Figures)
Item
Length
Diameter
Radius
Volume
cm
cm
cm
m3
m
m
m
cm3
Metal Cylinder
d) Wire Gauge* (3 sig. Figures) & Micrometer** (3 sig. Figures)
Wire Gauge
Diameter
(cm)
Item
(inches)
Micrometer
Diameter
(m)
(mm)
(m)
Wire
* 1.00 inch = 2.54 cm
** To the nearest 0.01 mm
II. MASS
Pan Balance: 4 significant Figures
Electronic Balance: 5 significant Figures
Pan Balance: 3 significant Figures
Electronic Balance: 4 significant Figures)
(For 100 gm. or more:
For less than 100 gm:
Item
Mass (g)
Pan Balance
Mass (kg)
Electronic Balance
Mass (g)
Mass (kg)
Slotted Mass
Rock
III. TIME
Object
Electric Stop clock (3 significant Figures)
# Beats
Time (s)
Beats/second
Beats/minute
Ave. Beats/min.
Metronome
____________
Object
# Beats
Time (s)
Beats/second
Pulse Rate =
Beats/minute
Ave.Pulse Rate
Beats/min.
Pulse
____________
-6-
Experiment 2: ACCELERATION DUE TO GRAVITY
PURPOSE:
In this experiment, the numerical value of the acceleration due to gravity will be
determined by a graphical technique.
INTRODUCTION:
If one neglects the effects of air friction, objects relatively close to the Earth's
surface undergo uniformly accelerated motion. For our purposes, we will take this value of
acceleration to be 9.80 m/s2.
In this experiment, the data are obtained by analyzing a wax paper tape that has a
series of spark dots. The apparatus that produces the tape sparks every 1/60 of a second
as the free-fall body descends. Thus a time-distance record of the object in free fall is
produced and the acceleration due to gravity can be calculated.
By definition, acceleration is the time rate of change of velocity, so a plot of the
instantaneous velocity vs. time should yield a straight line, the slope of which is the
acceleration. For each spark interval, the average velocity is readily calculated, being the
distance the object falls in the interval divided by time it takes to fall that interval distance.
Use the fact that the average velocity is equal to the instantaneous velocity at the midpoint
in time of the interval.
SUPPLIES & EQUIPMENT:
Demonstration free fall apparatus
Plastic triangle
x0
x1
t =
1
Spark tape
Masking tape
t0
0
Distance = x
Meter stick
Metric ruler
1
30
second
t1
2
-7-
x  xi
Average velocity = v = xt = f
tf
 ti
PROCEDURE:
1. Obtain a spark tape, secure it to the table with masking tape and draw a straight line
perpendicular to the long direction of the tape, through every other dot. Start at the
third or fourth dot down from the top. You should obtain about ten intervals. Number
the perpendicular lines 0 through 10.
t = 0
Spark
Tape
1/60
1/30
2/30
3/30
x
Interval #
0
1
2
3
2. Measure and record the interval distances between each three successive dots,
starting with 0 - 1, then 1 - 2 and so forth. Enter your data in the table.
3. Calculate the average velocity for each of your intervals by dividing the interval distance
x by the elapsed time for each interval. The elapsed time is 1/30 second. Plot these
average velocities on the y-axis vs the corresponding midpoint in time on the x-axis.
Draw the best straight line for the data points by fitting the line so that the line is the
closest it can be to all the data points. Note that the line does not necessarily have to
pass through any particular point.
Sample Graph
DESCRIPTIVE TITLE
Average
Velocity
(m/s)
v
x
0
|
|
|
|
1
2
3
4
|
|
5
|
6
7
Time (s) X 1/60 sec
4. Calculate the slope of your straight line. This is done by dividing the rise (change in y)
by the run (change in x) for any two points on the line, not necessarily data points, since
it is possible that no data points lie on the line. Choose the two points so that they are
widely separated. Slope = V/t = a = g (experimental)
5. Calculate the percent error of the value of g you obtain from your graph when
compared to the given value of 9.80 m/s2.
% error =
| gexp erimental  9.80 m / s2 |
9.80 m / s2
-8-
X 100 %
DATA: ACCELERATION DUE TO GRAVITY
Data and Calculations Table: (Measure to a fraction of a millimeter.)
Interval #
Interval Distance x
(m)
Interval Time t
(s)
Average Velocity
(m/s)
Time from Zero to
Midpoint in Time
(s)
0–1
1/30
1/60
1–2
1/30
3/60
2–3
1/30
5/60
3–4
1/30
7/60
4–5
1/30
9/60
5–6
1/30
11/60
6–7
1/30
13/60
7–8
1/30
15/60
8–9
1/30
17/60
9 – 10
1/30
19/60
Acceleration due to gravity (from graph) = ________ m/s2
% error ___________
-9-
Experiment 3: THE ADDITION OF VECTORS
PURPOSE:
To establish the condition for equilibrium of a suspended metal object.
INTRODUCTION:
The first condition for equilibrium is that the vector sum of the forces (the net force)
acting on an object is zero:
F = 0
in a two-dimensional problem this becomes:
Fx = 0, and Fy = 0.
SUPPLIES & EQUIPMENT:
Force table
Metal cube (brass or iron)
50-gram mass holder
Slotted masses
Metric ruler
Circular bubble level
Electronic balance
Protractor
PROCEDURE:
A. ADDING FORCES WITH THE SAME MAGNITUDE BUT DIFFERENT DIRECTIONS.
1. Level the force table using a spirit level.
2. Determine the mass of a metal cube ___________ grams, ___________ kg, on the
electronic balance.
3. Determine the weight of this metal object ___________ N. W = mg, where g = 9.80
m/s2
4. Clamp three pulleys along the edge of the force table as in fig. 1., with A = 10o, B = 10o, and c = 180o (position of cube).
5. Apply forces FA and FB of the same magnitude to balance the weight of
the metal cube (at C) by placing masses on the hangers at A and B.
6. Record FA, FB and  in data table 1.
7. Repeat steps 4 and 5, with: A = 30o, B = -30o and A = 50o, B = -50o
- 10 -
8. Using the graphical method, add the vectors head-to-tail to determine the resultant FR
of FA and FB. Use a ruler and protractor to draw the vectors to scale and be sure to
specify the scale you are using. (for example 1N = 3 cm). Enter the data in data table
1. (See Fig. 2.)
FA
Object
W
FB
A
180o
B
(Load)
0o
FR
Fig. 1
Fig. 2
DATA AND CALCULATIONS Table 1: VECTOR ADDITION
FR
A
B
+ 10o
- 10o
+ 30o
- 30o
+ 50o
- 50o
For Example :
FA
(N)
FB
(N)
(graphical method)
(N)
W
Object Weight
(N)
|f|
Frictional Force
(N)
FA = mA* g = ( ________ kg) X (9.80 m/s2) = ____________ N
* Note that mA includes the mass of the mass hanger.
At Equilibrium:
 F = FR + (-W) + Frictional Force = 0
| Frictional Force | = | f | = | FR - W |
- 11 -
B. Adding forces of different magnitudes and directions.
1. Balance the weight of the metal cube, W, with F A and FB of different magnitudes and
different angles A and B on the force table. See Fig. 3a.
2. Calculate the resultant of FA and FB using the components method. See Fig. 3b which
shows the components of FA.
FA
W
FA
FyA
A
180o
B
0o
FxA
FB
Fig. 3a
Fig. 3b
DATA AND CALCULATIONS Table 2: A = ______________
VECTOR ADDITON
FA = ______________
FA
FyA
Vector
FB
FyB
FAx = FA cos A
FR =
Fx2  Fy2 ;
FB = ______________
x-Component
(N)
FxA
FxB
B = ______________
y-Component
(N)
FA
FxA =
FyA =
FB
FxB =
FyB =
FR
Fx = FxA + FxB =
