# Calculating the elastic potential energy from a force-extension investigation

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LukeWatson4590

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#1

Hello, I have a question that I am struggling to answer. I have attempted all of the sections but would greatly appreciate any help or guidance in areas where I am clearly confused.

A student demonstrates an experiment to analyse the force-extension characteristics of a rubber band.

1.Describe the equipment the student could use to perform the experiment.

The student should use a rubber band, g-clamp, a retort stand, boss and clamp, a mass hanger, 100g masses and a metre rule.

The rubber band should be positioned to hang freely from the retort stand, held in place by a g-clamp to the laboratory bench. Measure the length of the rubber band using a ruler. At the base of the rubber band a mass holder should be looped across to hang below and a metre rule should be used to measure the new length of the band. The student should gradually increase the load on the band, recording the change in the length of the rubber band.

The student may wish to repeat the investigation with a selection of rubber bands.

2. Below are the students measurements, Use this information to plot a graph of force versus extension.

I have done so and will attach shortly.

3. The student wants to calculate the elastic potential energy stored by using the equation EPE = (1/2)kx^2. Why would she get the wrong answer? Would it be an underestimate or an overestimate?

This is where the greatest confusion lies. Since, I know that Eel = 1⁄2FΔx. Since Hooke’s law states ΔF = kΔx, substituting the expression for ΔF into the equation for the elastic strain energy equation produces the formula ΔEel = 1⁄2k(Δx)^2.

On the force extension graph, the area between the line on the force-extension graph and the extension axis will represent the work done, and thus the elastic strain energy. This is only true for linear relationships though, like Hooke's Law, as the area under the line is a triangle and the formula for the area of a triangle, 1/2 bh, is equivalent to the formula for the elastic strain energy, Eel=1/2Fx

Therefore, as a rubber bands follows a hysteresis loop and has a non-linear relationship, the work done (and hence the Eel) can be found from the area under the line. However, you cannot use the formula 1/2Fx to calculate this as this is the area of a triangle, where the area underneath the hysteresis loop of the rubber band is not a triangle. Usually, if a non-linear force-extension graph has a curved line, finding the area may require one to estimate or count the squares of graph paper under the line. In such an instance, one is further required to multiply the number of squares by the elastic strain energy value (Fx) for each square. The student wishes to calculate the elastic potential energy stored, which is equal to the area under the loading curve of the force-extension graph. I am not sure whether this will be an under or overestimate though?

4. Describe the energy transfer as the force is increased, then decreased. How is this represented in your graph?

I am not too sure how to answer here. Would it be that energyis absorbed and stored when a the force is increased and the rubber band is stretched but the energy is released when the force is decreased. Rubber absorbs more energy during loading than it releases in unloading. This difference is represented by the area of the hysteresis loop, between the curves of loading and unloading.

The effect of hysteresis in rubber is to transfer energy to its molecules, resulting in heating.

Could I elaborate or improve upon my answer to be more concise ?

I greatly appreciate any help 👍

A student demonstrates an experiment to analyse the force-extension characteristics of a rubber band.

1.Describe the equipment the student could use to perform the experiment.

The student should use a rubber band, g-clamp, a retort stand, boss and clamp, a mass hanger, 100g masses and a metre rule.

The rubber band should be positioned to hang freely from the retort stand, held in place by a g-clamp to the laboratory bench. Measure the length of the rubber band using a ruler. At the base of the rubber band a mass holder should be looped across to hang below and a metre rule should be used to measure the new length of the band. The student should gradually increase the load on the band, recording the change in the length of the rubber band.

The student may wish to repeat the investigation with a selection of rubber bands.

2. Below are the students measurements, Use this information to plot a graph of force versus extension.

I have done so and will attach shortly.

3. The student wants to calculate the elastic potential energy stored by using the equation EPE = (1/2)kx^2. Why would she get the wrong answer? Would it be an underestimate or an overestimate?

