PHYSICS
LAB
NOTES
FOR
MECHANICS
AND
HEAT
EXPERIMENTS
PHYSICS 37
Los Angeles Harbor College
J. C. FU
R. F. WHITING
© 1997
Eighth Edition (F)
February 2004
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 and limitations of certain equipment and procedures, and to think
analytically.
Each lab period will begin with a discussion of the experiment indicated in your lab notes.
Come prepared, having read the lab beforehand. An attempt has been made to have the labs
coordinated with the lecture material. Share work with your partner so that each person will have
an opportunity to have hands-on experience.
The laboratory reports that you turn in should have the following format:
Cover
page
Exp # ____
____ ____
Name
____
Title
Date
ABSTRACT
Successive
pages
PURPOSE
APPARATUS
DATA
GRAPHS
TABLE
INTRODUCTION
PROCEDURE
CALCULATIONS
TABLE
J. C. Fu, Ph.D
R. F. Whiting, M.S.
All rights reserved. No part of this book may be reproduced, in any form or by any means, without
permission in writing from the authors.
TABLE OF CONTENTS
1. Measurement ........................................................................................1
2. One-dimensional Non-Uniform Motion...................................................7
3. One-dimensional Free-Fall Motion....................................................... 10
4. Addition of Vectors .............................................................................. 13
5. Projectile Motion ................................................................................. 16
6. Newton's Second Law ........................................................................ 20
7. Simple Machines ................................................................................ 23
8. The Conservation of Mechanical Energy ............................................ 26
9. The Ballistic Pendulum ....................................................................... 30
10. Elastic and Inelastic Collisions ............................................................ 33
11. Center of Mass ................................................................................... 36
12. Torque and Equilibrium ....................................................................... 39
13. Centripetal Force ................................................................................ 43
14. Moment of Inertia ................................................................................ 48
15. The Gyroscope ................................................................................... 51
16. Archimedes' Principle ......................................................................... 55
17. The Coefficient of Linear Expansion ................................................... 60
18. Heats of Fusion and Vaporization ....................................................... 64
Experiment 1: MEASUREMENTS, COMPUTATION and DATA ANALYSIS
PURPOSE
The purpose of this experiment is to introduce:
a) the practice of making careful measurements with devices of varying precision,
b) the role of significant figures in the collection of raw data (measurements),
c) the role of significant figures in computation, and the rationale for rounding off computed
results to the appropriate number of significant digits, and
d) terms encountered in data analysis.
INTRODUCTION
The physics laboratory provides a useful introduction to the fundamental principles that
underlie experimentation and data analysis. These principles will be of value to the scientifically
literate citizen in evaluating information presented to him/her in both scientific and non-scientific
fields. The following is a list of definitions that will be useful in the computation and analysis of
data.
SI units: for length: the meter (m),
for mass: the kilogram (kg), and
for time: the second (s)
x i = individual measurement
x = average value
N = number of measurements made
Mean value = average value =
x
i
/N
Deviation from the mean = a measure of uncertainty = | x i  x |
Average deviation = a. d. =
x
i
 x /N
Standard deviation =  
| x
i
 x |2
N 1
Standard deviation of the mean = m =

N
APPARATUS & SUPPLIES
Electronic balance
Electric stop clock
Tape measure
Micrometer
-1-
Vernier caliper
100 copper pennies
Metronome
PROCEDURE
A. LENGTH
1. Determine the area of the classroom floor in square meters. Make three separate
measurements of both the length and the width and report your result with the
average deviation. Measurements taken with the tape measures and metric rulers
should be to the nearest millimeter.
2. Measure the diameter and thickness of a copper penny using the vernier caliper.
Calculate its volume = d2t/4. Observe the rules pertaining to significant figures in
computation. Measurements taken with a vernier caliper should be to the nearest
tenth of a millimeter.
3. Measure the diameter and thickness of a copper penny using the micrometer.
Calculate its volume = d2t/4. Observe the rules pertaining to significant figures in
computation. Measurements taken with a micrometer should be to the nearest
hundredth of a millimeter.
