The spring constant

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:
?( ?) = ? · sin⁡ ?? + ?)
(
?( ?) = −?? ∙ cos⁡ ?? + ?)
(
?( ?) = ⁡ −?2 ? ∙ sin( ?? + ?) = −? ? ? (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. ? 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,
? = −?? (2)
By Newton’s second law, we also know that:
? = ??

(3)

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

? = −??2 ?
−?? = −??2 ?

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

?
4? 2 ?
⁡⁡ → ⁡⁡? =
?
?2

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