Fy = FyA + FyB =
FBx = FB cos B
FAy = FA sin A
FBy = FB sin B
tan  = Fy / Fx   = tan-1 (Fy / Fx)
1. FR = _________________
at ____________ Degrees (from + x-axis)
2. Weight of Load (Metal Cube) = ___________ N at 180 o (from + x-axis)
- 12 -
Experiment 4: PROJECTILE MOTION
PURPOSE:
The object of this experiment is to determine the initial velocity of a projectile from
the range and time-of-flight measurements. Also, the equations of motion will be used to
predict the point of impact of a projectile.
INTRODUCTION:
A projectile is any object in motion through space, which no longer has a force
propelling it. Examples are: thrown balls, rifle bullets, falling bombs and rockets (after the
propelling force is gone).
In order to determine the initial velocity of a projectile fired horizontally, one first
makes use of the equation y = ½gt2 to calculate t, the time of flight; where t = 2y / g .
Then, from a measurement of the range (horizontal distance) the initial velocity, v o , can be
determined from the equation s = vot .
For a projectile fired at an angle, the range of a projectile can be determined if the
angle of elevation, the initial velocity and the initial height of the projectile above the
landing point are known.
SUPPLIES & EQUIPMENT:
Ballistic pendulum apparatus
Plain white and carbon paper
Spirit level
Short support rod
One and two meter sticks
Large cardboard
Metric ruler
Wooden board for inclined plane
Clamp and rod for inclined plane
vo
y
s
Fig. 1, Part A
- 13 -
Inclinometer
Catch box
"C" clamp
Plumb bob
PROCEDURE:
A. INITIAL VELOCITY
1. Be extremely careful not to hit anybody with a projectile during this experiment.
2. Clamp the gun (not too tightly) to the table, using the inclinometer to orient the gun to
fire horizontally and take a trial shot. Tape a large piece of cardboard to the floor
centered on the spot where the projectile landed. On top of the cardboard, tape a
carbon paper and a plain paper to record the point of impact. Use one of the boxes
supplied to catch the projectile (ball).
3. Take six shots. Measure the range of each shot accurately. Record your values for the
ranges in the data table.
4. Measure the height from the floor to the bottom of the ball, this is y and is the vertical
displacement of the projectile. Use a plumb bob to get the exact vertical direction.
Calculate the time of flight from this measurement:
t=
2y / g
5. Calculate the range s and the average initial velocity. vo = s / t .
DATA FOR PART A: PROJECTILE MOTION
Data and Calculations Table: Initial Velocity, vo
Trial
y*
(m)
Averagey
(m)
s**
(m)
s
(m)
1
2
_________________
3
4
5
6
* 3 sig. figs.
** 4 sig. figs.
- 14 -
_________________
Part A Calculations:
Time of Flight
t=
2y / g
Initial Velocity
vo = s / t .
Average Initial Velocity, vo ____________
Ballistic Gun # ____________
B. PREDICTION OF THE RANGE
1. Clamp the spring gun to a board at an arbitrary angle of between 10 o and 20o.
Measure this angle precisely with an inclinometer.
2. Measure the height of fall, y (= yf - yi).
3. Calculate the expected range. Fire the projectile and measure the range. Fire the
projectile five more times and determine an average measured range.
4. Calculate the percent difference between the measured and calculated range.
Compare the results.
y
Initial
Position
Of Projectile
vo
Gun
v oy = v ocos 
x
v ox = v ocos 
y
x
\
Fig. 2,
- 15 -
Part B
Final
Position
Of Projectile
Data for Part B:
Angle of Elevation () ________________
(degrees)
Height from floor to the bottom of ball, y ___________ m
Data and Calculations Table: Measured Range
Measured Range x
(m)
Average Measured Range
(m)
_____________________________________
Part B. Calculations:
Average Initial Velocity, vo = __________ (From part A)
vox = vo cos = __________
1
1.) y = voy t + 2 gt2
2.)
1
2
voy = vo sin = __________
gt2 + voy t - y = 0
3.) At2 + Bt + C = 0
1
Quadratic Equation
A=2 g
= ___________
B = voy
= ___________
(g = - 9.80 m/s2)
C = - (y) = ___________ (y is negative, therefore C is positive)
See Fig. 2
t
B 
B
2
 4 AC
2A
= __________ s (Choose t such that it is a positive number)
Expected Range:
x = vox t
Expected range, x _________________ m
Percent difference in measured and expected range _________________ %
- 16 -
Experiment 5: NEWTON'S SECOND LAW
INTRODUCTION:
The acceleration of an object is directly proportional to the resultant force acting on
it and inversely proportional to the mass being accelerated. Furthermore, the direction of
the acceleration is in the direction of the resultant force.
 F = ma
(Newton's Second Law)
Using an air track, the acceleration of masses due to an unbalanced applied force
will be determined, and compared with the acceleration calculated from the equation of
motion for a uniformly accelerated object.
From Newton's 2nd law:
F = (m1 + m2)a
m2g = (m1 + m2)a
Solving for a:
m1
a
m2g
m1  m 2
a
m2
From the equation of motion:
W = m2g
s = vot + ½at2. With
vo = 0,
2s
a 2 .
t
SUPPLIES & EQUIPMENT:
Air track and accessories
Thread and scissors
5 & 10-gram slotted masses
5-gram mass holder
Photogate #1
Electronic balance
Photogate #2
m1
m2
Fig. 1. Experimental Setup
- 17 -
PROCEDURE:
1. Level the air track by adjusting the leveling feet and balancing glider at the center of
the air track. Turn air supply off when this is accomplished. Do not lean on the air
track or the table (use another table for writing) during the experiment.
2. Determine the glider's mass m1 on a balance and convert this measurement to
kilograms.
3. Place photogate #1 at the position x1 = 80 cm and photogate #2 at the position x2 =
150 cm.
4. Place a 5-gram mass holder at the end of the thread running over the pulley. Add a 5gram mass onto the mass holder, so now, m2 = 10.0 grams = 0.0100 kg.
5. Set the photogate stop clock to the "pulse" mode. and push the "reset" button. Set
the resolution scale to 1ms. Set the “memory” switch to the “on” position. Make sure
the air is flowing steadily before you let go of the glider.
6. Turn on the air supply. Delicately hold the glider as close to the light beam of gate # 1
as possible (just before the LED on top of the gate lights up). Then release glider (do
not push or pertube the glider) and record the displayed time.
7. Reset the stop clock and repeat the procedure two more times. Average the three
values and record in the data table.
8. Add a 5-gram mass onto the mass holder. Repeat steps #5 and #6. (Remember that
m2 equals the mass of the holder plus the mass on the holder, so the total mass for
this step is 15 g.)
9. Repeat step #8 for m2 = 20 grams (including hanger). and m2 = 25 grams (including
hanger).
10. Calculate the acceleration of the masses by using the equation of motion:
1
s = vot + 2 at2 ,
s = | x2 - x1 |
2s
a= 2
t
11. Compare this calculated acceleration with the value calculated using Newton's law F
= ma.
with
vo = 0,
1
s = 2 at2
- 18 -