4. Convert the volume into units of cm3.
1mm3 = 1mm  1mm  1mm = 0.1cm  0.1cm  0.1cm = 0.001cm3, so the conversion is
accomplished by multiplying by 0.001.
B. MASS
1. Determine the mass of a group of 10 copper pennies on the electronic balance and
record your result on data table B. Repeat the measurement on nine separate
groups of 10 pennies.
2. Calculate the mean mass of one penny, and the standard deviation of the mean.
The standard deviation is considered to be a more accurate measure of the scatter,
but it is a more involved calculation.
C. Time
1. With the electric stop clock, time 50 beats of the metronome set at 120. Set the
metronome at 120 by sliding the pendulum weight so its top clicks into position just
beneath ’120’. Calculate the beats per second. 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 per minute). Repeat two
times for the same person and determine an average value, converted to beats per
minute.
-2-
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
-3-
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
+ 0.47
mm
0
4.97
mm
45
Read to the nearest hundredth of a millimeter.
-4-
4 7 div. X 0.01 mm/div. = 0.47 mm
DATA & CALCULATIONS: MEASUREMENT
A. LENGTH:
Table A1: Tape Measure*, Floor: Width deviation + |Width – Average Width|
Width, W
(m)
Deviation from the mean, W
(m)
Average Width, W
(m)
Average Deviation, W
(m)
 _______________
____________________
Table A2: Tape Measure*, Floor: Length deviation + |Length – Average Length|
Length, L
(m)
Deviation from the mean, L
(m)
Average Length, L
(m)
Average Deviation, L
(m)
 _______________
____________________
*Measure to the nearest millimeter: Less than 10m, 4 sig. Figs; Greater than 10m, 5 sig. Figs.
Area Calculations:
Average area:
Average deviation of area:
A = L X W
A =
= _______________ m
A  A
2
=  _____________ m2
Positive deviation of area: A + = ( W + W ) ( L + L ) = _______________ m2
Negative deviation of area: A  = ( W  W ) ( L  L )
= _______________ m2
Area of room = __________ _________ m2
Area of room = Average area  Average deviation
A
A
A
=

Table A3:
Copper penny
Diameter, d
(mm)
Thickness, t
(mm)
Vernier caliper
Micrometer
-5-
Volume
(mm3)
Volume
(cm3)
B. MASS:
Table B1
Group No.
Mass of 10
pennies
(g)
Mass per
penny, xi
(g)
Average
mass of one
Deviation from
penny, x
(g)
the mean |xi – x |
(g)
Standard
deviation, 
(g)
Standard
deviation of
the mean, m
(g)
1
2
3
4
5
6
7
8
9
10
Mass of one penny: Average  Standard deviation of the mean = x  m = ____________  _________ grams
C TIME:
Table C1,
Electric stop clock
Object
# of beats
Time (s)
Beats per
second
Average
beats/second
Average
beats/minute
Metronome
Table C2
Student name
# of beats
Time (s)
-6-
Beats / minute
Pulse rate =
Average beats/min.
Experiment 2: ONE DIMENSIONAL NON-UNIFORM MOTION
PURPOSE:
This experiment illustrates the definition of position, displacement, velocity and acceleration of an
object in non-uniform horizontal motion.
INTRODUCTION:
In the study of kinematics, an understanding of the terms that describe motion is of utmost
importance. Position describes the location of an object relative to a reference point.
Displacement is a change in position and / or direction. Average velocity is the displacement
divided by the amount of time it took the displacement to occur. Instantaneous velocity is the
velocity at a very small (infinitesimal) time interval. Average acceleration is the change of velocity
divided by the time interval. Instantaneous acceleration is the change in velocity over a very small
(infinitesimal) time interval. The scalar quantity distance is the length of the pathway taken by an
object as it traveled from one position to another. The scalar quantity average speed is the
magnitude of the distance traveled divided by the time of travel. As you observe and record the
motion of an object in horizontal motion in this lab experiment, you will perhaps appreciate the
physical meaning of these terms and be able to describe motion more accurately.
