Hooke's Law
I have designed the experiment to measure spring constants when the
springs are in series and in parallel. The theory is based on Hooke's
law which is:
F = kx
where F = Force, k = Constant and x = Extension [Ref. 1].
Unfortunately with the springs I have, I can only measure extension,
not compression for which Hooke's is also valid.
Prediction
Single Spring:
Hooke's law, where F = kx. I predict that I if I plot Force on the Y
axis and extension, x, on the X axis, it will be a straight line and
the gradient will be the spring constant.
Parallel Springs:
In the diagram above, T1 is the tension in spring 1, and T2 is the
tension is spring 2. So T1 + T2 has to equal the Force, F. I assume
the two springs are originally the same length and the extensions are
the same in both springs. So this means that T1 = k1x and T2 = k2x. So
by knowing this, you get the formula:
F = k1x + k2x = x (k1 +k2).
So overall, the spring constant for the two springs is k1 + k2. For
the graph, my prediction is the same as for a single spring as
mentioned earlier but with a different constant.
Springs In Series:
In this diagram, T1 is the tension in spring 1, and T2 is the tension
is spring 2. If I ignore the mass of the 2nd spring, the two tensions
have to be equal and both equal to the force. I have to find a final
expression in the form F = k (x1 + x2) because x1 + x2 is the total
extension. So from Hooke's law you can get x1 = F / k1 and x2 = F / k2.
Knowing this, gives me a new formula: F = k (F / k1 + F / k2). Here,
you can then take out the common factor, F and divide both sides by kF
which gives you a final formula: 1 / k = 1 / k1 + 1 / k2 .
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