What is the Relationship Between Force and Spring Constant?

[Perander]

Homework Statement

We have the task of calculating the spring constant in two ways. One way is to use the formula F = kx (1) and the other formula to use the formula for the oscillation time of a harmonic motion T=2pi*sqrt(m/k) = T^2 = (2pi)^2/k * m (2). And then we will answer the following questions:
1. Were the values of spring constant equal when using two ways calculating it?
2. What value should be used, i.e. the value in formula 1 or 2?
3. Can you analyze errors in measurements and calculation methods to arrive at which of the methods in formula 1 or 2 is better?

Relevant Equations

1. F = k*x
2. T ^ 2 = (2pi) ^ 2 / k * m.
T ^ 2 = K * m

In question 1, the spring constant from the two formulas was not the same. When we used the first formula, we got that the spring constant was 7.83 N / m. The second formula we got that the spring constant was 8,03 N / m.

In questions 2 and 3 I do not know and am unsure about how to answer. Using the first formula F=k*x, we found that the spring constant is equal to the force divided by the spring elongation in meters. . Using the second formula, we obtained that T ^ 2 = K * m (where K corresponds to the gradient of the graph) and T ^ 2 = (2pi) ^ 2 / k * m.
K * m = (2pi) ^ 2 / k * m and from that you can get small k (the spring constant).

 

[kuruman]

What is assumed known here? Is this an experiment for which you collected data? If so, please describe the experiment and what you did specifically.

 

[Perander]

What is assumed known here? Is this an experiment for which you collected data? If so, please describe the experiment and what you did specifically.

Please view this document: https://docs.google.com/document/d/1v_au2Pp7ipawJKw5MumKxdA4zkFZdPuk42b5-a2jgf4/edit
It says how the experiment was carried out and also measured values are there.
And it's "part 3 of the document" that I need help with.

 

[kuruman]

Thank you for providing the link to the document. If I were marking it, I would give you a very low grade because you have committed the unpardonable sin of producing a lab report showing numbers without units. So fix that and then we'll talk.

Also, any help you seek and get from us should be through Physics Forums - no direct comments in the document because it's your report and you are writing it, not we. I understand you are a new user, so maybe the mentors will also be understanding just this time.

 

[Perander]

Thank you for providing the link to the document. If I were marking it, I would give you a very low grade because you have committed the unpardonable sin of producing a lab report showing numbers without units. So fix that and then we'll talk.

Also, any help you seek and get from us should be through Physics Forums - no direct comments in the document because it's your report and you are writing it, not we. I understand you are a new user, so maybe the mentors will also be understanding just this time.

Okay, I have fixed it now. What I do not understand are questions 2 and 3, part 3 of the document. How do I know which of the two values of the spring constant I got should be used? And question 3, how to analyze errors in measurements and calculation methods and thus arrive at which method in part 1 or 2 is best to use when calculating the spring constant? Can you explain how it works?

 

[kuruman]

In part 3 you are expected to evaluate which method you trust better to give you a value that is closer to the average of many many measurements performed on the spring. Only you know what you did and how you performed your measurements. Evaluate how you determined the length of the spring elongation versus how you evaluated the period of one oscillation. Which one is more likely to have a higher percentage error? What did you do to limit that percentage in each case? Also, I cannot believe the accuracy to which you quote your numbers. Decide how many significant figures are appropriate for your measurements and be consistent.

How do I know which of the two values of the spring constant I got should be used?

This is what experimentation is all about. You use the value that you trust most. You gathered the evidence, analyzed it and came up with two numbers. You use the value you trust most given what you did. It's your call.

When I was teaching an intro physics course with 200 students, one experiment was determining the acceleration of gravity. I collected all the values by everybody (some were as low as 4.6 m/s² and some as high as 12.5 m/s²) and calculated the average of 9.68 m/s². I decreed to the class that from then on, this is the value that they should be using for all experiments that followed. To those who protested that this is not the "correct" value, I replied that this is the value that they and their peers measured in that lab at that location, therefore it is the correct value. Those who used 9,8 m/s² lost points. There is a lesson to be learned here about not rejecting your own observations and judgment in favor of somebody else's.

