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# Hookes Law- to determine the spring constant of a metal spring

Extracts from this document...

Introduction

IB Physics

IB Physics Investigation – Mechanics

Title: Hooke’s Law- to determine the spring constant of a metal spring

1. BACKGROUND/ INTRODUCTION

A branch of the mechanical energy, elastic potential energy is one of the fundamental base energy concepts that is very commonly used in the modern technology. Elastic potential energy deals with the elasticity of an object (e.g. to what extent a string can be stretched?), and it’s used in a wide range of everyday life situations; such as in the process of decelerating an airplane landing on an aircraft ship. And the type of strings used there, is regarded to the spring constant of that string, in which it may or may not be able to stop an airplane within a certain distance.

Under investigation of Hook’s Law, the aim of this experiment is to determine different spring constants of some springs, and that is done by measuring how far a spring stretches when a certain amount of mass added. In order to find the spring constant, we investigate it by rearranging energy formulas and basic knowledge:

• The general formula for Energy is , where  is force and  is displacement.
• The formula for Elastic Potential Energy is , where  is
•  the spring constant and  is

the extension of the spring.

• The relation between the two formulas is that  (displacement) is equal to  (extension) since both have the same units (measured in meters). Therefore, we can state that
• The area under the a Force-Extension () graph gives the work done on the spring and therefore, the Elastic Potential Energy (). Since it’s the area of a triangle, then
• From the graph, we know that , then . Therefore if we substitute this into the first formula () we get  and that is equal to

From this information we can extract that , the spring constant can be found as follows:

1. Measure the force and extension of the spring
2. Plot a  graph
3. Find the slop of the graph, which is , the spring constant
1. QUESTION

What is the role of the spring constant in the relationship of force and extension and how can it be determined?

1. HYPOTHESIS

The independent variable of this experiment is Mass (the number of masses added) and the dependent variable is Extension (the extension of the spring). The control variables are the same ruler used (the same scale) with the same initial positions (each spring was measured by the ruler sat on 4 cm), as well as the same masses are used for each spring.

The expected findings of this experiment is that the spring with the largest spring constant will extended least with the same amount of force applied. Therefore, the variation of the spring constant causes variation of extensions of different springs at the same amount of added-force.

1. EQUIPMENT & MATERIALS

The equipment used to conduct this experiment are:

• Rulers
• Masses
• Springs
• Retort Stand

Middle

350

78.2

78.1

78.2

400

87.6

87.6

87.6

450

97.2

97.2

97.2

Spring B (Pink)

 Mass, m (g) Length, L1 (cm) Length, L2 (cm) Length, L3 (cm) 0 12.3 12.3 12.2 50 17.3 17.3 17.3 100 22.3 22.2 22.3 150 27.2 27.1 27.2 200 32.1 32.1 32.1 250 37.0 37.1 37.1 300 42.0 42.0 42.1 350 47.0 46.9 47.0 400 51.9 51.8 51.9 450 56.7 56.6 56.7 500 61.6 61.6 61.6 550 66.5 66.4 66.5 600 71.4 71.4 71.3

Spring C (Silver)

 Mass, m (g) Length, L1 (cm) Length, L2 (cm) Length, L3 (cm) 0 12.3 12.3 12.4 50 15.8 15.7 15.8 100 19.1 19.1 19.1 150 22.5 22.6 22.5 200 25.8 25.8 25.8 250 29.1 29.1 29.1 300 32.4 32.4 32.4 350 35.7 35.7 35.7 400 40.0 40.0 40.0 450 42.4 42.4 42.4 500 45.6 45.7 45.7 550 49.0 49.1 49.1 600 52.3 52.3 52.3

Spring D (Gold)

 Mass, m (g) Length, L1 (cm) Length, L2 (cm) Length, L3 (cm) 0 11.8 11.8 11.8 50 13.6 13.6 13.6 100 15.4 15.4 15.4 150 17.3 17.3 17.3 200 19.2 19.2 19.2 250 21.0 21.0 21.0 300 22.9 22.9 22.9 350 24.8 24.8 24.8 400 26.7 26.7 26.7 450 28.5 28.5 28.5 500 30.4 30.4 30.4 550 32.3 32.3 32.3 600 34.1 34.1 34.1

The uncertainty in the Masses Measurement was + 1 g

The uncertainty in the Ruler Measurement was + 0.05 cm

1. Data Processing and Presentation:

1. CONCLUSIONS

The results support my hypothesis that states that the spring with the least spring constant will extend most. And that’s shown in the graphs, the most extended spring, which is spring A has the smallest slope (i.e. the spring constant). Whereas spring D, which is the steepest linear graph extended least among all 4 springs.

Possible sources of errors that may have limited the certainty of the results could be:

• Human errors (in recording measurements)
• Accuracy of the masses (do the masses exactly weigh as they’re labeled?)
• Each spring sat on an initial position on the ruler (at 4 cm), which may has slightly moved by the time.
• The springs possibly touched the ruler very rarely, which may caused less extension due to friction.

Conclusion

The gradient of spring C’s graph:

)

The percentage uncertainty of spring C is

Spring D:

The gradient of spring D’s graph:

)

The percentage uncertainty of spring D is 1.08

1. EVALUATION

The followed procedure achieved a good level of accuracy which was demonstrated in the results that were quite reasonable in comparison to reality. The equipment used in the procedure are not very sensitive however, hence the results weren’t very accurate because the range of uncertainty is not negligible. But the range could be narrowed by developing the design of the procedure, using some more sensitive equipments/better method.

Possible solutions to reduce sources of errors:

• Hanging the ruler and the spring into two separate Retort Stands, so we can more clearly see and measure both.
• Using light for example to determine the level between the spring and the ruler, or any better pointer for the spring. Because the current spring pointer keeps rotating which makes it hard for the observer to record accurately.
• Using bars for the Retort Stand instead of the current handles that hold the ruler; so the ruler will be attached/hanged to it. Because the handle is made to hold beakers often (it has a rounded shape which makes the ruler not very stable).

This student written piece of work is one of many that can be found in our International Baccalaureate Physics section.

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