Chapter 8

Hooke’s Law for Linear ElasticMaterials

8.1 Lecture - Force, Displacement, and Elastic Poten-tial Energy

In this lesson, we will study the engineering application core concept of experimental “mod-eling.” 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↵erentmaterials together to form alloys or composites. One type of material that we will use in animmense number of engineering applications includes a certain class of solids called “linearelastic 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 behaviorof 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 asstatics and strength of materials. In this class, we will learn the fundamental concepts andcharacteristics of linear elastic materials through the study of a simple coil spring.

In your secondary school chemistry and physics courses, you learned about another im-portant constitutive equation called the ideal gas law. The ideal gas law is a useful equationthat describes how certain kinds of gaseous materials will behave under varying environmen-tal conditions of temperature and pressure. We learned that the ideal gas law does not applyto all gases or ranges of environmental conditions. In the same way, Hooke’s Law helps usto describe the idealized response of certain solids, and the relationship that these solidsexhibit in the form of response to forces and deflections. Hooke’s Law does not apply to allsolids, 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 situationsinvolving 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 thespring constant K of the spring, and the change in energy stored within the spring as a resultof the work done by mass m upon the spring as the spring S moves from its initial positioni 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 = ? [N/m] ↵ spring constant of spring, S (8.2)

�SEi!f = ? [J ] ↵ change in elastic potential energy of S (8.3)

8.1.2 Assume

We will make several familiar assumptions for this analysis.

Identify Assumptions

The following assumptions may be employed during the analysis.

mspring = 0 [N ] � (8.4)

g = GmE

R2

E

⇡ 9.80665 [m/s2] � (8.5)

Qi!f = 0 for i ! f [J ] � (8.6)

We can make several assumptions about the initial and final displacements as:

z1i = 0 [m] � (8.7)

x1i = 0 [m] � (8.8)

x1f

= 0 [m] � (8.9)

z0i = z

0f= x

0i = x0f

= 0 [m] � (8.10)

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and we can also make assumptions about the initial and final components of velocity:

Vzi = 0 [m/s] � (8.11)

Vzf = 0 [m/s] � (8.12)

Vxi = 0 [m/s] � (8.13)

Vxf= 0 [m/s] � (8.14)

We will assume that the spring is made from a linear elastic material, and assume thatHooke’s Law is a valid material model to describe the behavior of the spring:

�!F

1

= K�!x1

[N ] � external force (8.15)�!F R

1

= �K�!x1

[N ] � restorative force (8.16)

Hooke’s Law is a vector equation. As we have seen in past problems, it is often convenient toanalyze vector problems by looking at each component individually. When analyzing helicalspring problems, it is a common practice to employ a local coordinate system whose axis isparallel 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, whichcauses a small displacement due to its own weight. However, since we will choose thecoordinate system origin from this location, the mass of the spring may be convenientlyneglected.

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 systemsand diagrams that we create. Equation 8.8 and 8.9 say that we are neglecting the lateraldisplacement of the bottom of the spring. Equation 8.10 indicates that the upper end of thespring is fixed to the mounting bracket, and does not move.

Equations 8.11 through 8.14 state the the vertical and horizontal components of velocityfor the spring are zero before the mass is placed on the end of the spring, and that we waituntil the spring and mass reach their equilibrium position before we consider the system tobe 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 propertiesof 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 toextend is shown in Figure 8.1. We place the coordinate system in alignment with the springaxis, with the origin located at the neutral position of the spring, prior to placing the massupon it. This provides a convenient way to express the initial condition of the system withno horizontal or vertical displacement. The first state i is considered to be the instant oftime just before the mass is placed upon the spring, and the final state f occurs after thespring 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 massexerts a force upon the spring due to the action of gravity upon the mass. As the massextends the spring, the mass exerts a force upon the spring, and the spring exerts an equaland opposite force upon the mass. The spring is supported in place by the reaction forcefrom the mounting bracket, which ultimately is transmitted through the lab apparatus tothe 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 freebody 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 shownbelow. Each row in the state table corresponds to one unique position of the spring as itis loaded by a unique mass, The symbols � and ↵ are used to indicate known and desiredinformation, respectively. The top header row of the state table includes the name of the