APPARATUS & SUPPLIES:
Stopwatch or stop clock
Clear plastic triangle
Tape measure
Chalk
Meter stick
French curve
Two-meter stick
Ruler
PROCEDURE:
1. Form a team of seven students.
2. With a piece of chalk, draw markers at one-meter intervals along the length of the
blackboard. Let the edge of the blackboard be xo, the starting point.
3. Let one member of the team volunteer to be the traveler. The traveler will start the trip
from position xo = 0.00 m.
4. Let each of the other members of the team reset a timer and locate themselves across
from the x1 = 2.00 m, x2 = 6.00 m, x3 = 10.00 m, x4 = 10.00 m, x5 = 8.00 m, and x6 = 2.00
m positions.
5. At the shout of “GO”, the traveler will walk forward at a non-uniform rate, and all the timers
will be started.
6. On the trip forward, students with timers t1, t2, and t3 at x1, x2, and x3 will each stop the
timer as the traveler walks past that position.
-7-
7. At position x3 = 10.00 m the traveler will pause for a short while. When the traveler turns
around and starts on the return trip, he will raise his hand as a signal for timer t 4 at x4 =
10.00 m to stop. As he walks backwards at a non-uniform rate, the timers at x5 = 8.00 m
and x6 at 2.00 m will be stopped when the traveler passes by each of those positions.
8. Record the data from each timer at each position in Table 1.
9. Plot a graph of position (on the vertical axis) vs. time (on the horizontal axis). Label both
axes, including units in brackets, and supply a title to the graph. Draw a smooth curve
through all these points.
10. Draw tangent lines to this curve at t1, t2, t3, t4, and t5, and calculate the slopes of these five
straight lines. These are the instantaneous velocities at these five times. Record these
results in Table 1.
11. Plot a graph of instantaneous velocity vs. time, labeling both axes and include a title for the
graph. Draw a smooth curve through these five points, draw tangent lines at t 2, t3, and t4,
and calculate the slopes of these three straight lines. These are the instantaneous
accelerations at these three times. Record in Table 1.
12. Calculate the average velocity x/t for each time interval between the observed times,
and place them in Table 2. Note that these are similar to, but not equal to, the
instantaneous velocities.
13. Calculate the average velocity of the trip forward, the trip backward, and the entire trip.
Record the results in Table 3.
14. Calculate the average speed of the trip forward, the trip backward, and the entire trip.
Record the results in Table 3.
15. Calculate the average acceleration v/t for each time interval between t1 and t5 by using
instantaneous velocities in Table 1. Record the results in Table 4. Notice the difference
between instantaneous acceleration and average acceleration.
16. Plot a graph of average acceleration versus time, with data from Table 4.
-8-
DATA & CALCULATIONS: One Dimensional Non-uniform Motion
Table 1: Non-uniform horizontal motion
Position, x
(m)
x0 = 0.00
to =
x1 = 2.00
t1 =
x2 = 6.00
t2 =
x3 = 10.00
t3 =
x4 = 10.00
t4 =
x5 = 8.00
t5 =
x6 = 2.00
t6 =
Instantaneous
velocity, v
(m/s)
Time
(s)
Instantaneous
acceleration, a
(m/s2)
Table 2: Average velocity
Time interval, t
(s)
t1 – t0 =
Displacement, x
(m)
x1 – x0 =
t2 – t1 =
x2 – x1 =
t3 – t2 =
x3 – x2 =
t4 – t3 =
x4 – x3 =
t5 – t4 =
x5 – x4 =
t6 – t5 =
x6 – x5 =
Table 3: Velocity and speed
Forward
(m/s)
Average Velocity
At rest at x3
(m/s)
Average velocity
in interval
(m/s)
Backwards
(m/s)
Entire trip
(m/s)
Average Speed
Table 4: Average acceleration
Time interval, t
(s)
t2 – t1 =
Change in velocity, v
(m/s)
v2 – v1 =
t3 – t2 =
v3 – v2 =
t4 – t3 =
v4 – v3 =
t5 – t4 =
v5 – v4 =
-9-
Average acceleration
(m/s2)
Experiment 3: ONE-DIMENSIONAL FREE-FALL MOTION
PURPOSE:
In this experiment, the numerical value of the acceleration due to gravity will be determined
by a graphical technique, and the kinematic equations will be applied in the study of an object in
free fall.
