Chapter 8
Hooke’s Law for Linear Elastic
Materials
8.1
Lecture - Force, Displacement, and Elastic Potential Energy
In this lesson, we will study the engineering application core concept of experimental “modeling.” There are numerous cases in our engineering curriculum, and professional practice,
when we desire to have a mathematical relationship to describe the behavior of materials.
As engineers, we use many di↵erent kinds of materials including solids, liquids, and gases.
Sometimes we use combinations of materials in di↵erent phases, or we even combine di↵erent
materials together to form alloys or composites. One type of material that we will use in an
immense number of engineering applications includes a certain class of solids called “linear
elastic materials.” These have all been observed to obey a particular relationship, called
“Hooke’s Law.” Hooke’s Law is a constitutive equation, meaning it describes the behavior
of a class of materials and how the material responds to certain environmental conditions.
A full understanding of the intricacies of Hooke’s Law will come in later classes such as
statics and strength of materials. In this class, we will learn the fundamental concepts and
characteristics of linear elastic materials through the study of a simple coil spring.
In your secondary school chemistry and physics courses, you learned about another important constitutive equation called the ideal gas law. The ideal gas law is a useful equation
that describes how certain kinds of gaseous materials will behave under varying environmental conditions of temperature and pressure. We learned that the ideal gas law does not apply
to all gases or ranges of environmental conditions. In the same way, Hooke’s Law helps us
to describe the idealized response of certain solids, and the relationship that these solids
exhibit in the form of response to forces and deflections. Hooke’s Law does not apply to all
solids, and it does not apply to all loading conditions of solids. However, when it does apply,
it is very convenient to use, just as the ideal gas law is very convenient in many situations
involving gasses. Hooke’s Law states that there is a linear relationship between the force
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applied to an elastic material and the deformation experienced by the material.
8.1.1
Formulate
State the Problem
Given the mass m of a dead weight suspended from a linear elastic spring, determine the
spring constant K of the spring, and the change in energy stored within the spring as a result
of the work done by mass m upon the spring as the spring S moves from its initial position
i to its final position f .
State the Known Information
The following information is provided
m = M1
[kg]
mass of dead weight
(8.1)
State the Desired Information
Upon conclusion of the experiment and analysis, we shall be required to report:
K =?
SEi!f = ?
8.1.2
[N/m] ↵
[J] ↵
spring constant of spring, S
change in elastic potential energy of S
(8.2)
(8.3)
Assume
We will make several familiar assumptions for this analysis.
Identify Assumptions
The following assumptions may be employed during the analysis.
mspring = 0
mE
g = G 2 ⇡ 9.80665
RE
Qi!f = 0
f or i ! f
[N ]
(8.4)
[m/s2 ]
(8.5)
[J]
(8.6)
We can make several assumptions about the initial and final displacements as:
z 1i = 0
x 1i = 0
x 1f = 0
[m]
[m]
[m]
(8.7)
(8.8)
(8.9)
z 0i = z 0f = x 0 i = x 0f = 0
[m]
(8.10)
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and we can also make assumptions about the initial and final components of velocity:
V zi = 0
V zf = 0
[m/s]
[m/s]
(8.11)
(8.12)
V xi = 0
V xf = 0
[m/s]
[m/s]
(8.13)
(8.14)
We will assume that the spring is made from a linear elastic material, and assume that
Hooke’s Law is a valid material model to describe the behavior of the spring:
!
F 1 = K!
x1
!
F R1 = K !
x1
[N ]
external force
(8.15)
[N ]
restorative force
(8.16)
Hooke’s Law is a vector equation. As we have seen in past problems, it is often convenient to
analyze vector problems by looking at each component individually. When analyzing helical
spring problems, it is a common practice to employ a local coordinate system whose axis is
parallel to the axis of the spring.
Justify Assumptions
We need to justify each assumption proposed for use in our analysis.
Equation 8.4 says that we are neglecting the mass of our spring as being negligible in
comparison to the external mass. In reality, the spring does have some finite mass, which
causes a small displacement due to its own weight. However, since we will choose the
coordinate system origin from this location, the mass of the spring may be conveniently
neglected.
Equation 8.5 says that the local acceleration of gravity, g, is a known value.
Equation 8.6 says that there is no heat transfer between the spring and its surroundings.
Equation 8.7 says that we are measuring the vertical position from the initial neutral
position of the spring. This assumption must be consistent with the coordinate systems
and diagrams that we create. Equation 8.8 and 8.9 say that we are neglecting the lateral
displacement of the bottom of the spring. Equation 8.10 indicates that the upper end of the
spring is fixed to the mounting bracket, and does not move.