State Applied Sensor Spring Displacement Applied Spring Spring Spring Spring Spring Spring

Mass Volts End From Force Mass SE PE KE Tip Momentum

me V Location Neutral, d Fd ms |V | |p|[kg] [V ] [inch] [m] [N ] [kg] [J] [J] [J] [m/s] [kgm/s]� � � ↵ ↵ � ↵ � � � �

1 0.0 N/A 0.0 0 0 0 0 0

2 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 entryin that column of the state table. The third row of header indicates that these values areunknown at the beginning of the problem, and must be determined during the laboratoryexperiment and subsequent analysis. In this case, several values have been filled in, as adirect consequence of the simplifying assumptions presented previously. Please review eachcompleted entry in the state table, to be certain that you understand why these values havebeen entered. We define state 1 as being the neutral condition of the spring, when suspendedfrom the hanger, and with no external mass applied. In this case, we take do not measure asensor 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 downwarddirection, 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, wehave our familiar set of Newton’s laws to draw upon, and we now include the work andenergy theorem as well.

If :�!F Net = 0 Then : �!a = 0 Newton’s 1st Law (8.17)

�!F Net = lim

�t!0

��!p�t

= lim�t!0

�(m�!V )

�t

=d(m

�!V )

dt

=d�!pdt

Newton’s 2nd Law (8.18)

�!F Action = ��!

F Reaction Newton’s 3rd Law (8.19)�!F g = g ·m # Newton’s Law of Gravity near Earth (8.20)

E2

� E1

= Q1!2

�W1!2

Work Energy Theorem (8.21)

Finally, let’s recall the definition of work

W1!2

=

Z z2

z1

�!F Net · dz (8.22)

Hooke’s Law

Hooke’s Law states that strain is directly related to stress. Unfortunately, we will not belearning the formal definitions of stress and strain until our later courses in statics andstrength 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 themagnitude of the displacement, and opposite in direction to this displacement. For a linearelastic material of uniform cross sectional area, Hooke’s Law may be stated mathematicallyas

�!F

1

= K�!x1

external force (8.23)�!F R

1

= �K�!x1

restorative force (8.24)

[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 thespring 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 �!x1

, thenthe spring will exert a restorative force

�!F R

1

(internal to the spring) that will attempt toreturn the spring to its original position. That is, the restorative force resists the action ofthe 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 endpoint of the relaxed spring as the origin of our coordinate system. If we “pull” on the springand place it in tension, then the spring will extend downwards in response to the forceapplied. On the other hand, if we “push” on the spring and place it in compression, thenthe spring will compress upwards in response to the force applied. This is consistent withHooke’s Law as given by 8.23. If we then release the external force from the spring, therestorative 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 com-pression.

There is a lot happening here. What is happening inside the spring as we apply thisforce? Why does the spring reach some displacement, and then stop moving? If there isan external force applied to the end of spring, why does it not keep on moving? Let’s usethis new constitutive relationship, Hooke’s Law, to fully understand the response of a springwhen it is subjected to an external force. In particular, we want to learn how to estimate thevalue 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-energytheorem during this action.

Newton’s laws, the work energy theorem, and Hooke’s Law for modeling the springmaterial 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 diagramfor 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. Thisrepresents the initial condition i. Now, we manually pick the mass m up with our handsand gently attach it to the bottom of the spring. Because the mass experiences the forceof gravity, it will extend the spring some distance. Eventually, we should observe that thespring comes to rest at a new equilibrium point. We call this new equilibrium point the finalstate f . At this final state f , Newton’s First Law as given by Equation 8.19 tells us that thenet force should be zero, since the observed acceleration is zero. Let us study the final statef in more detail.