m1
Fnet = W = m2g
Fnet = (m1 + m2)a
a
m2
m2 g
F

m1  m 2
m1  m 2
W
DATA: NEWTON'S SECOND LAW
Data and Calculations Table:
m1
(kg)
m2
(kg)
0.0100
0.0150
s*
(m)
Time, Trial 1
(s)
Time, Trial 2
(s)
Time, Trial 3
(s)
Average Time
(s)
Acceleration from: ak = 2s/t2
(m/s2)
Force from: F = m2g
(N)
Accelerated mass: m1 + m2
(kg)
Acceleration from Newton’s
2nd Law: aN = F/(m1 + m2)
(m/s2)
% difference =
ak  aN
aN
X
100%
* Measure carefully each time.
- 19 -
0.0200
0.0250
Experiment 6: CENTRIPETAL FORCE THISTLE TUBE METHOD
INTRODUCTION:
In this experiment we will study the motion of an object travelling in a circular path.
A small object of known mass will be rotated in a circular path. The centripetal force will be
determined directly and then calculated from measurements of the radius and the velocity.
The following relation will be verified:
2
Fc = mv
r
SUPPLIES & EQUIPMENT:
Thistle tube
String & scissors
# 5 Rubber stopper
Masking tape Red felt marker
Hooked masses, 50g, 100g & 200g
Stop clock
Meter stick
Electronic balance
PROCEDURE:
1. Determine the mass of a stopper. Tie a 1.5 m length of string to the stopper, then
thread it through the thistle tube. Tie a 0.150 kg mass to the other end of the string.
The weight of this mass creates the tension in the string that provides the centripetal
force on the stopper.
r
2. To help you maintain the radial
distance, use a dot of red ink as a
marker at the top edge of contact
with the thistle tube.
Revolving mass, m
Thistle Tube
3. Using the stop clock, measure the
total time for 25 revolutions for two
different values of radial distance.
Try values close to 0.500 m and
0.750 m.
The time for one
revolution is the total time divided
by 25.
Mark
4. Maintain a steady horizontal swing.
(Actual Centripetal Force = Mg)
Hanging mass, M
5. The velocity is given by the equation:
circumference
2r
v = time for 1 revolution = T
where r is the radius of the circular path and T is the time for one revolution.
6. Repeat the above procedure for a 0.200-kg mass attached to the string.
7. What factors contribute to error in this experiment?
- 20 -
DATA: CENTRIPETAL FORCE
Data and Calculations Table 1:
Mass of Stopper, m
(kg)
Radius, r
(m)
*
Time for 25 Revolutions
(s)
Time for 1 Revolution, T
(s)
Velocity, v = 2r/T
Velocity2 = v2
*
(m/s)
(m/s) 2
A. Experimental Fc = mv2/r
(N)
Hanging Mass, M
(kg)
B. Centripetal Force from Fc = Mg
0.150
(N)
Percent error
of centripetal force A relative to B
*Approximately 0.500 m
% error = (A – B) / A X 100 %
- 21 -
0.200
Data and Calculations Table 2:
Mass of Stopper, m
(kg)
Radius, r
(m)
**
Time for 25 Revolutions
(s)
Time for 1 Revolution, T
(s)
Velocity, v = 2r/T
Velocity2 = v2
**
(m/s)
(m/s) 2
A. Experimental Fc = mv2/r
(N)
Hanging Mass, M
(kg)
B. Centripetal Force from Fc = Mg
0.150
(N)
Percent error
of centripetal force A relative to B
**Approximately 0.750 m
- 22 -
0.200
Experiment 7: THE COEFFICIENT OF FRICTION
PURPOSE:
The object of this experiment is to demonstrate some of the principles of dry friction
and to determine the coefficients of kinetic and static friction for wood-on-wood surfaces.
INTRODUCTION:
In this experiment, we will investigate some of the principles of friction, such as:
1. The coefficient of static friction, s, is usually greater than the coefficient of kinetic
friction, k. 2.The frictional force, f, is proportional to the normal force, F N. 3. Friction
always acts in a direction opposite to the motion of an object.
SUPPLIES & EQUIPMENT:
Friction board
String & scissors
Electronic balance
Slotted masses
Friction Block
Meter stick
Inclinometer
Metric ruler
Masking tape Clamp & rod for inclined plane
Spirit level
Clamp-on pulley
PROCEDURE:
A. COEFFICIENT OF STATIC FRICTION
We will determine the coefficient of static friction by tilting the board at an angle. At
the point where the angle is just enough to cause the block to slip (overcome friction), we
have:
s = f / FN
FN
f
Fx = 0:
f + (-mg sin = 0
y
mg cos

f = mg sin
mg sin
Fy = 0:
mg sin

FN + (-mg cos = 0
FN = mg cos
x
mg sin
Fig.1: Experimental setup for Part A
with associated forces shown
s = mg cos
s = tan
- 23 -
1. Place the block at the top of the inclined board. Experimentally determine the angle at
which the block just breaks loose and starts sliding down the incline, using an inclinometer.
2. Repeat step 1-A five times and calculate an average value for the angle , and then
calculate the coefficient of static friction s = tangent 
Data For Part A:
Data and Calculations Table 1: Coefficient of Static Friction.
FN
f