Equations 8.11 through 8.14 state the the vertical and horizontal components of velocity
for the spring are zero before the mass is placed on the end of the spring, and that we wait
until the spring and mass reach their equilibrium position before we consider the system to
be in its final state. We will remove this assumption in a subsequent lesson.
Equations 8.15 through 8.16 states that the spring is made from linear elastic material,
and that Hooke’s Law may be used as a constitutive model to describe the physical properties
of this material. This assumption must be validated experimentally.
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8.1.3
Chart
Schematic Diagrams
A schematic diagram illustrating a mass suspended from the end of a spring, causing it to
extend is shown in Figure 8.1. We place the coordinate system in alignment with the spring
axis, with the origin located at the neutral position of the spring, prior to placing the mass
upon it. This provides a convenient way to express the initial condition of the system with
no horizontal or vertical displacement. The first state i is considered to be the instant of
time just before the mass is placed upon the spring, and the final state f occurs after the
spring and mass system has come to rest at a new equilibrium position.
Figure 8.1: Schematic diagram of a mass suspended from a spring in tension.
Free Body Diagrams
As the mass is suspended from the spring, the mass extends the spring, since the mass
exerts a force upon the spring due to the action of gravity upon the mass. As the mass
extends the spring, the mass exerts a force upon the spring, and the spring exerts an equal
and opposite force upon the mass. The spring is supported in place by the reaction force
from the mounting bracket, which ultimately is transmitted through the lab apparatus to
the table, the building, and finally the earth. Since our primary interest lies in the spring,
we chose to draw the system boundary (in the schematic) to include only the spring. A free
body diagrams for the spring and is shown in Figure 8.2.
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Figure 8.2: Free body diagram of a spring, with an external force applied.
Data Tables
As we complete our analysis, it will be helpful to record data in the state table as shown
below. Each row in the state table corresponds to one unique position of the spring as it
is loaded by a unique mass, The symbols and ↵ are used to indicate known and desired
information, respectively. The top header row of the state table includes the name of the
State
Applied
Mass
me
[kg]
Sensor
Volts
V
[V ]
1
2
0.0
N/A
Spring
End
Location
[inch]
Displacement
From
Neutral, d
[m]
↵
0.0
Applied
Force
Fd
[N ]
↵
.
.
.
Spring
Mass
ms
[kg]
0
0
0
Spring
SE
Spring
PE
Spring
KE
[J]
Spring
Tip
|V |
[m/s]
Spring
Momentum
|p|
[kgm/s]
[J]
↵
[J]
0
0
0
0
0
0
0
0
0
0
0
0
Table 8.1: State Table for spring, before and after displacement.
variable. The second row includes the correct engineering units associated with every entry
in that column of the state table. The third row of header indicates that these values are
unknown at the beginning of the problem, and must be determined during the laboratory
experiment and subsequent analysis. In this case, several values have been filled in, as a
direct consequence of the simplifying assumptions presented previously. Please review each
completed entry in the state table, to be certain that you understand why these values have
been entered. We define state 1 as being the neutral condition of the spring, when suspended
from the hanger, and with no external mass applied. In this case, we take do not measure a
sensor voltage, since there is no surface to measure against, and we define the displacement,
d = 0, at this condition. We will choose to measure the applied force Fd in a downward
direction, parallel to the displacement, d.
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8.1.4
Execute
Recall The Governing Equations
The first step in the execution phase is to recall the governing equations. In this case, we
have our familiar set of Newton’s laws to draw upon, and we now include the work and
energy theorem as well.
!
F N et = 0 T hen : !
a =0
!
!
p
F N et = lim
t!0
t
!
(m V )
= lim
t!0
t
!
d(m V )
=
dt
d!
p
=
dt
!
!
F Action = F Reaction
!
Fg =g·m#
E2 E1 = Q1!2 W1!2
Newton’s 1st Law
(8.17)
Newton’s 2nd Law
(8.18)
Newton’s 3rd Law
(8.19)
Newton’s Law of Gravity near Earth
Work Energy Theorem
(8.20)
(8.21)
Z
(8.22)
If :
Finally, let’s recall the definition of work
W1!2 =
z2
z1
!
F N et · dz
Hooke’s Law
Hooke’s Law states that strain is directly related to stress. Unfortunately, we will not be
learning the formal definitions of stress and strain until our later courses in statics and
strength of materials. For a solid linear elastic material of uniform cross sectional area,
Hooke’s Law says that the restoring force exerted by the material is proportional to the
magnitude of the displacement, and opposite in direction to this displacement. For a linear
elastic material of uniform cross sectional area, Hooke’s Law may be stated mathematically
as
!