Let’s apply Newton’s First Law, Equation 8.17 to the spring:

�!F Net = 0 (8.25)�!F Net = �Fmk̂ +Rk̂ (8.26)

[kgm/s2] = [kg][m/s2] + [kgm/s2] basic units

Now that we have a full understanding of the initial state i and the final state f , we canevaluate the process as the system moves from state i to state f . The work done by the massupon the spring from state i to state f is the integral of the force along the displacement.From the definition of work,

Wi!f =

Z zf

zi

�!F Net · dz (8.27)

The force exerted by the mass is�!F m = �mgk̂, by virtue of Newton’s Law of Gravity,

Equation 8.20. While the system is moving and before the final state is achieved, thedisplacement of the spring is not su�cient to balance the force due to the weight of themass. Assumptions 8.15 and 8.16 allow us to employ Hooke’s Law to tell us that, as thespring is being extended in the negative z direction, it opposes the motion of the masshanging from it with a restorative force in the positive z direction:

�!F S on m = +Kzk̂ 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 thenegative z direction:

�!F m on S = �mgk̂ force of mass on spring (8.29)

Newton’s Second Law, Equation 8.18 clearly tells us that the spring/mass system will con-tinue to move until the two forces come into balance. When the spring achieves its finalequilibrium position f , with the bottom of the spring deflected to length z

1fwe can state

by Newton’s First Law that

+Kz1fk̂ �mgk̂ = 0 or (8.30)

Kz1f

= mg (8.31)

[N/m][m] = [kg][m/s2] derived units

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We will look at that “transient response” of the system in more detail during our nextlesson. For this week, however, let’s use a work energy theorem analysis of the spring andmass system to gain insight into the system operation.

The work done on the spring by the mass can be found from the following:

Wm on Si!f =

Z zf

zi

(�Kzk̂) · (�dzk̂) (8.32)

=

Z zf

zi

+(Kz)dz (8.33)

= +

Z zf

zi

(Kz)dz (8.34)

= +K

2z2��zfzi

(8.35)

Wm on Si!f = +

K

2

�z2f � z2i

�(8.36)

[J ] = [N/m]�[m]2 � [m]2

�derived units

Note that the reaction force at the fixed mounting bracket is not included in the work doneon the spring, because that applied force does not give rise to a displacement in z. Thisreaction force keeps the spring in a static position, but does not exert work (�z

0

= 0).Conversely, let’s study the work done by the spring upon the mass.

W S on mi!f =

Z zf

zi

�!F S on m · dz (8.37)

Substitute Equation 8.28 into Equation 8.37

W S on mi!f =

Z zf

zi

(+Kzk̂) · (�dzk̂) (8.38)

=

Z zf

zi

�(Kz)dz (8.39)

= �Z zf

zi

(Kz)dz (8.40)

W S on mi!f = �K

2z2��zfzi

(8.41)

W S on mi!f = �K

2

�z2f � z2i

�(8.42)

[J ] = [N/m]�[m]2 � [m]2

�derived units

It is interesting to note that the work done by the mass upon the spring, as given by Equation8.36 is the negative of the work done by the spring upon the mass, as given by Equation8.42.

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Now, let’s recall and simplify the work-energy theorem given by Equation 8.21, as itapplies to the spring.

Ef � Ei = Qi!f �Wi!f or (8.43)

�Ei!f = Qi!f �Wi!f (8.44)

The energy stored in the spring may consist of three components: elastic potential energy,gravitational potential energy, and kinetic energy:

�SEi!f +

!0 by Eq.8.4z }| {�PEi!f +

!0 by Eq.8.4z }| {�KEi!f =

!0 by Eq.8.6z }| {Qi!f �Wi!f or (8.45)

�SEi!f = �Wi!f (8.46)

The mass m is the only object upon which the spring does work, since it is the only objectthat undergoes a displacement as a result of the force exerted by the spring. Thus, becauseW S

i!f = W S on mi!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

�SEi!f = �✓�K

2

�z2f � z2i

�◆(8.47)

�SEi!f ⌘ K

2

�z2f � z2i

�(8.48)

[J ] = [N/m][m]2 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 withinthe 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, wecannot completely solve the problem. We will need experimental data in order to estimatethe value of the spring constant K. Hooke’s Law is an empirical relationship, which meansthat it is based on experimental observations. The value of K cannot be determined fromfirst 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