This is where the greatest confusion lies. Since, I know that Eel = 1⁄2FΔx. Since Hooke’s law states ΔF = kΔx, substituting the expression for ΔF into the equation for the elastic strain energy equation produces the formula ΔEel = 1⁄2k(Δx)^2.

On the force extension graph, the area between the line on the force-extension graph and the extension axis will represent the work done, and thus the elastic strain energy. This is only true for linear relationships though, like Hooke's Law, as the area under the line is a triangle and the formula for the area of a triangle, 1/2 bh, is equivalent to the formula for the elastic strain energy, Eel=1/2Fx

Therefore, as a rubber bands follows a hysteresis loop and has a non-linear relationship, the work done (and hence the Eel) can be found from the area under the line. However, you cannot use the formula 1/2Fx to calculate this as this is the area of a triangle, where the area underneath the hysteresis loop of the rubber band is not a triangle. Usually, if a non-linear force-extension graph has a curved line, finding the area may require one to estimate or count the squares of graph paper under the line. In such an instance, one is further required to multiply the number of squares by the elastic strain energy value (Fx) for each square. The student wishes to calculate the elastic potential energy stored, which is equal to the area under the loading curve of the force-extension graph. I am not sure whether this will be an under or overestimate though?

4. Describe the energy transfer as the force is increased, then decreased. How is this represented in your graph?

I am not too sure how to answer here. Would it be that energyis absorbed and stored when a the force is increased and the rubber band is stretched but the energy is released when the force is decreased. Rubber absorbs more energy during loading than it releases in unloading. This difference is represented by the area of the hysteresis loop, between the curves of loading and unloading.

The effect of hysteresis in rubber is to transfer energy to its molecules, resulting in heating.

Could I elaborate or improve upon my answer to be more concise ?

I greatly appreciate any help 👍

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Joinedup

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#2

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#2

(Original post by

Hello, I have a question that I am struggling to answer. I have attempted all of the sections but would greatly appreciate any help or guidance in areas where I am clearly confused.

A student demonstrates an experiment to analyse the force-extension characteristics of a rubber band.

1.Describe the equipment the student could use to perform the experiment.

The student should use a rubber band, g-clamp, a retort stand, boss and clamp, a mass hanger, 100g masses and a metre rule.

The rubber band should be positioned to hang freely from the retort stand, held in place by a g-clamp to the laboratory bench. Measure the length of the rubber band using a ruler. At the base of the rubber band a mass holder should be looped across to hang below and a metre rule should be used to measure the new length of the band. The student should gradually increase the load on the band, recording the change in the length of the rubber band.

The student may wish to repeat the investigation with a selection of rubber bands.

2. Below are the students measurements, Use this information to plot a graph of force versus extension.

I have done so and will attach shortly.

3. The student wants to calculate the elastic potential energy stored by using the equation EPE = (1/2)kx^2. Why would she get the wrong answer? Would it be an underestimate or an overestimate?

This is where the greatest confusion lies. Since, I know that Eel = 1⁄2FΔx. Since Hooke’s law states ΔF = kΔx, substituting the expression for ΔF into the equation for the elastic strain energy equation produces the formula ΔEel = 1⁄2k(Δx)^2.

On the force extension graph, the area between the line on the force-extension graph and the extension axis will represent the work done, and thus the elastic strain energy. This is only true for linear relationships though, like Hooke's Law, as the area under the line is a triangle and the formula for the area of a triangle, 1/2 bh, is equivalent to the formula for the elastic strain energy, Eel=1/2Fx

Therefore, as a rubber bands follows a hysteresis loop and has a non-linear relationship, the work done (and hence the Eel) can be found from the area under the line. However, you cannot use the formula 1/2Fx to calculate this as this is the area of a triangle, where the area underneath the hysteresis loop of the rubber band is not a triangle. Usually, if a non-linear force-extension graph has a curved line, finding the area may require one to estimate or count the squares of graph paper under the line. In such an instance, one is further required to multiply the number of squares by the elastic strain energy value (Fx) for each square. The student wishes to calculate the elastic potential energy stored, which is equal to the area under the loading curve of the force-extension graph. I am not sure whether this will be an under or overestimate though?