F 1 = K!
x1
external force
(8.23)
!
F = K!
x
restorative force
(8.24)
R1
1
[N ] = [N/m][m]
units validation
!
Equation 8.23 says that, if we apply an external force F 1 to a linear elastic spring, then the
spring will experience a displacement an amount !
x 1 , in the same direction as the external
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force. Conversely, Equation 8.24 says that, if a spring is distorted by some amount !
x 1 , then
!
the spring will exert a restorative force F R1 (internal to the spring) that will attempt to
return the spring to its original position. That is, the restorative force resists the action of
the external force.
Consider a spring suspended from a fixed mounting bracket, as shown in Figure 8.3.
Under no-load conditions, we say that the spring is “relaxed”, and we often choose the end
point of the relaxed spring as the origin of our coordinate system. If we “pull” on the spring
and place it in tension, then the spring will extend downwards in response to the force
applied. On the other hand, if we “push” on the spring and place it in compression, then
the spring will compress upwards in response to the force applied. This is consistent with
Hooke’s Law as given by 8.23. If we then release the external force from the spring, the
restorative force will cause the spring to return to its original relaxed state.
Figure 8.3: Schematic diagram of a spring unloaded, loaded in tension, and loaded in compression.
There is a lot happening here. What is happening inside the spring as we apply this
force? Why does the spring reach some displacement, and then stop moving? If there is
an external force applied to the end of spring, why does it not keep on moving? Let’s use
this new constitutive relationship, Hooke’s Law, to fully understand the response of a spring
when it is subjected to an external force. In particular, we want to learn how to estimate the
value of the spring constant, K, estimate the work done when a force is a applied to a spring
(moving it from its rest position to a new extended position), and analyze the work-energy
theorem during this action.
Newton’s laws, the work energy theorem, and Hooke’s Law for modeling the spring
material may be used to achieve this understanding.
Simplify the Governing Equations
We used Newton’s Third Law, Equation 8.19, in order to construct the free body diagram
for the spring as shown in Figure 8.2. To begin the analysis, let’s imagine that the spring
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is initially at rest in its neutral state, and that the mass m is resting on the table. This
represents the initial condition i. Now, we manually pick the mass m up with our hands
and gently attach it to the bottom of the spring. Because the mass experiences the force
of gravity, it will extend the spring some distance. Eventually, we should observe that the
spring comes to rest at a new equilibrium point. We call this new equilibrium point the final
state f . At this final state f , Newton’s First Law as given by Equation 8.19 tells us that the
net force should be zero, since the observed acceleration is zero. Let us study the final state
f in more detail.
Let’s apply Newton’s First Law, Equation 8.17 to the spring:
!
F N et = 0
(8.25)
!
F N et = Fm k̂ + Rk̂
(8.26)
2
2
2
[kgm/s ] = [kg][m/s ] + [kgm/s ]
basic units
Now that we have a full understanding of the initial state i and the final state f , we can
evaluate the process as the system moves from state i to state f . The work done by the mass
upon the spring from state i to state f is the integral of the force along the displacement.
From the definition of work,
Z zf
!
Wi!f =
F N et · dz
(8.27)
zi
!
The force exerted by the mass is F m = mg k̂, by virtue of Newton’s Law of Gravity,
Equation 8.20. While the system is moving and before the final state is achieved, the
displacement of the spring is not sufficient to balance the force due to the weight of the
mass. Assumptions 8.15 and 8.16 allow us to employ Hooke’s Law to tell us that, as the
spring is being extended in the negative z direction, it opposes the motion of the mass
hanging from it with a restorative force in the positive z direction:
!
F S on m = +Kz k̂
force of spring on m
(8.28)
The force of the mass acting upon the spring is simply the weight of the mass, acting in the
negative z direction:
!
F m on S = mg k̂
force of mass on spring
(8.29)
Newton’s Second Law, Equation 8.18 clearly tells us that the spring/mass system will continue to move until the two forces come into balance. When the spring achieves its final
equilibrium position f , with the bottom of the spring deflected to length z1f we can state
by Newton’s First Law that
+Kz1f k̂
mg k̂ = 0
or
(8.30)
Kz1f = mg
(8.31)
[N/m][m] = [kg][m/s2 ]
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derived units
We will look at that “transient response” of the system in more detail during our next
lesson. For this week, however, let’s use a work energy theorem analysis of the spring and
mass system to gain insight into the system operation.