�SEi!f ⌘ K

2

�z2f � z2i

�(8.50)

[↵] = [�] ([↵]� [�]) inventory

and recall Equation 8.36

Wm on Si!f = +

K

2

�z2f � z2i

�(8.51)

[↵] = [�] ([↵]� [�]) inventory

Close inspection reveals that these two expressions are the negative of one another – thework 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 finitedisplacement when loaded with a deadweight, and come to rest. As the mass associated withthe dead weight is increased, we expect the spring displacement to increase, at least untilthe 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 thelab group should conduct an independent trial, using a unique mass for the deadweightand analyze a unique data set. When the results of all students’ trials are compiled on asingle graph, the team should compare their group estimate for the spring constant to theirindividual 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 aspring in response to a hanging dead weight that is suspended from the spring. You will alsocorrelate the manual displacement measurements to the measured output from the ultrasonictransducer. 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, themeasurements from the ultrasonic transducer will be used to calibrate the voltage output inpreparation 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 avariety of units.

Quantity Basic units Derived units

Length [m] [m]Mass [kg] [kg]Time [s] [s]

Velocity [m/s] [m/s]Force [kgm/s

2] [N ]Energy [kgm2

/s

2] [J ] or [N ][m]Spring Constant [kg/s2] [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 properusage. Review any previous materials related to the proper use of the data acquisitionsoftware 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 objectivesof this laboratory experiment.

Energy Displacement ForceKinetic Energy Velocity WorkGravitational Potential Energy Speed Spring ConstantElastic 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 YouTubefor this week’s exercise. Watch all of the available videos, and complete the online lab quizfor the week. The videos are your best reference for the specific tasks and procedures tofollow 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 ofhardware 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 unloadedspring 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 theinstructional 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 techniquesrequired to determine and document basic uncertainties associated with all measurements.When making the measurements in lab, be sure to record the uncertainties or instrumentleast counts associated with all measurements that are made. Also, make note of any otherpossible 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 withinone or both of the drawers attached to the lab bench or available from the laboratory instruc-tor. You are expected to bring all other necessary materials, particularly your logbook and aflash drive for storing electronic data as appropriate. You are to follow the general specifica-tions 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 ofthe 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 bereturned 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 theend of the spring and recording the corresponding displacements and steady state voltageoutputs from the ultrasonic sensor. Each group member is expected to use his/her own rangeof 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 setup the apparatus to make measurements, and for actually collecting data with the variousdevices and software interfaces. More generally, you should always observe the followinggeneral 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 asso-ciated 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 notesand 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 equipmentused, the actual data collected and/or notes related to the data collected (if doneelectronically for instance), and any other information relevant to the reporting andanalysis of the data and understanding of the exercise itself.

5. All students should create and complete a table indicating the sta�ng plan for theweek (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 exer-cise, 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 se-rial 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 specificidentifying 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 informationas possible in case someone was to try to reproduce or verify what you did. Refer toTable 1.3.

7. For the Lab Manager only: create a key sign-out/sign-in table for obtaining thekey 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/herown data for the exercise, as well as any specific notes related to the data collectionactivities. In those cases where data collection is done electronically, there may not beany data tables required.

9. Many of the laboratory exercises will require the use of a specific software interfacefor measurements and/or control. In all cases, these will be made available on themyCourses site unless stated otherwise.

10. The Scribe (or a designated alternative) should take a photo of each group memberperforming some aspect of the laboratory exercise for inclusion in the lab reportthat will be generated during the studio session. Refer to the example lab report formore details.

11. Record all relevant data and observations in your logbook, even those that may nothave been explicitly requested or indicated by the textbook or videos. If in doubtabout 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 beenreturned to the proper place. Log out of the computer, and straighten up everythingon the lab bench as you found it. Put the lab stools back under the bench and out ofthe way.

13. Prepare for the upcoming studio session for the week by carefully read and understandSection 4.3 of the textbook, and complete the Studio pre-work prior to your arrival atStudio.

