The spring constant

Topics: Oscillation, Measurement, Force Pages: 6 (808 words) Published: November 29, 2013
The spring constant
Manuel Cereijido Fernández – UO237242
PL-4
-----------------------------------------------------------------------------------1. Objectives The main objective of this experiment was to determine the spring constant using the dynamic method.

2. Theoretical fundamentals
On the one hand, when a spring oscillates, the
movement which describes can be classify as a
simple harmonic motion. Therefore, its position,
velocity and acceleration equations respect the time
will be:
𝑥( 𝑡) = 𝐴 · sin⁡ 𝜔𝑡 + 𝜑)
(
𝑣( 𝑡) = −𝐴𝜔 ∙ cos⁡ 𝜔𝑡 + 𝜑)
(
𝑎( 𝑡) = ⁡ −𝜔2 𝐴 ∙ sin( 𝜔𝑡 + 𝜑) = −𝝎 𝟐 𝒙 (1) On the other hand, if a weight W = mg is hung from one end of an ordinary spring, causing it to stretch a distance x, then, an equal and opposite force, F, is created in the spring which acts to oppose the pull of the weight. If W is not so large as to permanently distort the spring, then this force, F, will restore the spring to its original length after the load is removed. 𝐹 is called an elastic force, which is directly proportional to the stretch (Hooke’s Law). The minus sign means that the acceleration is in the direction opposite to the force. So, 𝐹 = −𝑘𝑥 (2)

By Newton’s second law, we also know that:
𝐹 = 𝑚𝑎

(3)

If we equal these functions, and substitute the value of 𝑎 by the acceleration in a simple harmonic motion, we obtain:
Substitute (4) in (3)
(2)= (3)

𝐹 = −𝑚𝜔2 𝑥
−𝑘𝑥 = −𝑚𝜔2 𝑥

Once we have that expression, we can compute the period of a spring and, consequently, the spring constant.
𝑇 = 2𝜋√

𝑚
4𝜋 2 𝑚
⁡⁡ → ⁡⁡𝑘 =
𝑘
𝑇2

3. Experimental procedure
Basically, the dynamic method we used to determine the spring constant consist in taking measures of the period of the spring when it is oscillating. To that end, we needed some instruments and material:

-

A spring
Different weights
Stop watch
Support

First of all, we had to choose a weight. Once the spring was hooked in the support, we had to hang the weight on the end of the spring. Because of that, the spring stretched until the equilibrium point.

Next, we threw down the weight a little and it started to oscillate. In this experiment we decided to measure 25 oscillations. In order to obtain a measure more homogeneous we repeated that process three times.

Time of 25 oscillations
1st measurement
2nd measurement
3rd measurement

Average time

15,65 ± 0,06 s
15,72 ± 0,01 s
15,75 ± 0,04 s
15,71 ± 0,06 s

Using these measurements, we could calculate the time of one oscillation, the period (𝜏), which gave us the following value:
𝜏 = 0,6283 ± 0,0024 s
Finally, we only had to apply the formula we deduced before and calculate the spring constant (𝑘), which was:
𝑘 = 20,00 ± 0,15 kg/s2

4. Results and discussion
The measurement of a physical quantity can never be made with perfect accuracy, there will always be some error or uncertainty present. If a measurement is to be useful, it is necessary to have some quantitative idea of the magnitude of the errors. Therefore, when experimental results are reported, they are accompanied by an estimate of the experimental error, called the uncertainty. This uncertainty indicates how reliable the experimenter believes the results to be.

To calculate all the errors, we followed the principles of the errors theory. When we calculated the error of the average time, we didn’t use the RMS error: 𝑅𝑀𝑆 = √

∑ 𝑖=𝑛 𝜖 𝑖 2
(𝜖⁡1)2 + (𝜖⁡2)2 + (𝜖⁡3)2
𝑖=1
=√
𝑛(𝑛 − 1)
𝑛(𝑛 − 1)

𝑅𝑀𝑆 = √

(0,06)2 +(0,01)2 +(0,04)2
3(3−1)

= ± 0,03 s

Instead of that, we chose the maximum error we had obtained in the measurements of the time of 25 oscillations (0,06s).
The period is defined as the time of one oscillation so, to get the error of the period, we used the next formula:
𝜖⁡𝜏 = |

𝜕𝜏
| ∙ 𝜖⁡𝑡
𝜕𝑡

1

𝜖⁡𝜏 = |25| ∙ 0,06 = ± 0,0024 s
The same process was done in order to compute the error of the spring constant. 𝜖⁡𝑘 = |

𝑑𝑘
| ∙ 𝜖⁡𝜏
𝑑𝜏

−8𝜋 2 𝑚
| ∙ 𝜖⁡𝜏
𝜖⁡𝑘 = |
𝜏3
−8𝜋2 0,200

𝜖⁡𝑘 =...
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