The work done on the spring by the mass can be found from the following:
Z zf
m on S
Wi!f
=
( Kz k̂) · ( dz k̂)
(8.32)
zi
Z zf
=
+(Kz)dz
(8.33)
zi
Z zf
=+
(Kz)dz
(8.34)
zi
K z
= + z 2 zfi
2
K 2
m on S
Wi!f
=+
z
zi2
2 f
[J] = [N/m] [m]2 [m]2
(8.35)
(8.36)
derived units
Note that the reaction force at the fixed mounting bracket is not included in the work done
on the spring, because that applied force does not give rise to a displacement in z. This
reaction force keeps the spring in a static position, but does not exert work ( z0 = 0).
Conversely, let’s study the work done by the spring upon the mass.
Z zf
!
S on m
Wi!f
=
F S on m · dz
(8.37)
zi
Substitute Equation 8.28 into Equation 8.37
Z zf
S on m
Wi!f
=
(+Kz k̂) · ( dz k̂)
zi
Z zf
=
(Kz)dz
zi
Z zf
=
(Kz)dz
(8.38)
(8.39)
(8.40)
zi
K 2 zf
z zi
2
K 2
S on m
Wi!f
=
zf zi2
2
[J] = [N/m] [m]2 [m]2
S on
Wi!f
m
(8.41)
=
(8.42)
derived units
It is interesting to note that the work done by the mass upon the spring, as given by Equation
8.36 is the negative of the work done by the spring upon the mass, as given by Equation
8.42.
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Now, let’s recall and simplify the work-energy theorem given by Equation 8.21, as it
applies to the spring.
Ef
Ei = Qi!f
Ei!f = Qi!f
Wi!f
Wi!f
or
(8.43)
(8.44)
The energy stored in the spring may consist of three components: elastic potential energy,
gravitational potential energy, and kinetic energy:
!0 by Eq.8.4
SEi!f
!0 by Eq.8.4
z }| {
z }| {
+ P Ei!f + KEi!f =
SEi!f =
!0 by Eq.8.6
z }| {
Qi!f
Wi!f
Wi!f
or
(8.45)
(8.46)
The mass m is the only object upon which the spring does work, since it is the only object
that undergoes a displacement as a result of the force exerted by the spring. Thus, because
S
S on m
Wi!f
= Wi!f
we can substitute Equation 8.42 into Equation 8.46. We define the change
in the elastic potential energy of an ideal spring (one that is made of a linear elastic material)
as
✓
◆
K 2
2
SEi!f =
z
zi
(8.47)
2 f
K 2
zf zi2
2
[J] = [N/m][m]2
SEi!f ⌘
(8.48)
derived units
The mass does positive work on the spring. The spring does negative work on the mass.
The work done on the spring causes a change in the elastic potential energy stored within
the spring.
Inventory the Governing Equations, Known, and Desired Information
Recall Equation 8.31
Kzf = mg
[↵][↵] = [ ][ ]
(8.49)
inventory
This equation has two unique unknown values (K and zf ). In order to solve this equation,
we need at least one additional piece of information. Without additional information, we
cannot completely solve the problem. We will need experimental data in order to estimate
the value of the spring constant K. Hooke’s Law is an empirical relationship, which means
that it is based on experimental observations. The value of K cannot be determined from
first principles.
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Solve
Now, assuming that we have experimental data available to estimate the spring constant K,
we can recall Equation 8.48
K 2
z
zi2
2 f
[↵] = [ ] ([↵] [ ])
SEi!f ⌘
(8.50)
inventory
and recall Equation 8.36
K 2
z
2 f
[↵] = [ ] ([↵]
m on
Wi!f
S
=+
zi2
(8.51)
[ ])
inventory
Close inspection reveals that these two expressions are the negative of one another – the
work done by the spring upon the mass is stored as elastic potential energy in the spring.
8.1.5
Test
Validate
In lab, we will conduct an experimental validation of Hooke’s Law.
Verify
We have verified that the units on each result are correct.
Apply Intuition
The results are consistent with our intuition. We expect that a spring will undergo a finite
displacement when loaded with a deadweight, and come to rest. As the mass associated with
the dead weight is increased, we expect the spring displacement to increase, at least until
the point of permanent damage to the spring is observed.
8.1.6
Iterate
In lab, we will conduct multiple trials of this experiment. Each student member of the
lab group should conduct an independent trial, using a unique mass for the deadweight
and analyze a unique data set. When the results of all students’ trials are compiled on a
single graph, the team should compare their group estimate for the spring constant to their
individual estimate of the spring constant. Explain any discrepancies.
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8.2
Lab - Stationary Spring Mass Systems
8.2.1
Scope
This week in lab you will use a manual measurement system to record the displacement of a
spring in response to a hanging dead weight that is suspended from the spring. You will also
correlate the manual displacement measurements to the measured output from the ultrasonic
transducer. The primary purpose of this experiment is to estimate the spring constant, K,
of the spring] using the manual displacement measurements. As a secondary purpose, the
measurements from the ultrasonic transducer will be used to calibrate the voltage output in
preparation for next week’s experiment.