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8.3 Studio - Spring Constant and Elastic Potential En-ergy

This week in Studio you complete an analysis of Hook’s Law. Using the data acquired inlab, you will estimate the spring constant for a small spring. In addition, you will apply yourknowledge of sensor calibration to derive a calibration equation for the ultrasonic sensorwhich can be applied again in next weeks analysis. You will create a state table whichincludes the spring energy at each state of the spring under various applied loads. Theseresults 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 thatyou will need to complete the Studio. Section 8.2.9 Measurement Uncertainty describes theprocess for analyzing the experimental errors. There will be uncertainty equation derivationsthat you need to complete prior to arriving at studio and these are listed in the MeasurementUncertainty section.

Record all observations and notes about your studio procedures inyour logbook.

8.3.1 Calculation and Interpretation of Results

The following equations may be helpful in the context of the Studio Procedure.

E = KE + PE + SE total energy of a state (8.52)

[J ] = [J ] + [J ] + [J ] units validation

SE =1

2Kd2 definition of SE (8.53)

[J ] = [N/m][m]2 units validation

FS = Kd Hooke’s Law (8.54)

[N ] = [N/m][m] units validation

8.3.2 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 acorresponding schematic that clearly identifies where each measurement was made in sym-bolic notation.

In addition, each of you will complete several steps of the Studio exercise. This will allowmore 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 correspondingweek. 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 toyour individual drop-box, before coming to coming to Studio. You can work on theremaining portions of the exercise during Studio. All steps with the exception of the reportare 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 completethe steps simply by following the written instructions. For those procedures that do nothave videos, you should rely on previously developed skills. You may want to review videosfrom 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 Figure8.4, 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 fromyour experiment. The remaining columns, E through H, will be completed during theanalysis. Enter the standard value for the acceleration of gravity in cell G1.

2. CONVERT SPRING-END LOCATION DATA TO DISPLACEMENT FROM NEU-TRAL POSITION: The neutral position of the spring is defined as the spring-endlocation when there is no load applied. In other words, for the case where zero loadis applied, the spring-end displacement from the neutral position is zero. We definethe neutral position as state 1. Enter an equation in Column E to compute the dis-placement of the spring-end from the neutral position for each applied weight. Expressyour displacement in units of [m]. For this exercise, use the sign convention that thedownward deflection is positive.

3. CREATE THE CALIBRATION CURVE: Create a calibration curve that relates volt-age to displacement from the neutral axis. Use good graphing practices and fullydocument your plot. Your graph will be a plot of displacement d, on the vertical axisand Sensor Voltage, V on the horizontal axis. Use the line fitting tool to estimate theslope and intercept for the calibration curve, and enter the appropriate values in cellsD1 and D2 of your spreadsheet. This spreadsheet can be used to analyze data for nextweek. An example calibration plot of is presented in Figure 8.5.

4. COMPUTE THE DISPLACING EXTERNAL FORCE: Compute the force appliedby the suspended mass upon the spring, using your knowledge of Newton’s Law ofGravity. 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 themass is small enough to be ignored.

VIDEO RESOURCE: Studio 08 Video - Force and Displacement

5. PLOT THE COMPUTED FORCE AND DISPLACEMENT DATA: Using good graph-ing practices, create a plot of External Force Fd on the vertical axis vs. DisplacementFrom Neutral Position, d on the horizontal axis. An example plot is presented in Figure8.6.

6. ESTIMATE THE SPRING CONSTANT: Use the line fitting feature to create a linearfit of force versus displacement data. Include the equation Fd = Kd and the correlationcoe�cient on the plot. An example plot is presented in Figure 8.7. Think about whatthe uncertainty for your spring constant estimate is. Since the estimate is from the slopeof the Fdvs.d line, the uncertainty can be determined from the ”Quotient” uncertaintyexpression 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 re-maining columns of the state table, Columns G though H, using your knowledge of theexperiment, 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 yourlogbook. 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 cross-reference equations from Sections 8.1, 8.3.1 and 8.2.9 in your work.