INTRODUCTION:
If the effect of air friction is neglected, 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 which has a series
of spark holes. The apparatus which 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 graphical 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
Meter stick
Metric ruler
Spark tape
Masking tape
PROCEDURE:
1. Obtain a spark tape, secure it to the table
with masking tape. Starting with holes that
are about 1 centimeter apart, number each
hole from 1 to 21. Draw a straight line
perpendicular to the length of the tape
through every other hole.
y1
1
Distance = y
t1
t = 1/30 second
y2
t2
3
2. Measure and record the interval distances
between each three successive holes,
starting with 1 - 3, then 3 - 5 and so forth.
Enter your data in the table.
5
3. Calculate the average velocity for each of
your intervals by dividing the interval
distance y by the elapsed time for each
interval. The elapsed time is 1/30 second.
4. The average velocities of each time
interval is also the instantaneous velocity
- 10 -
Average velocity
=
v
=
y
t
=
yf  yi
tf  ti
at the midpoint in time of that interval. For example, the average velocity between holes
1 and 3 is the instantaneous velocity of the free fall object when it is at hole 2. Plot a graph of
the instantaneous velocities of the midpoint in time of each interval (that is, at holes 2, 4, 6, 8
etc.) on the vertical axis versus time on the horizontal 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.
5. Calculate the slope of your straight line. This is the rate at which the velocity changes as the
free fall object accelerates downwards due to gravity. Divide the rise (v) by the run (t) for
any two points on the line, not necessarily data points. Choose the two points so that they are
widely separated.
Slope = v/t = a = g (measured value)
5. The local acceleration of g has been well-determined to be 9.80 m/s2, and this given value
may be compared to your measured value by calculating the percent error:
% error =
measured value  given value
given value
X 100 %
The given value may be a value that has been determined by very precise experiments or it
may be a value expected from theoretical considerations.
7. From the graph determine the instantaneous velocity the free fall object has attained by the
time it reached hole #1. Calculate the time it took to reach this position from the moment it
was released, using v1 = vo + gt0 – 1. This time interval can also be determined by
extrapolating the velocity versus time graph to vo. Tape a graph paper to the left edge of the
graph for extrapolating to the moment of release when vo = 0. Label the horizontal axis
negative time (- t) as it indicates the time before the object reached hole #1.
t0 – 1 = ____________ s;
from the equation v1 = vo + gt0 – 1 and using g = 9.80 m/s2.
t0 – 1 = ____________ s;
from the graph.
8. Determine the displacement of the object from the point of release to hole #1, from the
equation y = vo t + ½ gt2, with t0 – 1 from vo + gt0 – 1 and g = 9.80 m/s2; and also with t0 – 1
extrapolated from the graph.
y0 – 1 = ____________ m;
with t0 – 1 from vo + gt0 – 1, and g = 9.80 m/s2.
y0 – 1 = ____________ m;
with t0 – 1 extrapolated from the graph.
- 11 -
DATA & CALCULATIONS: One-dimensional Free-Fall Motion
Data and Calculations Table: (Measure to a fraction of a millimeter.