8.2.2
Goal
The goals of this laboratory experiment are to
1. demonstrate the concept of Hooke’s Law as a constitutive equation, and
2. demonstrate the concept of elastic potential energy.
8.2.3
Units of Measurement to use
All reports shall be presented in the SI system of units. Raw data may be collected in a
variety of units.
Quantity
Length
Mass
Time
Velocity
Force
Energy
Spring Constant
Basic units
[m]
[kg]
[s]
[m/s]
[kgm/s2 ]
[kgm2 /s2 ]
[kg/s2 ]
Derived units
[m]
[kg]
[s]
[m/s]
[N ]
[J] or [N ][m]
[N ]/[m]
Table 8.2: Units of Measurement to be used for stationary spring mass system.
8.2.4
Reference Documents
Review the materials from previous chapters related to the ultrasonic sensor and its proper
usage. Review any previous materials related to the proper use of the data acquisition
software that accompanies the ultrasonic sensor.
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8.2.5
Terminology
The following terms must be fully understood in order to achieve the educational objectives
of this laboratory experiment.
Energy
Displacement Force
Kinetic Energy
Velocity
Work
Gravitational Potential Energy Speed
Spring Constant
Elastic Potential Energy
Acceleration Hooke’s Law
8.2.6
Summary of Test Method
On the myCourses site for this course you will find links to one or more videos on YouTube
for this week’s exercise. Watch all of the available videos, and complete the online lab quiz
for the week. The videos are your best reference for the specific tasks and procedures to
follow for completing the laboratory exercise.
8.2.7
Calibration and Standardization
By now in this course, students should be in a position to conduct independent calibrations of
hardware and properly configure the use of all hardware without having detailed instructions.
Please note, however, that it is very important to treat the neutral position of the unloaded
spring as an elevation of zero for all subsequent measurements and analyses.
8.2.8
Apparatus
All required apparatus and equipment components are described and demonstrated in the
instructional videos for this exercise, or will be familiar from common or previous use.
8.2.9
Measurement Uncertainty
By now in this course, students should be well aware of the proper procedures and techniques
required to determine and document basic uncertainties associated with all measurements.
When making the measurements in lab, be sure to record the uncertainties or instrument
least counts associated with all measurements that are made. Also, make note of any other
possible sources or uncertainty that you observe while performing the experiment.
8.2.10
Preparation of Apparatus
All required equipment for conducting the laboratory exercise is made available either within
one or both of the drawers attached to the lab bench or available from the laboratory instructor. You are expected to bring all other necessary materials, particularly your logbook and a
flash drive for storing electronic data as appropriate. You are to follow the general specifications for team roles within the lab. Although there are specific, individual expectations for
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each role, you are each responsible overall to ensure that the objectives and requirements of
the laboratory exercise are met and that all rules and procedures are followed at all times,
especially any that are related to safety in the lab. When finished, all equipment is to be
returned to the proper location, in proper working order.
8.2.11
Sampling, Test Specimens
The basic procedure for this experiment consists of applying a sequential set of masses to the
end of the spring and recording the corresponding displacements and steady state voltage
outputs from the ultrasonic sensor. Each group member is expected to use his/her own range
of masses and collect his/her own set of measurements for each.
8.2.12
Procedure - Lab Portion
The instructional videos for this exercise cover the specific procedures to follow as you set
up the apparatus to make measurements, and for actually collecting data with the various
devices and software interfaces. More generally, you should always observe the following
general procedures as you conduct any of the exercises in this laboratory.
1. Come prepared to lab, having watched the videos in detail, then completing the associated lab quiz and preparing your logbook before you arrive to class.
2. Follow the basic outline of elements to include in your logbook related to headers,
footer, and signatures.
3. As you conduct the exercise, please pay attention to the following safety concerns:
• Watch for tripping hazards, due to cables and moving elements.
• Watch for pinch points, during assembling and disassembly.
• Be careful of shock hazards while connecting and operating electrical components
4. Every week, for every exercise, your logbook will minimally contain background notes
and information that you collect before the lab, at least one schematic of the apparatus,
various standard tables for recording the organization of your roles and equipment
used, the actual data collected and/or notes related to the data collected (if done
electronically for instance), and any other information relevant to the reporting and
analysis of the data and understanding of the exercise itself.
5. All students should create and complete a table indicating the staffing plan for the
week (that is, the roles assumed by each group member), as shown in Table 1.2.