Trial
Average 
tan  = s
y
mg cos

mg sin

mg sin
1
2
x
fs = s FN
s =
3
fs
mg sin 
= mg cos  = tan
N
4
5
Average value of coefficient of static friction: s = ________________
B. COEFFICIENT OF KINETIC FRICTION
The coefficient of kinetic friction will be determined by making use of the fact that
the frictional force is proportional to the normal force, f = kFN .
FN
T
f
T
mg ( = W)
m2
F (Applied Force) = m2g
Fig. 2. Experimental setup for Part B
with associated forces shown.
- 24 -
1. Determine the mass of the friction block and record its mass on the data sheet.
2. Level the friction board on the table. Clamp a pulley on one end. Tie a string and mass
hanger to the block. Place slotted masses on the hanger until the block starts moving
with constant velocity once given a slight push. The force pulling on the block is the
applied force to overcome kinetic friction and is equal and opposite to the kinetic friction
force. Mark the place on the board with a piece of tape where you start the block in
order to start the block at the same place each time.
3. Repeat step 2-B four more times, each time adding 100 additional grams to the top of
the block.
4. Plot a graph of the magnitude of the force of friction, | f |, on the y-axis vs. normal force
on the x-axis. The slope of the graph can be used to calculate the coefficient of kinetic
friction.
f
Descriptive Title
(Newtons)
f
Slope =
f
= k
FN
FN
FN
(Newtons)
Fig. 3: Sample graph
Data For Part B:
Data and Calculations Table 2: Coefficient of Kinetic Friction.
f
m1 T
Trial
T
Total Sliding Mass
m1
1
2
3
4
(kg)
m 2g
At constant velocity,
Fx = 0
f + T = 0
Normal Force
FN = m1g
Hanging Mass m2
Applied Force m2g
(N)
(kg)
N)
Fy = 0
T – m2g = 0
Magnitude of
Frictional Force
(N)
Value of coefficient of kinetic friction from graph,
- 25 -
k = _____________
5
Experiment 8: THE CONSERVATION OF
MECHANICAL ENERGY
INTRODUCTION:
Though conservation of energy is one of the most powerful laws of physics, it is not
an easy principle to verify. If a boulder is rolling down a hill, for example, it is constantly
converting gravitational potential energy into kinetic energy (linear and rotational), and into
heat energy due to the friction between it and the hillside. It also loses energy as it strikes
other objects along the way, imparting to them a certain portion of its kinetic energy.
Measuring all these energy changes is no simple task.
This kind of difficulty exists throughout physics, and physicists meet this problem by
creating simplified situations in which they can focus on a particular aspect of the problem.
In this experiment you will examine the transformation of energy that occurs as an air track
glider moves down an inclined track. Since there are no objects to interfere with the
motion and there is minimal friction between the track and glider, the loss in gravitational
potential energy as the glider moves down the track should be very nearly equal to the gain
in kinetic energy. In the form of an equation, we have:
KE = (mgh) = mgh
where KE is the change in kinetic energy of the glider, KE = ½mv22 – ½mv12 and
(mgh) is the change in its gravitational potential energy (m is the mass of the glider, g is
the acceleration of gravity, and h is the change in the vertical position of the glider).
SUPPLIES & EQUIPMENT:
Air Track & accessory kit
2 Shim blocks, about 1 cm thick
Accessory photogate timer
Photogate timer transformer
Meter stick
Vernier caliper
Glider
Electronic balance
Photogate timer
Air supply
PROCEDURE:
PART A:
1. Level the air track as accurately as possible by setting the glider at the middle of the
track and adjusting the leveling screws until there is no movement of the glider. Once
leveled, do not lean on the table or push down on the glider.
2. Measure D, the distance between the air track support legs. Record the distance
above table A to the nearest millimeter.
3. Place a block of known thickness, H, under the support leg of the track. For greater
accuracy, the thickness of the block should be measured with a vernier caliper. Record
the thickness of the block above table A to the nearest tenth of a millimeter.
- 26 -
4.
Set up a photogate timer and an accessory photogate as shown in the figure below.
d
L
H
D
Fig. 1: Equipment Setup.
5. Measure and record d, the distance the glider moves on the air track from where it
first triggers the first photogate, to where it triggers the second photogate. You can
tell where the photogates are triggered by watching the LED on top of each
photogate. When the LED lights up, the photogate has been triggered. As always
when measuring with a metric ruler, your measurement should be to the nearest
millimeter.
6. Measure and record L, the length of the glider. (The best technique is to move the
glider slowly through one of the photogates, and measure the distance it travels from
where the LED first lights up to where it just goes off.)
7. Measure and record m, the mass of the glider.
8. Set the photogate timer to GATE mode, leave the memory function in the "off"
position, and press the RESET button.
9. Hold the glider steady near the end of the air track, then release it, (don't push), so it
glides freely through the photogates. Record t1 the time during which the glider
blocks the first photogate and t2 the time during which it blocks the second photogate.
Notice that t2 = ttotal - t1. (Photogate timer first displays t1 , then ttotal = t1 + t2 , and
does not display t2 by itself.)
10. Repeat the measurement four times and record your data in table A. You need not
release the glider from the same point on the air track for each trial, but it must be
gliding freely and smoothly (minimum wobble) as it passes through the photogate.
PART B:
1. Repeat procedure A with a block of greater thickness, H '. Record data in Table B.
- 27 -
CALCULATIONS:
1. Calculate , the angle of incline for the air track, using the equation  = sin-1(h/d).
Since sin  = h/d = H/D, you can calculate h = d (H/D), which is the distance through
which the glider drops vertically in passing between the two photogates.
2. For each set of time measurements:
a. Divide L by t1 and t2 to determine v1 and v2, the velocity of the glider as it passed
through each photogate.
b. Use the equation KE = ½mv2 to calculate the kinetic energy of the glider as it
passed through each photogate.
c. Calculate the change in kinetic energy, KE = KE2 - KE1.
D = distance between
supports
H
h
d
D
d = distance between
photogates


H = block thickness (distance
air track leg raised)
Fig. 2: Elevations
d. Calculate the average value of KE = KE2 - KE1, and calculate mgh. Find the
percent difference between them. A small value of this percent difference is
expected from the law of conservation of energy.
- 28 -
DATA SHEET: CONSERVATION OF MECHANICAL ENERGY
Part A:


D = ____________
h = ____________
H = ____________
 = ____________
L = ____________
d =____________
m =____________
Data and Calculations Table A
Trial
1
t1
(s)
t1
(s)
v1
(m/s)
v2
(m/s)
KE1
(J)
KE2
(J)
KE2 - KE1
(J)
2
3
4
5
Average KE = ____________ mgh = ____________ % difference = ____________
PART B:


D = ____________
h = ____________
H = ____________
= ____________
L = ____________
d =____________
m =____________
Data and Calculations Table B:
Trial
1
t1
(s)
t1
(s)
v1
(m/s)
v2
(m/s)
KE1
(J)
KE2
(J)
KE2 - KE1
(J)
2
3
4
5
Average KE = ____________ mgh = ____________ % difference = ____________
- 29 -
Experiment 9: THE BALLISTIC PENDULUM
In this experiment we will determine the initial velocity of a projectile by using the
principles of the conservation of momentum and the conservation of energy.
INTRODUCTION:
A device called a ballistic pendulum will be used in this experiment to determine the
initial velocity of a projectile. The device consists of a spring gun that propels a metal ball
of mass m into a pendulum bob of mass M. This pendulum-ball combination then swings
up onto a rack with a velocity v just after impact. The change in height h through which it
rises depends directly on the initial velocity vo of the ball.
In order to derive an expression for the initial velocity vo of the projectile, we can
make use of the law of conservation of linear momentum, expressed as:
Momentum Before Impact = Momentum After Impact
mvo
mvo = (m + M) V
m  M
vo = 
V
 m 
Before Impact
Eq.
1
The second part of the process involves the pendulum-ball combination emerging
with initial velocity v, then rising from h1 to h2. The conservation of energy for this part can
be expressed as:
KE1 + PE1 (at h1)
(m+M)V
= KE2 + PE2 (at h2)
KE = 0
0
KE1 - KE2 = PE2 - PE1 ; since v2 = 0
½(m + M)V2 = (m + M)gh2 - (m + M)gh1
PE = (m+M)gh
h2
h1 h
½(m + M)V2 = (m + M)gh
Immediately
After Impact
½v2 = gh
- 30 -
At Rest
So
V=
2gh
Eq. 2
Substituting the expression for V from Eq. 2 into the momentum Eq. 1, we have:
m  M
vo = 
 2gh
 m 
Eq. 3
SUPPLIES & EQUIPMENT:
Ballistic pendulum apparatus
Electronic balance
Ruler
Spirit level
C-clamp
PROCEDURE:
1. Level the apparatus on the lab table using a spirit level. You may need to shim the
apparatus. Lightly clamp the apparatus to the table using a C-clamp. Once leveled
and clamped, do not lean on the table or otherwise disturb the level of the apparatus.
2. Determine the position (hi) of the center of mass of the stationary pendulum relative to
the base plate. The center of mass is indicated by the pointed projection on the side of
the pendulum.
3. Determine the mass of the ball and record it on the data sheet.
4.
Fire the gun six times, each time recording the number of the notch in which the
pendulum comes to rest.
5. Calculate the average notch number. Place the pendulum at this average position and
determine the height (hf) from the base plate to the pendulum center of mass.
Calculate h = hf - hi.
6. Calculate the velocity of the ball and pendulum just after impact. V =
2gh .
m  M
7. Calculate the initial velocity of the ball: vo = 
 V.
 m 
8. Calculate the energy loss in Joules. The kinetic energy before impact is ½mv o2, and
immediately after impact the kinetic energy is ½(m+M)V2. What percent of the original
kinetic energy was "lost" to non-conservative work? Where did this energy go?
- 31 -
DATA SHEET: BALLISTIC PENDULUM
Ballistic pendulum number ______________ (See label on equipment)
Mass of Pendulum
______________ (See label on equipment)
Mass of Ball
______________ kg
Data Table 1: Pendulum Height Measurements
Trial
Notch #
Trial
Notch #
Trial
1
3
5
2
4
6
Notch #
Average Notch
#
hi = height of pendulum when hanging freely
hf = height of pendulum at average notch number
h = hf - hi
m  M
Initial velocity of ball: vo = 
 2gh
 m 
vo from Experiment 5, Projectile Motion
(Eq. 3)
______________ m
______________ m
______________ m
____________ m/s
____________ m/s
% difference between the two vo
____________
Velocity of pendulum & ball after impact, V =
2gh (Eq. 2)
____________ m/s
Momentum before collision: mvo =
____________ kg-m/s
Momentum after collision: (m+M)V =
Is momentum conserved in this inelastic collision?
____________ kg-m/s
____________
KEi before collision: ½mvo2 =
KEf after collision: ½(m + M)V2 =
____________ J
Is kinetic energy conserved in this inelastic collision?
____________
____________ J
Energy loss: W nc = KE + PE
W nc = KEf – KEi) + PEf - PEi)
W nc = KEf – KEi) + m + M)gh
% energy loss:
Wnc
X 100% =
(1 / 2)mv o2
____________ J
____________ %
- 32 -
Experiment 10: TORQUE AND CENTER OF MASS
PURPOSE:
The object of this experiment is to use the method of balancing torques to
determine the center of mass of a non-homogeneous meter stick, and to determine the
unknown mass of an object.
INTRODUCTION:
If a rigid object is in rotational equilibrium, the net torque acting on it, about an axis,
is zero. This equilibrium condition can be stated as:
 =0
where = Fd, F is the applied force, and d is lever arm. The lever arm is the distance from
the axis of rotation (the fulcrum) to the point where the downward force is applied. The
plus sign {+} corresponds to a counter-clockwise torque and the negative sign {-}
corresponds to a clockwise torque.
The center of mass, denoted here by CM, is the point at which the mass of the
object can be considered to be concentrated. The position x of the CM of a nonhomogeneous meter stick can be determined by balancing the torque of the stick on one
side of the fulcrum with the torque of a known mass on the other side of the fulcrum.
Having established the position of the CM and knowing the mass of the stick, the
same procedure can be used to determine the unknown mass of another object.
SUPPLIES & EQUIPMENT:
Weighted meter stick
Electronic balance
Knife edge clamp
Knife-edge stand
Scissors
Hooked masses
Metal cube
Light string
PROCEDURE:
A.
CENTER OF MASS OF A NON-UNIFORM METER STICK
1. Record the mass of the non-uniform meter stick m1 indicated on the electronic
balance.
2. Set up the apparatus as shown in Fig. 1, making sure that the fulcrum is at the
midpoint of the stick. Slide m2 in along the stick until the stick is in equilibrium.
Record m2 and d2. Be sure to include the mass of the string in the mass of m 2.
- 33 -

3. Use Eq. 1 to estimate the lever arm, d1, the distance of the meter stick CM from
fulcrum. Then calculate x, the position of the meter stick’s CM relative to the
weighted end of the stick. This equation is obtained from the equilibrium condition:
counter-clockwise+ clockwise = F1d1 - F2d2 = 0
d1
=
m2g
d =
m1g 2
F1d1
=
F2d2
d1
=
F2
d
F1 2
=
m2g
d
m1g 2
distance of CM from fulcrum
Eq. 1
x = position of CM from weighted end
= fulcrum position minus d1
Fulcrum at midpoint of stick
x
d1
d2
m2
counter-clockwise
is a positive torque
F1 = m1g
F2 = m2g
clockwise
is a negative torque
 = F1d1 + (F2d2) = 0
Fig. 1
4. Move the fulcrum 5.0 cm away from the midpoint, toward the weighted end of the
stick as shown in Fig. 2. Slide m2 to establish equilibrium. Record m1 and m2 and
the new value of d2, the lever arm measured from the new fulcrum position. Use
Eq. 1 to calculate the new value of d1 from the fulcrum position to obtain your
second estimate of x. The position of the CM of the stick is x.
Fulcrum
x
Midpoint of stick
d1
d2
m2
F 2 = m 2g
F 1 = m 1g
Fig. 2
5. To obtain your third estimate of x, remove m 2 and balance the stick on a knife edge
clamp. The meter stick is balanced because its CM is resting on the knife-edge
clamp which is at the fulcrum of the system. Record x, the position of the CM from
the weighted end of the stick.
- 34 -
B. DETERMINATION OF AN UNKNOWN MASS
1. Having calculated the position of the center of mass on the previous page, set up
the apparatus as shown in Fig. 1 by moving the fulcrum back to the midpoint of the
stick. m2, a metal cube, will be the unknown mass.
2. Using a string, hang the unknown mass on the stick and slide it along the stick to
balance.
3. Record the new value of d2. Use Eq. 2 to obtain your estimate of the unknown
mass, m2.
  = 0 , so
F1d1 + (F2d2) =
m2
0
1
=m
d
2
and
=
F2
F1
d
d2 1
,
giving
m 2g
=
m1g
d .
d2 1
Eq. 2
d1
4. Weigh the metal cube on the electronic balance and find the percent difference
between the two measurements of m 2.
C. MULTIPLE-TORQUE SYSTEM: FINDING THE MASS OF A METAL CUBE
(Use the same m2 as in Part B)
1.
The equilibrium condition can be used even when there are several torques
involved. Set up the apparatus as shown below:
Fulcrum at midpoint of stick
d2
d4
d3
d1
x
m2
F2
m4
m3
F1
F2
F3
Fig. 3
2. Use Eq. 3 to obtain another estimate of the unknown mass m 2.
 = 0 , so
F1d1 + F2d2  F3d3 - F4d4
m2 =
m3d3  m4d4  m1d1
d2
=0
giving
F2 =
F3d3  F4d4  F1d1
d2
Eq. 3
3. Find the percent difference between this measurement and the value obtained
directly from the electronic balance.
- 35 -
DATA SHEET: TORQUE AND CENTER OF GRAVITY
Data Table A: Determination of the Center of Gravity
m1 (stick)
(kg)
Fulcrum Position
m2
(kg)
d1
(m)
d2
(m)
x
(m)
*
At midpoint
(Steps 1 – 3) Fig. 1
At 5.0 cm from midpoint
(Step 4) Fig. 2
At CM
(Step 5)
Data Table B: Unknown Mass m2
m1 (stick)
(kg)
d1
(m)
d2
(m)
m2
(from Eq. 2)
(kg)
*
 = 0
(Steps 1-3) Fig. 1
Unknown
mass
from weighing
(Step 4)
Percent Difference _________________
Data Table C: Multiple Torque System. (Unknown mass m 2, same mass as in Part B.)
m1
(kg)
d1
(m)
m3
(kg)
d3
(m)
*
Fig. 3
 = 0
Percent Difference ___________________
- 36 -
m4
(kg)
d4
(m)
d2
(m)
m2 (from
Eq. 3)
(kg)
Experiment 11: ARCHIMEDES' PRINCIPLE
PURPOSE:
Archimedes' Principle will be used to determine: a) the density of a symmetricallyshaped object; b) the density of an irregularly-shaped object; and c) the specific gravity of a
liquid.
INTRODUCTION:
Archimedes' Principle states that an object that is submerged in a fluid is buoyed up
by a force that is equal in magnitude to the weight of the fluid displaced by the object. This
force is called the buoyant force, or the buoyancy. The buoyant force can be determined
experimentally with the following setup:
Beam Balance
Beam Balance
Paper Clip
T1
T2
B
m
ma
mg
mg
Lab Jack
Fig. 1
T1 = W o (Weight of object in air)
W o = mog
Fig. 2
T2 = W aw (Apparent weight of object in water)
T2 = W o  FB
W aw = W o  FB
Therefore
FB = W o  W aw
(Eq. 1)
According to Archimedes' principle, the buoyant force,
FB = W w
or
FB = mwg
Since
mw = wVw
then
FB = wVwg , and the volume of water displaced
by the immersed object can be expressed as
Vw = FB /wg
Key to symbols:
mo = mass of object (in air) ,
(Eq. 2)
W o= weight of object in air
maw = apparent mass of object in water (fluid)
mw = mass of water displaced
w = density of water = 1000 kg/m3
- 37 -
Vo = volume of object
Waw = apparent weight of object in water
Ww = weight of water displaced
Vw = volume of water displaced
THE DENSITY OF AN OBJECT
When an object is totally submerged in water (a fluid) , the volume of water
displaced is equal to the volume of the object.
(Volume of submerged object)
Vo = V w
(Volume of water displaced)
(Eq. 3)
Since the volume of an object is  Vo = mo / o , and volume of the fluid displaced is
Vw = FB /wg., then (Eq. 3) becomes
mo / o = FB /wg
Density of the object can be expressed as
m g