Interval
Between Holes
Interval Distance y
(m)
Interval Time t
(s)
Time from Hole #1
(s)
Instantaneous Velocity
(m/s)
1– 3
1/30
1/60
3– 5
1/30
3/60
5–7
1/30
5/60
7–9
1/30
7/60
9 – 11
1/30
9/60
11 – 13
1/30
11/60
13 – 15
1/30
13/60
15 – 17
1/30
15/60
17 – 19
1/30
17/60
19 – 21
1/30
19/60
At: t = 0
Spark
Tape
1/60
1/30
2/30
3/30
x
Hole # :
1
2
3
Sample Graph
4
5
6
7
DESCRIPTIVE TITLE
Instantaneous
Velocity
(m/s)
v
t
|
|
0
60
1
60
2
60
Time at Hole #: 1
2
3
|
3
60
4
|
4
60
5
|
5
60
5
60
Acceleration due to gravity (from graph) = ________ m/s 2
- 12 -
6
|
6
60
5 7
60
|
7
60
5
60
Time (s)
8
% error ________
Experiment 4: 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:
Fx = 0
Fy = 0
SUPPLIES & EQUIPMENT:
Force table
Metal cube (brass or iron)
50-gram mass holder
Slotted masses
Metric ruler
Spirit level
Electronic balance
Protractor
PROCEDURE:
A. GRAPHICAL METHOD
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: m c = ___________ grams = ___________ kg, on the
electronic balance.
3. Determine the weight of the metal cube: W = ___________ N. W = mg, where g = 9.80 m/s2.
Enter this value in Table 1.
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 until the ring remains
centered after the restraining pin is removed.
mA = ___________ grams = _________ kg
mB = ___________ grams = _________ kg
6. Determine FA (= FB) and record in Table 1.
FA = ________ kg X 9.8 m/s2 = ________N
FB = ________ kg X 9.8 m/s2 = ________N
7. Repeat steps 4, 5, and 6, with: A = 30o, B = -30o and A = 50o, B = -50o.
- 13 -
8. Using the graphical method (see Figure 2), add the vectors F A and FB head-to-tail to
determine the resultant FR. 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 magnitude of F R
(converted back into Newtons) in Table 1.
9. At equilibrium, F = FR + (W) + Frictional force = 0. Calculate the magnitude of the frictional
force f as the difference between the FR and W: |f| = |FR – W|.
Object
(load)
W
180o
B
FB
FA
A
0o
FR
FB
Fig. 1
Fig. 2
Table A: GRAPHICAL METHOD OF VECTOR ADDITION
FR
A
B
+ 10o
- 10o
+ 30o
- 30o
+ 50o
- 50o
For Example :
|FA| (= |FB|)
(N)
(graphical method)
(N)
W
Object Weight
(N)
|f|
Frictional Force
(N)
FA = mA* g = ( ________ kg) X (9.80 m/s2) = ____________ N
|f| = |FR – W|
* Note that mA includes the mass of the mass hanger.
B. COMPONENTS METHOD
Adding forces with different magnitudes and different directions.
- 14 -
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
FA
FyA
A
W
180o
B
0o
FxA
FB
Fig. 3a
Fig. 3b
Table B: COMPONENTS METHOD OF VECTOR ADDITON
A = __________
mA = ___________ grams = _________ kg
FA = __________
B = __________
mB = ___________ grams = _________ kg
FB = __________
FA
FyA
Vector
x-Component
(N)
FxA
FxB
FB
FyB
FxA = FA cos A
FR =
Fx2  Fy2 ;
y-Component
(N)
FA
FxA =
FyA =
FB
FxB =
FyB =
FR
Fx = FxA + FxB =
Fy = FyA + FyB =
FxB = FB cos B
FyA = FA sin A
FyB = 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)
- 15 -
Experiment 5: 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 which has its weight as the only significant force acting upon it.
Examples are: thrown balls, fired 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 the time of flight as t = 2y / g . Then, from a measurement
of the range (horizontal distance), the initial velocity vo 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
White paper and carbon paper
One and two meter sticks
Short support rod
Table Clamp
2 Large pieces of cardboard
Metric ruler
Wooden board for angle shot
Clamp and rod for wooden board
vo
y
s
Fig. 1, Part A
- 16 -
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. Place a piece
of cardboard against the wall, in the path of the projectile. Clamp the gun (not too tightly) to
the table, at least six feet from the wall. Use the inclinometer to orient the gun to fire
horizontally, and take a trial shot. Tape a second piece of cardboard to the floor centered on
the spot where the projectile landed. On top of the cardboard, tape a piece of carbon paper
face-down above a piece of plain paper to record the point of impact. Place the wooden catch
box just beyond the carbon paper, to catch the projectile.