6. All students should create and complete a table listing all equipment used for the exercise, the location (from where was it obtained: top drawer, bottom drawer, instructor?)
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and all identifying information that is readily available. If the manufacturer and serial number are available, then record both (this would be an ideal scenario). If not,
record whatever you can about the component. In some, cases, there will be no specific
identifying information whatsoever either because of the simplicity of the component,
or because of its origin. In these cases, just identify the component as best you can,
perhaps as “Manufactured by RITME.” The point here is to give as much information
as possible in case someone was to try to reproduce or verify what you did. Refer to
Table 1.3.
7. For the Lab Manager only: create a key sign-out/sign-in table for obtaining the
key to the equipment drawers, as shown in Table 1.4.
8. All students should create a table or series of tables as appropriate to collect his/her
own data for the exercise, as well as any specific notes related to the data collection
activities. In those cases where data collection is done electronically, there may not be
any data tables required.
9. Many of the laboratory exercises will require the use of a specific software interface
for measurements and/or control. In all cases, these will be made available on the
myCourses site unless stated otherwise.
10. The Scribe (or a designated alternative) should take a photo of each group member
performing some aspect of the laboratory exercise for inclusion in the lab report
that will be generated during the studio session. Refer to the example lab report for
more details.
11. Record all relevant data and observations in your logbook, even those that may not
have been explicitly requested or indicated by the textbook or videos. If in doubt
about any measurements, it is better to make the measurement rather than not.
12. When you are finished with all lab activities, make sure that all equipment has been
returned to the proper place. Log out of the computer, and straighten up everything
on the lab bench as you found it. Put the lab stools back under the bench and out of
the way.
13. Prepare for the upcoming studio session for the week by carefully read and understand
Section 4.3 of the textbook, and complete the Studio pre-work prior to your arrival at
Studio.
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8.3
Studio - Spring Constant and Elastic Potential Energy
This week in Studio you complete an analysis of Hook’s Law. Using the data acquired in
lab, you will estimate the spring constant for a small spring. In addition, you will apply your
knowledge of sensor calibration to derive a calibration equation for the ultrasonic sensor
which can be applied again in next weeks analysis. You will create a state table which
includes the spring energy at each state of the spring under various applied loads. These
results lead into next week’s analysis of oscillation utilizing the same experimental set-up.
The theory needed to analysis the data is discussed discussed in Section 8.1 of the text.
Section 8.3.1 Calculation and Interpretation of Results provides a summary of equations that
you will need to complete the Studio. Section 8.2.9 Measurement Uncertainty describes the
process for analyzing the experimental errors. There will be uncertainty equation derivations
that you need to complete prior to arriving at studio and these are listed in the Measurement
Uncertainty section.
Record all observations and notes about your studio procedures in
your logbook.
8.3.1
Calculation and Interpretation of Results
The following equations may be helpful in the context of the Studio Procedure.
E = KE + P E + SE
[J] = [J] + [J] + [J]
1
SE = Kd2
2
[J] = [N/m][m]2
FS = Kd
[N ] = [N/m][m]
8.3.2
total energy of a state
units validation
(8.52)
definition of SE
(8.53)
units validation
Hooke’s Law
units validation
(8.54)
Procedure - Studio Portion
Studio Pre-work
Prior to arriving at Studio, each student should have acquired the necessary data in lab,
recorded data in your notebook and stored data on a thumb drive. You should also have a
corresponding schematic that clearly identifies where each measurement was made in symbolic notation.
In addition, each of you will complete several steps of the Studio exercise. This will allow
more quality time with the instructor to discuss the physical meaning of the analysis results.
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You will upload your studio pre-work to your individual drop-box for the corresponding
week. You will receive a quiz grade based on the completeness of your submission.
Please complete at least steps 1-6 and upload your pre-work spreadsheet to
your individual drop-box, before coming to coming to Studio. You can work on the
remaining portions of the exercise during Studio. All steps with the exception of the report
are due within 24 hrs after leaving Studio.
Videos
There are videos available to help with some of the excel techniques that may be new to you.
We have highlighted steps where videos might be helpful. However, you can also complete
the steps simply by following the written instructions. For those procedures that do not
have videos, you should rely on previously developed skills. You may want to review videos
from previous weeks if you feel that you need a refresher on some of the techniques.
Steps to Complete the Analysis
1. CREATE A STATE TABLE: Create a state table, similar to that illustrated in Figure
8.4, in your spreadsheet. Columns A, B, C, and D should contain the state number,
suspended mass, observed sensor voltage, and observed spring-end position data from
your experiment. The remaining columns, E through H, will be completed during the
analysis. Enter the standard value for the acceleration of gravity in cell G1.