 

[Perander]

In part 3 you are expected to evaluate which method you trust better to give you a value that is closer to the average of many many measurements performed on the spring. Only you know what you did and how you performed your measurements. Evaluate how you determined the length of the spring elongation versus how you evaluated the period of one oscillation. Which one is more likely to have a higher percentage error? What did you do to limit that percentage in each case? Also, I cannot believe the accuracy to which you quote your numbers. Decide how many significant figures are appropriate for your measurements and be consistent.This is what experimentation is all about. You use the value that you trust most. You gathered the evidence, analyzed it and came up with two numbers. You use the value you trust most given what you did. It's your call.

When I was teaching an intro physics course with 200 students, one experiment was determining the acceleration of gravity. I collected all the values by everybody (some were as low as 4.6 m/s² and some as high as 12.5 m/s²) and calculated the average of 9.68 m/s². I decreed to the class that from then on, this is the value that they should be using for all experiments that followed. To those who protested that this is not the "correct" value, I replied that this is the value that they and their peers measured in that lab at that location, therefore it is the correct value. Those who used 9,8 m/s² lost points. There is a lesson to be learned here about not rejecting your own observations and judgment in favor of somebody else's.

Can the fact that in part 2 the formula for the oscillation time in square and 2pi was used affect the result and thus method 1 in part 1 is more reliable?

 

[haruspex]

Please view this document: https://docs.google.com/document/d/1v_au2Pp7ipawJKw5MumKxdA4zkFZdPuk42b5-a2jgf4/edit
It says how the experiment was carried out and also measured values are there.
And it's "part 3 of the document" that I need help with.

Google says I don't have access to it.
Have you gone through the error analysis for each method, estimating the error range in each reading and how that affects the result?
Did the two methods test the same range of extension of the spring, e.g., multiple measures using method 1 across the range of oscillation that occurred in method 2?

 

[Perander]

Google says I don't have access to it.
Have you gone through the error analysis for each method, estimating the error range in each reading and how that affects the result?
Did the two methods test the same range of extension of the spring, e.g., multiple measures using method 1 across the range of oscillation that occurred in method 2?

You have access now.
No, that's what I'm not sure how to do.
How should I analyze the error for each method and estimate the error interval in each reading? This is what I think about but do not figure out how to do an error analysis and error estimate.

 

[kuruman]

You have access now.
No, that's what I'm not sure how to do.
How should I analyze the error for each method and estimate the error interval in each reading? This is what I think about but do not figure out how to do an error analysis and error estimate.

You begin by estimating the error in the measurements on which the calculations are based. For example, you say that when a mass of 0.15 kg was attached, the spring was extended by 0.165 m. Did you use a meter-stick or something else? If you used a meter-stick, how confident were you that you used reproducibly from one measurement to another the same spots on the physical spring as the "origin" and "end" to determine the elongation? In short, how careful were you determining your fiducial points? That will determine the extent of your uncertainty in the measurement of the length.

Similar ideas apply for the method of finding the spring constant using the period of oscillations. What method did you use to determine that? Did you release the mass from rest starting a clock at the same time and click stop when it came back up after one round trip? Well, that method has the problem of not being very accurate because the oscillator spends relatively more time at the turning points than at the equilibrium position when its speed is maximum. You get better accuracy if you draw a line or put some tape that marks the equilibrium point and start and stop the clock when the mass crosses that point. You get even better accuracy if you start the clock then stop after 10 (or 20 or 30) oscillations and divide the total time by the number of oscillations. That minimizes the error introduced by your reaction time.

As I said, only you know what you did and how and only you can estimate your uncertainty in your measurements. My usual advice to beginners is to consider which of the variables that you measured has the largest uncertainty and ignore the other ones. In the first method is it the mass or the length that you have measured most accurately? What about the second method, mass or period? The answer depends on what you did and how. A very rough calculation of how the uncertainty in the measurement affects the value of the spring constant might be as follows

• Decide which variable has the largest uncertainty.
• Estimate the uncertainty as a ±δ of the measured value.
• Do two separate calculations of the spring constant adding and subtracting δ from the variable.
• Declare that the uncertainty in the spring constant is ± half the difference between the high and low calculated values.