o = o w
FB



(Eq. 4)
The buoyant force FB can be determined from the apparent weight loss, FB = (W o - W aw).
It can also be determined from the weight of the water displaced.
SUPPLIES & EQUIPMENT:
Double pan balance 150-ml beaker
Lab jack
600-ml beaker
250-ml graduated cylinder
Rock sample
Unknown fluid
Vernier caliper
Hydrometer
Short support rod
Table clamp
Overflow can
String & scissors
Small paper clips
Metal cube
PROCEDURE:
A. DENSITY OF A METAL CUBE
1. Measure the length of one side of the metal cube. Calculate the volume of the cube V o.
2. Suspend the cube from a beam balance mounted on a support rod as in Figure 1 and
determine its mass, mo.
3. Immerse the suspended cube in a beaker of water as in Figure 2. Determine the
apparent mass of the cube in water, maw.
4. Determine the buoyant force (FB = W o  W aw) in newtons. (Eq. 1)
5. Determine the density of the cube from o =
- 38 -
Wo
w. (Eq. 4)
FB
B. DENSITY OF AN IRREGULARLY-SHAPED ROCK
1. Suspend a rock from the beam balance and determine its mass, m o in kilograms.
2. Immerse the suspended rock in a beaker of water as in Figure 2.
3. Determine the apparent mass of the rock immersed in the fluid, m aw in kilograms.
4. Determine the buoyant force, FB = W o  W aw, in newtons (N). (Eq. 1)
m g
5. Determine the density of the rock from o = o w. (Eq. 4)
FB
6 Determine the mass of a 150-ml beaker mb = _______________ kg.
7. Place the displacement can on a level surface near the edge of a sink. Fill it with water,
and let the excess drain off into the sink.
8. Slowly lower the rock into the water, allowing the displaced water to flow into the small
beaker. Weigh the beaker with displaced water. m b+w = _______________ kg.
9.
Determine the mass of the water displaced,( m w = mb+w – mb).__________kg
10. Determine the weight of water displaced, W w= mwg = ____________N. This is equal
in magnitude to the buoyant force FB, according to Archimedes principle.
11 .Determine the density of the rock by applying (Eq.4), o =
mo g
w. _________kg/m3
FB
C. SPECIFIC GRAVITY OF AN UNKNOWN LIQUID.
1 With the same cube used in Part A, determine the buoyant force, FB (fluid) on the metal
cube by immersing it in an unknown fluid.
FB (fluid) = W o  W af.
W o = weight of object in air.
FB (fluid) = mog  maf g
W af = apparent weight of object in fluid.
2. Calculate the density of the fluid using equation (5).
Wo  w
mo g  w
In Water: o =
=
In Fluid:
FB( water )
FB
Therefore,
Wo  f
Wo  w
=
,
FB( fluid )
FB
and
f =
o =
Wo  f
mo g  f
=
FB( fluid )
FB
FB( fluid )  w
FB( water )
Eq. 5)
where o = density of metal cube in air, w = density of water and f = density of Fluid
f
.
w
4.
Fill a tall measuring cylinder with the “unknown” fluid. Use a hydrometer to measure
the specific gravity of the fluid.
3. Calculate the specific gravity, S.G. =
- 39 -
DATA SHEET: ARCHIMEDES' PRINCIPLE
A. Metal Cube
Mass of cube
mo = __________ kg
(i) Apparent mass of cube in water
maw = __________ kg
FB = mog  mawg
W
Density of cube o = o w
FB
(ii) Length of side
FB = __________ N
Buoyancy
 = __________ kg/ m3
L = __________ m
V = __________ m3
Volume of cube
Density of cubeo =
mo
Vo
 = __________ kg / m3
 = __________ kg / m3
(iii) Density (known)
B. Rock
Mass of rock
mo = __________ kg
(i) Apparent mass of rock in water
maw = __________ kg
FB = mog  mawg
W
Density of rock o = o w
FB
(ii) Mass of water displaced
FB = __________ N
Buoyancy
=__________kg/ m3
mw = __________ kg
Weight of water displaced
Density of rock
o =
W w = mwg = FB = __________ N
Wo
w
FB
=__________ kg/ m3
C. Specific Gravity
Mass of cube (from part A)
mo = __________ kg
Apparent mass of cube in fluid
maf = __________ kg
FB(fluid) = mog  mafg
FB( fluid )
Density of fluid f =
w
FB( water )
Buoyancy
FB = __________ N
=__________kg/ m3
f / w
= __________
Specific gravity measured with hydrometer
= __________
Specific gravity
- 40 -
Experiment 12: T HE COEFFICIENT OF
LINEAR EXPANSION
PURPOSE:
The purpose of this experiment is to measure the coefficient of linear expansion for
various metals and to compare the results with the known values.
INTRODUCTION:
In most cases, when materials are heated or cooled, they undergo expansion or
contraction respectively. From the standpoint of materials science, this process must be
taken into account when designing structures that are subjected to temperature variations.
Otherwise, tensile or compressive stresses might develop which could destroy the
structure.
The linear (one-dimensional) coefficient of expansion is defined as the fractional
increase in length divided by the temperature change. This coefficient is designated by the
Greek letter alpha (), and is found to be almost constant over a wide range in
temperature. In equation form, the definition of  is:
L
L o T
where L is the change in length, Lo is the original length, and T is the temperature
change in degrees Celsius.
In this experiment, the value of the linear coefficient of expansion of several rods of
common metals will be determined. The length of the rod is measured at room
temperature, then steam is passed over the rod with the resulting temperature increase
causing it to expand. The amount of expansion is measured with a dial indicator. The
coefficient is then determined using the data gathered.
=
SUPPLIES & EQUIPMENT:
Linear expansion apparatus
Aluminum, copper and steel rods
0 - 100 oC Thermometer
Meter stick
Dial indicator
Electric steam generator
Glycerine
PROCEDURE:
1. Measure and record the initial length of the rod L o, to the nearest millimeter. Determine
and record the ambient temperature (room temperature).
2. Set up the apparatus as shown in Figure 1. The steam jacket for the rod has an
opening for steam, thermometer, and rod ends, and an outlet for the condensed steam.
Fill the steam generator about 2/3 full of water and turn on the generator, but do not
connect the generator to the expansion apparatus as yet. Insert the rod in the
apparatus until it just makes contact with the dial indicator probe and is in firm contact
with the screw at the other end.
- 41 -
Steam inlet tube
Thermometer
Rod
Steam
Generator
Dial indicator
Steam outlet tube
To sink
Fig. 1. Linear Expansion Apparatus with Associated Equipment
3. Make sure that the dial indicator is firmly screwed onto its holder and that the graduated
ring is tightened down. See Fig. 2. Record the initial reading of the dial indicator, to
the nearest 0.01 mm (= 0.00001m).
Secure movable ring firmly
READING THE DIAL INDICATOR:
20
0.01 mm
per div .
Example:
30
40
10
1 mm/div
0
90
1 2
9
0
1
8 7
70
to
the
left
The gauge below indicates
0.14 mm
50
6
5
4
2 3 60
20
0
80
The gauge
indicates 0.07 mm
1 cm/div
10
0
1
0
0
123
4
Fig. 2 Dial Indicator (Micrometer Gauge)
4. When the generator is generating steam briskly, connect the steam tube to the inlet on
the apparatus. Warning! Be careful not to scald yourself.
5. Allow the steam to warm up the rod to a constant maximum temperature, T max. When
the rod stops expanding, record the final reading of the dial indicator.
Calculate T = (Tmax  T ambient).
6. Calculate the change in length, L = Final reading - Initial reading.
7. Calculate the coefficient of expansion and record it on the data sheet. Compare your
values with the known values of the coefficient of linear expansion by calculating the
percent difference.
8. Repeat the above procedure for two other rods. Be careful not to burn yourself on the
hot metal. When finished, dry the equipment thoroughly.
- 42 -
DATA SHEET: COEFFICIENT OF LINEAR EXPANSION
Ambient Temperature _________________ oC
Data and Calculations Table:
Type of Rod
Lo
Copper
Steel
2.4 X 10-5
1.7 X 10-5
1.1 X 10-5
(m)
Initial Reading of Dial Indicator
(m)
Final Reading of Dial Indicator
(m)
Tmax
Aluminum
(oC)
T
(oC)
L
(m)