3. Take six shots. Measure the range of each shot to the nearest millimeter. To accomplish this,
use a plumb bob to get the exact vertical drop from the end of the gun to the floor. Record
your values for the range in the data table.
4. Measure the height from the floor to the bottom of the ball to the nearest millimeter. 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  2y / g .
5. Calculate the range s and the average initial velocity: vo = s / t .
Data for Part A.
Data and Calculations Table: Initial Velocity, vo
Trial
y
(m)
s
(m)
s
(m)
1
2
3
_____________________
4
5
6
- 17 -
_____________________
Part A Calculations:
Time of Flight
t  2y / g =
Initial Velocity
vo = s / t =
B. PREDICTION OF THE RANGE
1. Clamp the spring gun to a board at an angle between 10 o and 20o. Measure this angle
precisely with an inclinometer.
2. Measure the height of fall, y = yf - yi.
3. Fire the projectile and measure the range. Fire the projectile five more times and determine
an average measured range.
4. Calculate the time of flight and the expected range. Calculate the percent difference between
the measured and expected range.
y
Initial
Position
Of Projectile
vo

Gun
v oy = v ocos 
x
v ox = v ocos 
y
x
\
Fig. 2,
- 18 -
Part B
Final
Position
Of Projectile
Data for Part B:
Angle of Elevation:  = ________________ degrees
Height from floor to the bottom of ball: y = ________________ m
Six values of the measured range:
________________ m,
________________ m,
________________ m,
________________ m,
________________ m,
________________ m
Average value of the measured range, x = ________________ m
Part B. Calculations:
Average Initial Velocity, vo = ________________ (From part A)
vox = vo cos  = ________________
voy = vo sin  = ________________
The equation y = voy t + ½gt2 may be rewritten as ½gt2 + voy t  y = 0.
This equation has the form At2 + Bt + C = 0. Solve this quadratic equation for t by making
the following substitutions:
A = ½ g = ________________
(g = - 9.80 m/s2)
B = voy = ________________
C =  y = ________________ (y is negative, therefore C is positive; see Fig. 2)
t
B 
B
2
2A
 4 AC
= ________________ s (Choose t such that it is a positive number)
Expected Range:
Expected range: x = vox t = ________________ m
Percent difference between measured and expected range ________________ %
- 19 -
Experiment 6: 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, so
 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,
a
2s
.
t2
SUPPLIES & EQUIPMENT:
Air track and accessory kit
5-gram mass holder
Four 5-gram slotted masses
Accessory photogate timer
Masking tape
Electronic balance
Air track air supply
Photogate timer
Photogate transformer
Photogate #1
Scissors
Fine string
Glider
Cardboard flag
Photogate #2
m1
m2
Fig. 1. Experimental Setup
- 20 -
PROCEDURE:
1. Connect the air supply to the air track and turn it on. Level the air track by adjusting the
leveling feet so the glider is balanced at the center of the air track. Do not lean on the air
track or the table (use another table for writing) during the experiment.
2. Tape the cardboard flag to the top of the glider, so the flag juts a couple of centimeters
above the top of the glider. Determine the glider's mass m1 on a balance and convert this
measurement to kilograms.
3. Place photogate #1 near the position 70 cm and photogate #2 near the position 140 cm.
Slowly slide the glider towards photogate #1 until the LED on top of the gate lights up
because the flag breaks the light beam. Back up the glider and move it forwards several
times until you can determine the position x1 of the front of the flag, to the nearest millimeter.
Do the same with photogate #2 to get x2, and calculate s = x2 - x1.
4. Place a 5-gram mass holder at the end of the string running over the pulley. Add a 5-gram
mass onto the mass holder, so m2 = 10.0 grams = 0.0100 kg.
5. Set the photogate timer to the "pulse" mode. Set the resolution scale to 1ms.
6. Push the “reset” button. 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 perturb the glider) and record the displayed time.
