Experiment Hooke's Law - Mountain ... - Mountain View · PDF fileDiscussion and Review Hooke’s law is named after the seventeenth century physicist Robert Hooke and relates the

Hooke’s LawPeter Jeschofnig, Ph.D. Version 42-0264-00-01

Review the safety materials and wear goggles when working with chemicals. Read the entire exercise before you begin. Take time to organize the materials you will need and set aside a safe work space in which to complete the exercise.

Experiment Summary:

Students will investigate Hooke’s law and determine the spring constant for two springs and a rubber band. Students will stretch two different springs and a rubber band, while measuring both the distance elongated and the force required to extend the springs. From this data, students will calculate the elastic potential energy in joules and the spring constant. Students will then compare the spring data to the rubber band.

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EXPERIMENT

Objectives ● To investigate Hooke’s law and to determine the spring constant for two springs and a rubber

band

Time Allocation: 1–2 hours

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Experiment Hooke’s Law

Materials

MATERIALS FROM LABEL OR BOX/BAG QTY ITEM DESCRIPTION

Student Provides 1 Small rubber band 1 Computer and spreadsheet software

From LabPaq 1 Scale-Spring-500-g 1 Tape measure, 1.5-m 1 Springs, 2 sizes-PK

Note: The packaging and/or materials in this LabPaq may differ slightly from that which is listed above. For an exact listing of materials, refer to the Contents List form included in the LabPaq.



Discussion and Review Hooke’s law is named after the seventeenth century physicist Robert Hooke and relates the force pulling or pushing on a spring (or other elastic material) to the amount the spring stretches or compresses. The force exerted by a spring to restore itself to its natural length is referred to as the restoring force. When a spring is stretched, as in this experiment, the restoring force is exerted inward; if a spring is compressed, the restoring force is exerted outward. Mathematically, the restoring force of a spring is expressed as:

F kx= −

where F = restoring force k = proportionality constant, called the spring constant x = distance the spring has been stretched or compressedThe negative sign indicates that the restoring force acts in the direction opposite of the displacement direction.

Depending on material, length, diameter, and number of coils, each spring has its unique spring constant. The greater the spring constant, the stiffer the spring (the more difficult it is to stretch it or compress it).

The elastic limit is the maximum extension to which a spring can be stretched without permanent deformation and still return to its original shape. If a spring is stretched beyond its elastic limit, it will not return to its original shape and will remain deformed.

On a force versus elongation graph, the elastic limit will show up as the point where the slope of the line changes or where the straight-line portion of the graph ends.

Not all elastic materials obey Hooke’s law. For example, rubber is generally considered a hyperelastic, neo-Hookean material because its elastic behavior varies with loading rate and temperature. Under simple experimental conditions, rubber bands seem to follow Hooke’s law for a limited range. Depending on the latex and rubber, a rubber band may not return to its exact original shape after stretching.

Hooke’s law can be used in two ways. The first is to find the force exerted by a spring. The second is to derive the period of oscillating motion for a mass connected to a spring. The two related equations are: Equation 1: spring F k x= − ∆

Equation 2: 2 mTk

π=

Where T is the time period of one oscillation cycle (a complete up and down movement of the weighted spring) and m is the mass on the end of the spring.



Commonly, a Hooke’s law experiment is conducted by adding increasing masses to a spring and recording the cumulative stretch (elongation) of the spring.

This experiment will use a spring scale in place of calibrated weights to increase the force on a spring. However, this method will add an additional step to the experiment. To cancel out the effect of the internal spring of the spring scale, you need to measure the elongation of the spring for each force increment by recording the position of the top and bottom of the spring. If you only record the bottom position of the spring, you would measure the combined spring constant of the spring and the internal spring of the spring scale.

Procedure Ensure that you do not stretch a spring beyond its capacity to recover by first performing a stretch test to estimate the spring’s full-elongation capacity:

● Hold the spring at both ends and pull it apart with only moderate force, not with so much force that you permanently distort it.

● Estimate how many centimeters (cm) you were able to stretch the spring and mentally divide that stretch by the number of experimental increments you wish to test.

● For each test step, you will add only sufficient weight or force to increase the stretch by one increment. For example, if you can stretch a spring 10 cm and need 10 measurements, each experimental increment should be 1 cm and you should add enough force at each step to stretch the spring by only 1 cm.

1. Suspend the spring scale from a wall hook, doorknob, or something similar with a flat surface behind it to which you can tape a meter tape.

2. Perform a stretch test as described above on the first spring and then suspend it from the scale as shown at right.

3. Position and affix the meter tape along the side of the spring. The location of the beginning of the tape is not important as you will record the top and bottom measurement for each force addition.