(oC-1)
Known
(oC-1)
Percent difference
- 43 -
Experiment 13: THE HEAT OF FUSION OF ICE
PURPOSE:
The value of the latent heat of fusion for water will be determined by the method of
calorimetry.
INTRODUCTION:
When a substance such as water undergoes a change of state from the solid phase
to the liquid phase, not all of the heat energy that is added to the system is reflected in a
change of temperature of the substance. Some energy is needed to break the bonds
between the molecules of the substance and this energy is called the latent heat of fusion
of the substance.
In today's experiment the latent heat of fusion will be determined by the method of
mixtures and by applying the principle that the heat lost is equal to the heat gained
(conservation of energy).
In this experiment, an ice cube is placed into a measured amount of warmed water
and is left to melt, cooling the water in the process. By noting the temperatures before and
after melting, the heat of fusion is then calculated as follows:
HEAT GAINED:
by ice cube = (heat needed to melt the ice) + (heat for warming the melted ice)
Qi = mi Lf + micw(Tf - 0oC)
HEAT LOST:
by water = (mass of water) X (1.00 cal / g.oC) X (temperature change)
Qw = mwcw(To - Tf)
cw = Specific heat of water.
by calorimeter = (mass of calorimeter) X (0.22 cal / g.oC) X (temperature change)
Qc = mccc(To - Tf)
cc = Specific heat of calorimeter.
CONSERVATION OF ENERGY:
Heat Gained = Heat Lost
Qi = Q w + Q c
mi Lf + micw(Tf - 0oC) = mwcw(To - Tf) + mccc(To - Tf)
mi Lf = mwcw(To - Tf) + mccc(To - Tf) - micw(Tf - 0oC)
- 44 -
Eq. (1)
Eq. (2)
m w c w (To  Tf )  m c c c (To  Tf )  mi c w (Tf  0 o C)
mi
SUPPLIES & EQUIPMENT:
Lf =
Double-walled calorimeter
Thermometer
Forceps/Tongs
Ice cubes
Electronic balance
Eq. (3)
Steam Generator
Beaker
PROCEDURE:
1. Determine the mass of the plastic collar. Slip the collar back on the inner cup.
2. Determine the mass of the inner cup, stirrer and plastic collar of the calorimeter.
3. Fill the inner cup of the calorimeter to about 2/3 full with warm water at about 40o.
4. Re-determine the mass of the inner cup, stirrer, collar, and water. Calculate the mass
of water in the cup.
5. Place the cup, stirrer, collar, and water into the outer calorimeter jacket and record the
exact temperature just before the ice cube is placed in the water.
6. Wipe any excess water from an ice cube and place it carefully into the calorimeter cup.
7. Stir the contents occasionally while constantly observing the ice cube. As soon as the
ice cube is completely melted, record the temperature. This temperature is T f.
8. Re-determine the mass of the cup, stirrer, collar, and contents. The mass of the ice
cube can now be calculated.
9. Compute the heat of fusion of ice and compare this to the accepted value by
calculating the percent error.
10. Repeat the experiment. Comment on the reproducibility of the results.
- 45 -
DATA: THE HEAT OF FUSION OF ICE
Data and Calculations Table:
Trial
Mass of inner cup,collar and
stirrer of calorimeter
(g)
Mass of inner cup, collar,
stirrer and water
(g)
Mass of water
(g)
Initial temperature
of water
(oC)
Final temperature
of contents
(oC)
Mass of inner cup, collar
stirrer and contents
(g)
Mass of ice cube
(g)
by water
(cal)
by calorimeter
(cal)
by ice cube
(cal)
1
2
79.7 cal/gram
79.7 cal/gram
HEAT
LOST:
HEAT
GAINED:
Heat of fusion
(cal/gram)
Known value of
Heat of fusion
(cal/gram)
% error
- 46 -
Experiment 14: STANDING WAVES ON STRINGS
PURPOSE:
In this experiment we will study the relationship between tension in a stretched string
and the wavelength and frequency of the standing waves produced in it.
INTRODUCTION:
Standing waves are produced by the interference between two traveling waves with
the same wavelength, velocity, frequency and amplitude traveling in opposite directions.
The equation for the velocity of propagation of transverse waves on a stretched string is:
T
.

v
(Eq. 1)
where T is the tension in the string and  is the linear density (the mass per unit length of
the string). The velocity of propagation v, the frequency of vibration f, and the wavelength
 are related this way:
v = f
(Eq. 2)
A stretched string has many modes of vibration. It may vibrate as a single segment,
in which case its length is half of a wavelength. It may vibrate in two segments with a node
(zero displacement) at the center as well as at each end; then the wavelength is equal to
the length of the string. The wavelengths of the many modes of vibration are given by the
relation:
L
so,  
n
,
2
2L
n
(Eq. 3)
where L is the length of the string, is the wavelength, and n is an integer called the
harmonic number, indicating the number of segments.
SUPPLIES & EQUIPMENT:
Electric tuning fork
Stroboscope
Electronic balance
Power supply for tuning fork
Rod pulley
5-gram mass hanger