7. Reset the timer and repeat the procedure two more times. Average the three values and
record in the table.
8. Add a 5-gram mass onto the mass holder. Repeat steps 6 and 7. Remember that m 2
equals the mass of the holder plus the mass on the holder, so the total mass for this step is
15 g.
9. Repeat steps 6 and 7 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:
s = vot + ½at2 ,
with
vo = 0,
s = x2 - x1
s = ½at2
a = 2s/t2
11. Compute the percent difference between the acceleration a k measured kinematically with
the photogates and the acceleration aN from Newton’s law F = ma, as shown below:
F = W = m2g
F = (m1 + m2)a
- 21 -
a
m2 g
F

m1  m 2
m1  m 2
DATA SHEET: NEWTON'S SECOND LAW
Data and Calculations Table:
m1
(kg)
m2
(kg)
S
(m)
Time, Trial 1
(s)
Time, Trial 2
(s)
Time, Trial 3
(s)
Average Time
(s)
Acceleration from:
ak = 2s/t2
Force from:
F = m2g
0.0100
0.0150
(m/s2)
(N)
Accelerated mass:
m1 + m2
(kg)
Acceleration from Newton’s
2nd Law:
aN = F/(m1 + m2)
(m/s2)
% difference
- 22 -
0.0200
0.0250
Experiment 7:
SIMPLE MACHINES
PURPOSE:
In this experiment the principle of work is studied, using the inclined plane and pulleys as
examples of simple machines.
INTRODUCTION:
A machine is any device used to do work by changing the magnitude or direction of a
force. A simple mechanical machine exerts an output force which is greater than the applied
force. A machine makes work easier by multiplying the applied force or increasing the speed of
doing work. Work or energy is not multiplied by a machine, force is multiplied. In practical
applications the work output is always less than the energy input. The efficiency of a machine is
defined as:
F d
F  d 
Work Output
Efficiency = Work Input = out out =  out   out 
Fin din
 Fin   din 
F 
where:  out  = AMA = Actual Mechanical Advantage,
 Fin 
 din 
d 
1
,

 = IMA = Ideal Mechanical Advantage and  out  = IMA
d
d
 out 
 in 
therefore
Efficiency = AMA
IMA
EQUIPMENT & SUPPLIES:
Single pulleys
Double tandem pulleys
Mass hanger
Slotted Masses
Hall's carriage
Inclined plane board
Table clamp
Short support rod
Masking tape Double pan balance Pendulum Clamp Clamp & rod for board
Inclinometer
Meter stick
Ruler
String & scissors
Clamp-on pulley
PROCEDURE:
m1 = ________ kg
I. The Inclined Plane
din = _________ m
Fin = m1g
dout
m2 = ________ kg
d in

Fin = m1g = __________ N
Load = Fout = m2g
dout = din sin = _________ m
Fig. 1.
Fout = m2g = _________ N
- 23 -
1. Set up the apparatus as in Figure 1, with  = 30o.
2. Add mass to the mass holder until the cart (m 2) moves up the incline at a constant speed.
Measure din , and calculate dout = din sin . Calculate Fin = m1g and Fout = m2g.
3. Calculate AMA = _________, IMA = _________ and the percent efficiency = __________ %.
4. Repeat steps 1 through 3 for a less-steep angle.
 = ____________ ;
AMA = ____________ ;
IMA = ____________ .
 AMA 
% Efficiency = 
 X 100% = _______________ %.
 IMA 
II. Pulley Systems
1. Set up the pulley system as in System A.
2. Add slotted masses to the mass holder to move the cart up at a constant velocity.
3. Measure din (the distance the mass holder travels), d out (the distance the cart travels), m1
(the mass of the mass holder and slotted masses) and m 2 (the mass of the cart and any
pulleys that move up or down). System A has no moving pulleys.