4. Hold the bottom hook of the spring and gently pull straight down with sufficient force to stretch the spring 1/10 of its elongation capacity. Now measure and record the position of the top and the bottom of the spring. The difference will be the exact elongation of the spring. Also, record the force required to create that elongation.

Continue to stretch the spring and record data in steps, which add sufficient force to achieve an additional 1/10 elongation. You will record 10 sets of force and elongation data. The elongations recorded at each step are already cumulative elongations.

Figure 1.



5. Repeat the above steps with a second spring of a different stiffness.

6. Finally repeat the above procedures using a small rubber band. Then continue adding weight until the rubber band breaks or is on the verge of breaking or nearly stops stretching with added force.

Optional exerciseAs an optional exercise, you can determine the spring constant k of the internal spring of the spring scale.

Calculations and Analysis

1. Before beginning, set up data tables similar to Data Tables 1 and 2.

Data Table 1

Force (N) Top position of spring, cm

Bottom position of spring, cm

Elongation, cmBottom reading – top

readingData Point 1Data Point 2Data Point 3Etc.

Data Table 2

Force (N) Accumulated (cm) Elongation (stretch)

Accumulated (m) Elongation (stretch)

Elastic PE(Joules)

Data Point 1Data Point 2Data Point 3Etc.

2. For each data row in each of your tables calculate: Elastic PE = 212

kx

3. For each spring and the rubber band, plot the accumulated elongation (x-axis) versus the applied force (y-axis) on a computer spreadsheet.

4. Find the spring constant for the springs in Newtons per meter from the slope of each graph. (Refer to the Excel tutorial in the Introduction section.) Spring constant, Fk

x= , where F is in

Newton and x is in meters. Therefore, the units are N/m.

5. Find the “spring” constant for the rubber band from the slope of the curve using the linear portion of the graph.

Sample graph. Rubber band.





Force vs Cumulative Elongation (rubber band)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Cumulative elongation, m

Forc

e, N

Series1

QuestionsA. How does the relative stiffness of a spring relate to its spring constant?

B. How does PE change relative to the stretch of the spring?

C. Indicate on your graph for the rubber band where the linear behavior stops. What does this mean?

D. Which is stronger in the region where Hooke’s law is obeyed, the spring or the rubber band? Explain.

E. Explain what happens to the “spring constant” of the rubber band for the nonlinear part of your curve.

Hooke’s LawPeter Jeschofnig, Ph.D. Version 42-0264-00-01

Lab Report AssistantThis document is not meant to be a substitute for a formal laboratory report. The Lab Report Assistant is simply a summary of the experiment’s questions, diagrams if needed, and data tables that should be addressed in a formal lab report. The intent is to facilitate students’ writing of lab reports by providing this information in an editable file which can be sent to an instructor.

Observations

Data Table 1

Force (N) Top position of spring, cm

Bottom position of spring, cm

Elongation, cmBottom reading – top

readingData Point 1Data Point 2Data Point 3Data Point 4Data Point 5Data Point 6Data Point 7Data Point 8Data Point 9Data Point 10



Data Table 2

Force (N) Accumulated (cm) Elongation (stretch)

Accumulated (m) Elongation (stretch)

Elastic PE(Joules)

Data Point 1Data Point 2Data Point 3Data Point 4Data Point 5Data Point 6Data Point 7Data Point 8Data Point 9Data Point 10



Calculations and Analysis1. For each data row in each of your tables calculate: Elastic PE = 21

2kx

2. For each spring and the rubber band, plot the accumulated elongation (x-axis) versus the applied force (y-axis) on a computer spreadsheet.

3. Find the spring constant for the springs in Newton/meters from the slope of each graph. (Refer to the Excel tutorial in the Introduction section.) Spring constant, Fk

x= , where F is in

Newton and x is in meters. Therefore, the units are N/m.

4. Find the “spring” constant for the rubber band from the slope of the curve using the linear portion of the graph.



A. How does the relative stiffness of a spring relate to its spring constant?

B. How does PE change relative to the stretch of the spring?

C. Indicate on your graph for the rubber band where the linear behavior stops. What does this mean?

D. Which is stronger in the region where Hooke’s law is obeyed, the spring or the rubber band? Explain.

Sample graph. Rubber band.

Force vs Cumulative Elongation (rubber band)

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Cumulative elongation, m

Forc

e, N

Series1



Questions

E. Explain what happens to the “spring constant” of the rubber band for the nonlinear part of your curve.



Documents

Experiment Hooke's Law - Mountain ... - Mountain View · PDF fileDiscussion and Review Hooke’s law is named after the seventeenth century physicist Robert Hooke and relates the