(a) Record the value of the curve fit coe�cients for the spring constant graph. Whatis the physical significance of the slope coe�cient in the linear curve fit? Useyour engineering judgement to interpret your results, and place your numericalestimate 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 measurevalue 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 reportsubmission. Before leaving Studio, decide on a date and time to meet up with yourteam 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 Week1. 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 andthe conclusion. However, all team members should contribute ideas to drafting theconclusion.

Prepare a report to include only the following components:

• TITLE PAGE: Include the title of your experiment, “Hook’s Law”, Team Num-ber, 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 labelbelow each photo providing team member’s name.

• PAGE 1: The heading on this page should read Experimental Set-up. Createa diagram of the experimental set-up. This week we will include only the diagramand 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. Theimportant information to communicate are the variable names, distances, axisand datums that relate to your measurements and results. It is a good practiceto add a legend that defines any variables or components of the schematic thatare not obvious. At the bottom of the figure include a figure caption, for exampleFigure 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 arenumbered sequentially beginning with Figure 1 for the first figure in the report.

• PAGE 2: The heading on this page should read Results. Include the tableshown in Table 8.3 summarizing each team member’s results. Remember that anymeasured 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 brieffigure 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 thereader what the data represents.

Note: Table captions are required for every table in the report, except for thetitle page. Unlike figure captions, table captions are placed above the tables, andare numbered sequentially (independent of figure caption numbering) beginningwith Table 1 for the first table in the report.

Each team member should report their value for K and one of their example load-ing conditions with the corresponding displacement and spring elastic potentialenergy. Be sure to include uncertainties for all results.

Table 8.3: Summary data from Lab 8.Team Estimated Suspended Observed Spring Elastic

Member Spring Constant Mass Displacement Potential EnergyName K ± ✏K m± ✏m d± ✏d SE ± ✏SE

[N/m] [kg] [m] [J ]Member 1 NameMember 2 NameMember 3 NameMember 4 Name

• PAGES 3: No heading is needed on this page, since it is a continuation of theResults section. On a single page, include External Force vs. Displacement plotsfor each member of the team. Each plot should show the linear curves fit withthe correlation coe�cient. Format the plot according to the guidelines shownin previous chapters. Arrange the plots so that they are easily compared on toanother.

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• PAGES 4: The heading on this page should read Conclusions. Here you willstate the major conclusions that can be drawn from this analysis. In other words,you will qualitatively and quantitatively answer the questions posed by the exper-iment. Consider the following guiding questions when preparing your conclusion.Do any of your results violate Newton’s Laws or the Hooke’s Law, within un-certainty limits? What are the most significant contributors to uncertainty, andhow would you mitigate them? Finally, comment on whether your experimentalresults support the Hooke’s Law within reasonable uncertainty?

Your conclusion should be NO LONGER than 1/2 a page when typed in 12 ptfont.

• The final report should be collated into one document with page numbers and aconsistent formatting style for sections, subsections and captions. Before upload-ing the file, you must convert it to a pdf. Non-pdf version files may not appearthe same in di↵erent viewers. Be sure to check the pdf file to make sure it appearsas you intend.

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8.4 Recitation

Recitation this week will focus on problem solving. Please come prepared, with yourattempts 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 gen-erated by incrementally adding mass to a suspended spring.Mass on Spring [grams] Spring Extension [mm]

5 1510 3015 4520 6025 7530 9035 10040 120

8.5.2 Consider a mass, m, is dropped a vertical distance, z , onto a compression springhaving spring constant K. Solve SYMBOLICALLY for the distance x, that thespring 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 incompressed 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 velocityof the riders if they each have a mass of 80[kg] and the structure holding theriders 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]. Calculatethe spring constant.

8.5.5 A compressive force of 35[N ] is applied to a spring which causes its length tocompress from 300[mm] to 250[mm]. Calculate the spring constant.

8.5.6 Given a spring with a constant of 3, 000[lb/ft], how much force is required toextend 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 of3, 550[N/m], what is the new length of the spring if the following force is ap-plied to the spring?

A) 250 [N ] compressive force C) 500 [N ] compressive forceB) 250 [N ] tensile force D) 500 [N ] tensile force

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