4. Calculate Fin, Fout, IMA, AMA and the efficiency and enter the results in the data table.
5. Repeat the procedure for pulley systems B, C, D, E, and F.
Fin = Weight of mass holder
and slotted masses
F
F
out
in
din = Input displacement
Fout = Weight of cart and
moving pulley
d out
Original
Position
of cart
din
m2
dout = Output displacement
m1
Final
Position
of slotted
masses
System A: Fixed Pulley
IMA should be ≈ 1
System B: Fixed Pulley
IMA should be ≈ 2
- 24 -
System C
System D
System E
System F
Data and Calculations Table:
System
m1
(kg)
din
(m)
m2
(kg)
dout
(m)
Fin
(N)
Fout
(N)
A
Mass of cart = ____________ kg
B
C
AMA
IMA
% Efficiency
- 25 -
D
E
F
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 = ½mv22 – ½mv12 is the change in kinetic energy of the glider, and (mgh) = mgh 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
Air supply
Meter stick
Vernier caliper
Glider
Electronic balance
Photogate timer
Photogate timer transformer
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 shim block of known thickness, H, under the single 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.
4. Set up a photogate timer and an accessory photogate as shown in the figure below.
- 26 -
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 for this 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. That is, the photogate timer first displays t1 , then ttotal = t1 + t2 ,
and does not display t2 by itself.
10. Repeat the measurement four more 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 photogates.
PART B:
1. Repeat procedure A with a block of greater thickness, H ' by using two shim blocks. Record
the data in Table B.
- 27 -
CALCULATIONS:
1. Calculate , the angle of incline for the air track, by 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
support legs
H
h
d
D
d = distance between
photogates


H = block thickness (distance
the air track leg is 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 = ____________
 = ____________
d = ____________
L =____________
m =____________
Data and Calculations Table A:
Trial
1
t1
(s)
t2
(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 = ____________
= ____________
d = ____________
L =____________
m =____________
Data and Calculations Table B:
Trial
1
t1
(s)
t2
(s)
v1
(m/s)
v2
(m/s)
KE1
(J)
KE2
(J)
KE2 - KE1
(J)
2
3
4
Average KE = ____________ mgh = ____________ % difference = ____________
- 29 -
5
Experiment 9: THE BALLISTIC PENDULUM
PURPOSE:
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
vo
m  M
vo = 
V
 m 
Eq. 1
Before Impact
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) = KE2 + PE2 (at h2)
KE = 0
(m+M)V
V
Immediately
After Impact
0
KE1  KE2 = PE2  PE1 ; since v2 = 0
½(m + M)V2 = (m + M)gh2  (m +
M)gh1
½ (m + M)V2 = (m + M)gh
½ V2 = gh
- 30 -
PE = (m+M)gh
h2
h1 h
At Rest
So
V=
2gh
Eq. 2
Substituting the expression for V from Eq. 2 into Eq. 1 gives:
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 (h1) 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 (h2) from the base plate to the pendulum center of mass. Calculate
h = h2 - h1.
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 ½mvo2, 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
Mass of Ball
______________ (See label on equipment)
______________ kg
Data Table 1: Pendulum Height Measurements
Trial
Notch #
Trial
Notch #
Trial
1
3
5
2
4
6
Notch #
Average Notch #
h1 = height of pendulum when hanging freely
______________ m
h2 = height of pendulum at average notch number
h = h2  h1
______________ m
______________ m
m  M
Initial velocity of ball: vo = 
 2gh
 m 
vo from Experiment 5, Projectile Motion
% difference between the two vo
Velocity of pendulum & ball after impact, V =
(Eq. 3)
____________ m/s
____________ m/s
____________
2gh (Eq. 2)
____________ m/s
Momentum before collision: mvo =
Momentum after collision: (m + M)V =
Is momentum conserved in this inelastic collision?
____________ kgm/s
KE before collision: ½ mvo2 =
____________ J
KE after collision: ½ (m + M)V2 =
Is kinetic energy conserved in this inelastic collision?
____________ J
____________
Energy loss: W nc = ½ mvo2  ½ (m + M)V2
% energy loss:
____________ kgm/s
____________
____________ J
Wnc
X 100% =
(1 / 2)mv o2
____________ %
- 32 -