In both methods you found the gradient by considering two data points. Fitting a straight line to the data is a good way to average results, however the way you did it may be problematic. There are 6 ways to select pairs when you have 4 data points. What made you choose that particular pair? If those two points are off in opposite ways, you could get a significant misrepresentation of the gradient. Perhaps you can repeat the gradient calculation using additional pairs if not all 6 of them. Then you can do a high-low calculation for the value of the spring constant based on the high-low values of the gradient.

Other PF users may different suggestions from me. I think this would make sense to a beginner who is not conversant with the issues involved.

 

[Perander]

You begin by estimating the error in the measurements on which the calculations are based. For example, you say that when a mass of 0.15 kg was attached, the spring was extended by 0.165 m. Did you use a meter-stick or something else? If you used a meter-stick, how confident were you that you used reproducibly from one measurement to another the same spots on the physical spring as the "origin" and "end" to determine the elongation? In short, how careful were you determining your fiducial points? That will determine the extent of your uncertainty in the measurement of the length.

Similar ideas apply for the method of finding the spring constant using the period of oscillations. What method did you use to determine that? Did you release the mass from rest starting a clock at the same time and click stop when it came back up after one round trip? Well, that method has the problem of not being very accurate because the oscillator spends relatively more time at the turning points than at the equilibrium position when its speed is maximum. You get better accuracy if you draw a line or put some tape that marks the equilibrium point and start and stop the clock when the mass crosses that point. You get even better accuracy if you start the clock then stop after 10 (or 20 or 30) oscillations and divide the total time by the number of oscillations. That minimizes the error introduced by your reaction time.

As I said, only you know what you did and how and only you can estimate your uncertainty in your measurements. My usual advice to beginners is to consider which of the variables that you measured has the largest uncertainty and ignore the other ones. In the first method is it the mass or the length that you have measured most accurately? What about the second method, mass or period? The answer depends on what you did and how. A very rough calculation of how the uncertainty in the measurement affects the value of the spring constant might be as follows

• Decide which variable has the largest uncertainty.
• Estimate the uncertainty as a ±δ of the measured value.
• Do two separate calculations of the spring constant adding and subtracting δ from the variable.
• Declare that the uncertainty in the spring constant is ± half the difference between the high and low calculated values.

In both methods you found the gradient by considering two data points. Fitting a straight line to the data is a good way to average results, however the way you did it may be problematic. There are 6 ways to select pairs when you have 4 data points. What made you choose that particular pair? If those two points are off in opposite ways, you could get a significant misrepresentation of the gradient. Perhaps you can repeat the gradient calculation using additional pairs if not all 6 of them. Then you can do a high-low calculation for the value of the spring constant based on the high-low values of the gradient.

Other PF users may different suggestions from me. I think this would make sense to a beginner who is not conversant with the issues involved.

Okay, I understand.
But I wonder one thing, and that is whether using this formula ( T ^ 2 = (2pi) ^ 2 / k * m) used in part 2 could affect the results, or what I mean is that the fact that we used the oscillation time squared could affect the result in some way when you got the spring constant?

 

[haruspex]

Okay, I understand.
But I wonder one thing, and that is whether using this formula ( T ^ 2 = (2pi) ^ 2 / k * m) used in part 2 could affect the results, or what I mean is that the fact that we used the oscillation time squared could affect the result in some way when you got the spring constant?

The rules of error propagation say that squaring a value doubles the fractional error, so a 1% error in the estimate of T leads to a 2% error in $T^2$.

 

[Perander]

Have a question regarding making error bars. Should I calculate the mean of the spring constant (by taking the largest value plus the smallest value of the spring constant divided by 2), and then calculate the standard deviation and then make a diagram?

 

[haruspex]

by taking the largest value minus the smallest value of the spring constant divided by 2

That would be an unusual way to find a mean.

 

[Perander]

That would be an unusual way to find a mean.

I mean plus