2. CONVERT SPRING-END LOCATION DATA TO DISPLACEMENT FROM NEUTRAL POSITION: The neutral position of the spring is defined as the spring-end
location when there is no load applied. In other words, for the case where zero load
is applied, the spring-end displacement from the neutral position is zero. We define
the neutral position as state 1. Enter an equation in Column E to compute the displacement of the spring-end from the neutral position for each applied weight. Express
your displacement in units of [m]. For this exercise, use the sign convention that the
downward deflection is positive.
3. CREATE THE CALIBRATION CURVE: Create a calibration curve that relates voltage to displacement from the neutral axis. Use good graphing practices and fully
document your plot. Your graph will be a plot of displacement d, on the vertical axis
and Sensor Voltage, V on the horizontal axis. Use the line fitting tool to estimate the
slope and intercept for the calibration curve, and enter the appropriate values in cells
D1 and D2 of your spreadsheet. This spreadsheet can be used to analyze data for next
week. An example calibration plot of is presented in Figure 8.5.
4. COMPUTE THE DISPLACING EXTERNAL FORCE: Compute the force applied
by the suspended mass upon the spring, using your knowledge of Newton’s Law of
Gravity. Enter the Displacing Force Fd in Column F, expressing the force in units of
[N ]. For this exercise, use the sign convention that the downward force is positive.
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Figure 8.4: Screen Capture of a State Table for Hooke’s Law Analysis.
Also, consider what is the uncertainty of this force if we assume the uncertainty of the
mass is small enough to be ignored.
VIDEO RESOURCE: Studio 08 Video - Force and Displacement
5. PLOT THE COMPUTED FORCE AND DISPLACEMENT DATA: Using good graphing practices, create a plot of External Force Fd on the vertical axis vs. Displacement
From Neutral Position, d on the horizontal axis. An example plot is presented in Figure
8.6.
6. ESTIMATE THE SPRING CONSTANT: Use the line fitting feature to create a linear
fit of force versus displacement data. Include the equation Fd = Kd and the correlation
coefficient on the plot. An example plot is presented in Figure 8.7. Think about what
the uncertainty for your spring constant estimate is. Since the estimate is from the slope
of the Fd vs.d line, the uncertainty can be determined from the ”Quotient” uncertainty
expression from the equation reference sheet. Details of this will be discussed in Studio.
VIDEO RESOURCE: Studio 08 Video - Spring Constant
7. COMPLETE THE REMAINING ENTRIES IN THE STATE TABLE: Fill in the remaining columns of the state table, Columns G though H, using your knowledge of the
experiment, the simplifying assumptions, and the results computed up to this point.
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Figure 8.5: Calibration Plot for Ultrasonic Transducer.
8. OBSERVATIONS AND ANALYSIS: Write responses to the following questions in your
logbook. Be sure to include a justification for your answer by referring to the data,
plots, and derivations that are contained within your logbook. You may want to crossreference equations from Sections 8.1, 8.3.1 and 8.2.9 in your work.
(a) Record the value of the curve fit coefficients for the spring constant graph. What
is the physical significance of the slope coefficient in the linear curve fit? Use
your engineering judgement to interpret your results, and place your numerical
estimate for the spring constant K in cell G2 of your spreadsheet.
(b) What were the main two objectives of this analysis?
(c) How well do your results reproduce linear elastic behavior for the spring?
(d) What is the range of potential values for spring constant? How does you measure
value compare to the value provided in the lab video for the spring?
9. CONGRATULATIONS! You have just completed the Studio portion for week 8.
10. WRITE THE REPORT: Please refer to section 8.3.3 Report on details for the report
submission. Before leaving Studio, decide on a date and time to meet up with your
team mates to prepare the report. Reports are due Monday by 6 pm.
8.3.3
Report
Please use the same task distribution for writing the report that was outlined in Week
1. This week we have added a conclusion section. This section is to be no more than
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Figure 8.6: Data showing External Force vs. Displacement.
Figure 8.7: Curve Fit correlation for External Force vs. Displacement.
1/2 a page long. The scribe is responsible for compiling both the results section and
the conclusion. However, all team members should contribute ideas to drafting the
conclusion.
Prepare a report to include only the following components:
• TITLE PAGE: Include the title of your experiment, “Hook’s Law”, Team Number, date, authors, with the scribe first, the team member’s role for the week,
and a photograph of each person beginning to initiate their trial, with a label
below each photo providing team member’s name.