4. Describe the energy transfer as the force is increased, then decreased. How is this represented in your graph?

I am not too sure how to answer here. Would it be that energyis absorbed and stored when a the force is increased and the rubber band is stretched but the energy is released when the force is decreased. Rubber absorbs more energy during loading than it releases in unloading. This difference is represented by the area of the hysteresis loop, between the curves of loading and unloading.

The effect of hysteresis in rubber is to transfer energy to its molecules, resulting in heating.

Could I elaborate or improve upon my answer to be more concise ?

I greatly appreciate any help 👍

**LukeWatson4590**)Hello, I have a question that I am struggling to answer. I have attempted all of the sections but would greatly appreciate any help or guidance in areas where I am clearly confused.

A student demonstrates an experiment to analyse the force-extension characteristics of a rubber band.

1.Describe the equipment the student could use to perform the experiment.

The student should use a rubber band, g-clamp, a retort stand, boss and clamp, a mass hanger, 100g masses and a metre rule.

The rubber band should be positioned to hang freely from the retort stand, held in place by a g-clamp to the laboratory bench. Measure the length of the rubber band using a ruler. At the base of the rubber band a mass holder should be looped across to hang below and a metre rule should be used to measure the new length of the band. The student should gradually increase the load on the band, recording the change in the length of the rubber band.

The student may wish to repeat the investigation with a selection of rubber bands.

2. Below are the students measurements, Use this information to plot a graph of force versus extension.

I have done so and will attach shortly.

3. The student wants to calculate the elastic potential energy stored by using the equation EPE = (1/2)kx^2. Why would she get the wrong answer? Would it be an underestimate or an overestimate?

This is where the greatest confusion lies. Since, I know that Eel = 1⁄2FΔx. Since Hooke’s law states ΔF = kΔx, substituting the expression for ΔF into the equation for the elastic strain energy equation produces the formula ΔEel = 1⁄2k(Δx)^2.

On the force extension graph, the area between the line on the force-extension graph and the extension axis will represent the work done, and thus the elastic strain energy. This is only true for linear relationships though, like Hooke's Law, as the area under the line is a triangle and the formula for the area of a triangle, 1/2 bh, is equivalent to the formula for the elastic strain energy, Eel=1/2Fx

Therefore, as a rubber bands follows a hysteresis loop and has a non-linear relationship, the work done (and hence the Eel) can be found from the area under the line. However, you cannot use the formula 1/2Fx to calculate this as this is the area of a triangle, where the area underneath the hysteresis loop of the rubber band is not a triangle. Usually, if a non-linear force-extension graph has a curved line, finding the area may require one to estimate or count the squares of graph paper under the line. In such an instance, one is further required to multiply the number of squares by the elastic strain energy value (Fx) for each square. The student wishes to calculate the elastic potential energy stored, which is equal to the area under the loading curve of the force-extension graph. I am not sure whether this will be an under or overestimate though?

4. Describe the energy transfer as the force is increased, then decreased. How is this represented in your graph?

I am not too sure how to answer here. Would it be that energyis absorbed and stored when a the force is increased and the rubber band is stretched but the energy is released when the force is decreased. Rubber absorbs more energy during loading than it releases in unloading. This difference is represented by the area of the hysteresis loop, between the curves of loading and unloading.

The effect of hysteresis in rubber is to transfer energy to its molecules, resulting in heating.

Could I elaborate or improve upon my answer to be more concise ?

I greatly appreciate any help 👍

IMO you should allow time for the band to reach equilibrium temperature between changing the loading and making the measurement and make a not that you're doing it in your experimental method.

more on the thermal effects of changing the loading on rubber bands https://www.youtube.com/watch?v=lfmrvxB154w

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LukeWatson4590

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#3

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As you've probably noticed rubber bands increase in temperature while you extend them - if you're doing an experiment in a fairly slow and methodical way the heat will be transferred to the environment and the band will return to thermal equilibrium with it's environment. if you unload a stretched elastic band that is the same temperature as it's environment it drops in temperature to below the temperature of it's surroundings. This is more noticeable on thicker rubber bands.

