MEASUREMENT OF THE SPRING CONSTANT
TWO METHODS COMPARED

ABSTRACT
A spring is an elastic object, which can be extended and contracted by forces with a constant elastic coefficient. This report assesses two methods for determining the spring constant of a specific spring. One method uses the length that the spring extends and the masses that are attached to the spring. The other method measuring the spring constant is based on the harmonic motion.

INTRODUCTION
Spring constant is an instinct property of a spring. The magnitude of a spring constant depends on the material of the spring but not the lengths it extends or contracts or the forces that are applied to the spring. Hook’s law predicts that the length the spring extends or contracts is directly proportional to the force applied by the coefficient of the spring constant. The aim of this experiment is to compare the two measures of finding the value of the spring constant of a spring.

MATERIALS AND METHODS

METHOD A
The spring is hanged on an iron stand. The lengths of the spring, L, are measured when the spring is attached to different masses, M, of metals vertically.
The lengths are measured by using a ruler with scales, while the mass attached to the spring can be read from the metal. For better accuracy, five masses are attached during this method.

METHOD B
The spring is hanged on an iron stand. Different masses, m, attached to the spring can be read from the metals and the time period, t, of the spring finishing one period of oscillation.
Attach a mass to the spring and extend a short distance vertically. A stopwatch is used to measure the time intervals when the spring experiences full cycles of oscillation when the mass reaches one end point of its trajectory, at the maximum extension or compression of the spring. For better accuracy, time intervals when the spring experiences several cycles rather than just one cycle are measured and the period of one cycle can be determined by dividing the time intervals with times of cycles.

RESULTS

METHOD A
Table 1 shows the data gathered for the measurement of the spring constant based on the Hook’s law.

Table 1
|Mass (kg) |Length (m) | |
|0 |0.055 | |
|0.02 |0.0688 | |
|0.04 |0.083 | |
|0.05 |0.0916 | |
|0.07 |0.1054 | |
|0.1 |0.126 | |

The relationship between the mass, M, and the length, L, is given by Hook’s law

M*g=kL

where g is the acceleration due to gravity and k is the spring constant.
This equation is of the form y=ax+c, so a given by

a=k/g

Figure 1 shows a plot of length versus mass and the regression line including its formula found by the Microsoft Excel Software

Figure 1
Length versus mass for a spring

[pic]

The slope is therefore found be a=0.7153±0.01222. As, k=a*g
We can get the value of the spring constant, assuming g=9.8, that is k=7.00994±0.119756 METHOD B
Table 2 shows the data gathered for the period as a function of square mass

Table 2
|mass |SQRTmass |period |
|0.02 |0.141421356 |0.34 |
|0.04 |0.2 |0.468 |
|0.05 |0.223606798 |0.47 |
|0.07 |0.264575131 |0.63 |
|0.1 |0.316227766 |0.73 |

The relationship between the period, T, and the SQRTmass, SQRT(M), is

T=[2π/SQRT(k)]SQRT(M)

This equation is of the form y=ax+c. And the spring constant can be found to be k=(4 π2)/(a2)

Figure 2 shows a plot of periods versus SQRTmass and the regression line including its formula found by the Microsoft Excel Software

Figure 2
Period versus SQRTmass
[pic]
The slope is found to be a=2.2878±0.14365 Therefore, we can find the spring constant to be k=7.534998603± DISCUSSION
Even though we can get similar values by using two method, the value gotten by using method A is superior to that found by method B because the uncertainty gotten from that is smaller. Value found by using method B is not very precise since mistakes may occur when our eyes are observing whether the mass has reached the end point or not.

CONCLUSION
A value is determined for the spring constant by adopting method A, while a value is found by employing method B. Method A, which is based on the Hook’s law is superior to method B.
REFERENCE

1) Conceptual Physics the eleventh international edition; 2) Hand outs by Dr. Friis.