• PAGE 1: The heading on this page should read Experimental Set-up. Create
a diagram of the experimental set-up. This week we will include only the diagram
and its caption. Thus, is it important that your diagram clearly communicate the
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set-up, including each key component and where measurements were taken. The
important information to communicate are the variable names, distances, axis
and datums that relate to your measurements and results. It is a good practice
to add a legend that defines any variables or components of the schematic that
are not obvious. At the bottom of the figure include a figure caption, for example
Figure 1. A brief figure caption. Refer to the text for examples.
Note: Figure captions are required for every plot and diagram in the report,
except for the title page. Figure captions are placed below the figures, and are
numbered sequentially beginning with Figure 1 for the first figure in the report.
• PAGE 2: The heading on this page should read Results. Include the table
shown in Table 8.3 summarizing each team member’s results. Remember that any
measured data point or value calculated from measure data has an uncertainty.
At the top of the table, include a table caption, for example Table 1. A brief
figure caption. Refer to the text for examples.
This week we include only tables and plots with no accompanying text. Thus,
it is important that your tables, graphs and captions clearly communicate to the
reader what the data represents.
Note: Table captions are required for every table in the report, except for the
title page. Unlike figure captions, table captions are placed above the tables, and
are numbered sequentially (independent of figure caption numbering) beginning
with Table 1 for the first table in the report.
Each team member should report their value for K and one of their example loading conditions with the corresponding displacement and spring elastic potential
energy. Be sure to include uncertainties for all results.
Team
Member
Name
Member
Member
Member
Member
1
2
3
4
Table 8.3: Summary data from Lab 8.
Estimated
Suspended
Observed
Spring Constant
Mass
Displacement
K ± ✏K
m ± ✏m
d ± ✏d
[N/m]
[kg]
[m]
Spring Elastic
Potential Energy
SE ± ✏SE
[J]
Name
Name
Name
Name
• PAGES 3: No heading is needed on this page, since it is a continuation of the
Results section. On a single page, include External Force vs. Displacement plots
for each member of the team. Each plot should show the linear curves fit with
the correlation coefficient. Format the plot according to the guidelines shown
in previous chapters. Arrange the plots so that they are easily compared on to
another.
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• PAGES 4: The heading on this page should read Conclusions. Here you will
state the major conclusions that can be drawn from this analysis. In other words,
you will qualitatively and quantitatively answer the questions posed by the experiment. Consider the following guiding questions when preparing your conclusion.
Do any of your results violate Newton’s Laws or the Hooke’s Law, within uncertainty limits? What are the most significant contributors to uncertainty, and
how would you mitigate them? Finally, comment on whether your experimental
results support the Hooke’s Law within reasonable uncertainty?
Your conclusion should be NO LONGER than 1/2 a page when typed in 12 pt
font.
• The final report should be collated into one document with page numbers and a
consistent formatting style for sections, subsections and captions. Before uploading the file, you must convert it to a pdf. Non-pdf version files may not appear
the same in di↵erent viewers. Be sure to check the pdf file to make sure it appears
as you intend.
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8.4
Recitation
Recitation this week will focus on problem solving. Please come prepared, with your
attempts at the homework problem already in your logbooks.
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8.5
Homework Problems
Complete all assigned homework problems in your logbook.
8.5.1 Determine the spring constant, K[N/m], based on the following set of data generated by incrementally adding mass to a suspended spring.
Mass on Spring [grams] Spring Extension [mm]
5
15
10
30
15
45
20
60
25
75
30
90
35
100
40
120
8.5.2 Consider a mass, m, is dropped a vertical distance, z , onto a compression spring
having spring constant K. Solve SYMBOLICALLY for the distance x, that the
spring is compressed.
8.5.3 An amusement park ride similar to one at Darien Lake called “The Slingshot”
launches two riders hundreds of feet into the air by utilizing the energy stored in
compressed springs. Assuming the e↵ective spring rate for the ride is 100, 000[N/m]
and the spring is compressed 2.5[m], calculate the approximate launch velocity
of the riders if they each have a mass of 80[kg] and the structure holding the
riders has a mass of 100[kg].
8.5.4 A tensile force of 50[lbs] causes a spring to extend from 1[in] to 5[in]. Calculate
the spring constant.
8.5.5 A compressive force of 35[N ] is applied to a spring which causes its length to
compress from 300[mm] to 250[mm]. Calculate the spring constant.
8.5.6 Given a spring with a constant of 3, 000[lb/f t], how much force is required to
extend the spring by:
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A) 6 inches
B) 18 inches
C) 1.75 feet
D) 3 yards
8.5.7 Given a spring with an unloaded length of 30[cm] and a spring constant of
3, 550[N/m], what is the new length of the spring if the following force is applied to the spring?
A) 250 [N ] compressive force
C) 500 [N ] compressive force
B) 250 [N ] tensile force
D) 500 [N ] tensile force
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