IMO you should allow time for the band to reach equilibrium temperature between changing the loading and making the measurement and make a not that you're doing it in your experimental method.

more on the thermal effects of changing the loading on rubber bands https://www.youtube.com/watch?v=lfmrvxB154w

**Joinedup**)As you've probably noticed rubber bands increase in temperature while you extend them - if you're doing an experiment in a fairly slow and methodical way the heat will be transferred to the environment and the band will return to thermal equilibrium with it's environment. if you unload a stretched elastic band that is the same temperature as it's environment it drops in temperature to below the temperature of it's surroundings. This is more noticeable on thicker rubber bands.

IMO you should allow time for the band to reach equilibrium temperature between changing the loading and making the measurement and make a not that you're doing it in your experimental method.

more on the thermal effects of changing the loading on rubber bands https://www.youtube.com/watch?v=lfmrvxB154w

Moreover, could you offer any guidance on my answers to the other questions, especially part 3 which has confused me.

Thank you very much again I greatly appreciate it 👍

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Joinedup

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#4

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Thank you for your reply I will absolutely take note of your advice. I must say, that the method I included was a brief outline to show how the equipment should be utilised and arranged. Although, none of the questions actaully ask me to outline the methodology. Are you suggesting that in question 4 I include more detail to explain how the band requires sufficient time to reach equilibrium temperature as it undergoes energy transfer as the force is increased?

Moreover, could you offer any guidance on my answers to the other questions, especially part 3 which has confused me.

Thank you very much again I greatly appreciate it 👍

**LukeWatson4590**)Thank you for your reply I will absolutely take note of your advice. I must say, that the method I included was a brief outline to show how the equipment should be utilised and arranged. Although, none of the questions actaully ask me to outline the methodology. Are you suggesting that in question 4 I include more detail to explain how the band requires sufficient time to reach equilibrium temperature as it undergoes energy transfer as the force is increased?

Moreover, could you offer any guidance on my answers to the other questions, especially part 3 which has confused me.

Thank you very much again I greatly appreciate it 👍

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LukeWatson4590

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#5

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I don't know how many marks it's for, but probably showing the linear relationship F=kx does not apply to rubber bands and which way the Fx curve deviates from the linear relationship would be sufficient without getting into discussion of hysteresis

**Joinedup**)I don't know how many marks it's for, but probably showing the linear relationship F=kx does not apply to rubber bands and which way the Fx curve deviates from the linear relationship would be sufficient without getting into discussion of hysteresis

Do you know where would be a good place to begin? Sorry to ask, my brain has frozen.

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Joinedup

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#6

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Thank you for your reply. I am very grateful for your help. I understand what you are saying but I do not know where to begin. Sorry I feel I am being very silly, I just cannot seem to find a starting point.

Do you know where would be a good place to begin? Sorry to ask, my brain has frozen.

**LukeWatson4590**)Thank you for your reply. I am very grateful for your help. I understand what you are saying but I do not know where to begin. Sorry I feel I am being very silly, I just cannot seem to find a starting point.

Do you know where would be a good place to begin? Sorry to ask, my brain has frozen.

You can describe what the graph is saying first and then (if required to) attempt to explain it

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#7

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Well I suppose I'm quite visual so I'd always start by putting the points on a graph as soon as possible - have a think about what a steep gradient means on a force extension graph and what a shallow gradient means.

You can describe what the graph is saying first and then (if required to) attempt to explain it

**Joinedup**)Well I suppose I'm quite visual so I'd always start by putting the points on a graph as soon as possible - have a think about what a steep gradient means on a force extension graph and what a shallow gradient means.

You can describe what the graph is saying first and then (if required to) attempt to explain it

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