5- rheostat
Ruler
Meter stick
Leads & connectors
50-gram mass hanger
6-inch "C" Clamp
- 47 -
Thick string
Thin string
Slotted masses
Table clamp
Scissors
PROCEDURE:
1. Cut off a piece of the string about 2 meters long and determine its length, mass and
linear density.
2. Clamp the apparatus to one end of your table and clamp the pulley to the other end, as
shown in Figure 1. Knot the string to one end of the tuning fork and knot the other end
to the mass hanger. Suspend the string over the pulley, and adjust the pulley until the
string is horizontal. Write down the mass of the hanger.
3. Connect the fork to the rheostat, as your source of current as shown in Figure 1, with
the tap on top of the rheostat set very close to the positive end. Use no more than a 6
V setting. Set the fork into vibration by adjusting the contact point screw above and to
the left of the two terminals of the tuning fork apparatus. The rheostat may need to be
adjusted to create noticeable but not violent vibrations.
4. Measure the frequency of the tuning fork using a stroboscope. Start with the highest
strobe frequency possible and lower it until one stationary image of the tuning fork is
obtained. When lowering the frequency of the strobe, also observe that a stationary
image is obtained when the strobe frequency is ½, ⅓, ¼, etc., times that of the tuning
fork. Divide the number that appears on the stroboscope by 60 to get the frequency of
the tuning fork in cycles per second (Hertz).
5. Vary the tension of the string by adding masses to the hanger until the string vibrates in
five segments with maximum amplitude. Measure the length of one segment from a
point vertically over the center of the pulley wheel to a node (zero amplitude). The
wavelength will be twice the length of one segment. Record in the data table the added
mass in kilograms. Then record the total mass (added mass plus the mass hanger) in
the data table. Record the resulting tension T = mg in Newtons.
6. Repeat the procedure for 4, 3 and 2 segments by adding more mass to the pulley.
7. Compare the experimental velocity (v = f) with the theoretical velocity ( v  T /  ) by
computing the percent difference.
Fig. 1: Standing Waves on Strings Apparatus
- 48 -
LABORATORY REPORT: STANDING WAVES ON STRINGS
Length of string ___________ m
Mass of string ___________ kg
 = Linear Density of String __________ kg / m
Mass of hanger ___________ kg
f = Frequency of vibrating tuning fork __________ Hz
Number of
Segments
Length of one
segment
5
4
(m)
Wavelength  (m)
Velocity from
v = f
(m/s)
Added mass
(kg)
Total mass
(kg)
Tension T
(N)
Velocity from
v=
T/
(m/s)
% difference
- 49 -
3
2
Experiment 15: THE SPEED OF SOUND IN AIR
PURPOSE:
The speed of sound in air will be determined by means of an adjustable air column
resonance tube, driven by a known frequency.
INTRODUCTION:
When a tuning fork is set into vibration over the open end of a tube which is closed
at the other end, a series of compressions and rarefactions occur within the length of the
tube. If the length of the tube is such that an odd number of quarter wavelengths just fit into
the tube, a condition known as resonance occurs and the sound heard emanating from the
tube is greatly enhanced.
When resonance occurs, the gas particles just outside the mouth of the open tube
are oscillating up and down with their maximum amplitude. This location is called the
displacement antinode, and is symbolized in Figure 1 by curved lines showing a maximum
displacement from the center of the column (although the actual displacement is vertical,
not horizontal). There must be a displacement node at the bottom of the air column, as the
water prevents the gas particles from oscillating up and down.
Since resonance occurs at odd multiples of quarter wavelengths, one is able to
determine the wavelength, , by noting the length of the tube at which resonance occurs.
Thus:
L1
L2
L3
L1= (1/4)
L2 = (3/4)
L3 = (5/4)
Fig. 1.
From the above equations, the wavelength () can be calculated as:
 = (L3 - L1)
(Eq. 1)
 = 2(L2 - L1)
(Eq. 2)


- 50 -

Once the wavelength is determined, the speed can be calculated from the
relationship:
Speed = Wavelength X Frequency
or,
v =  f.
(Eq. 3)
The theoretical speed of sound in meters per second is given by the relation:
v = 331.5 + 0.607 T.
(Eq. 4)
where 331.5 m/s is the speed of sound at 0O C in meters per second and T is the
temperature in Celsius degrees.
SUPPLIES & EQUIPMENT:
Resonance apparatus
480 Hz tuning fork
Thermometer
Turkey baster
512 Hz tuning fork
Rubber mallet
Beaker
PROCEDURE:
1. Adjust the water level in the tube by raising the reservoir until the water level is about 10
cm from the top of the tube. See Figure 2.
Fig. 2.
2. Strike the fork with the rubber mallet and hold it (horizontally) close to the top of the
tube, with the prongs vibrating vertically. Lower the reservoir until the first resonance is
heard. Record this position as L1.
3. Determine the lengths for the other two resonance positions as in step 2.
4. Take the average of three readings at each resonance position for calculating the
speed of sound in air. Compare this to the theoretical value.
5. Repeat the procedure for another fork of a different frequency.
- 51 -
DATA: THE SPEED OF SOUND IN AIR
Data for tuning fork of frequency 512 Hz
Averages
L1 = _________ m, _________ m, _________ m
L1 = _________
L2 = _________ m, _________ m, _________ m
L2 = _________
L3 = _________ m, _________ m, _________ m
L3 = _________
 = L3 - L1 = ____________ m
Average  ____________
 = 2 (L2 - L1) = __________ m
Experimental value for the speed of sound, v =  f = ___________ m/s
Theoretical value for the speed of sound = ___________ m/s (Eq. 4)
% Error = ___________
Data for tuning fork of frequency 480 Hz
Averages
L1 = _________ m, _________ m, _________ m
L1 = _________
L2 = _________ m, _________ m, _________ m
L2 = _________
 = 2 (L2 - L1) = __________ m
Experimental value for the speed of sound = _______________ m/s
Theoretical value for the speed of sound
= ________________ m/s (Eq. 4)
% Error = __________________
- 52 -
Physics 6 Laboratory Assignment
Part I.
View the video produced by the Jet Propulsion Laboratory, NASA Space Agency,
Pasadena California. The segments that will be shown are:
a) Space Flight Operations Facility
b) Magellan: Exploration of Venus
c) Tracking and Data Acquisition
d) L.A. The Movie
e) Miranda The Movie
f) Earth The Movie
g) Mars The Movie
Write one or two sentences on each segment. Make sure that your report is neat and
legible.
Part II. EXTRA CREDIT
Find an article in a periodical or book (not an encyclopedia ) that is related to
satellites, space craft or space technology. You may want to consider using the cumulative
index in the library. The following are some examples of the type of literature that you may
consider, but are certainly not limited to:













National Geographic
Discover Magazine
Popular Science
Physics Today
Time Magazine
Newsweek
US News Report
Newspapers
Scientific American
Science Digest
Various textbooks
Books
Etc.
Write a summary of the article. The length of this summary should be about one
page of neatly and legibly hand-written text or half page of type-written text.
Attach the article (a Xerox copy is okay) to your report.
This assignment can earn up to 10 points extra credit depending on the quality of
the work.