The helical spring

The animation shows the oscillation of a helical spring. The motion is simple harmonic, the greater the mass on the end of the spring the longer the period of the motion.

Consider a mass m suspended at rest from a spiral spring and let the extension produced be e. If the spring constant is k we have: mg = ke The mass is then pulled down a small distance x and released.

The mass will oscillate due to both the effect of the gravitational attraction (mg) and the varying force in the spring (k(e + x)).

At any point distance x from the midpoint:

restoring force = k(e + x) - mg

But F = ma, so ma = - kx and this shows that the acceleration is directly proportional to the displacement, the equation for s.h.m. The negative sign shows that the acceleration acts in the opposite direction to increasing x.

From the defining equation for s.h.m. :

(a = - w²x) we have

w² = k/m = g/e and therefore the period of the motion T is given by:

T = 2p(e/g)^1/2 = 2p(m/k)^1/2

A graph of e against T² can be used to determine g.

If the mass of the spring is significant we can allow for it and the corrected equation becomes:
T = 2p[(M + m)/k]^1/2 where M is one-third of the mass of the spring. The mass must be sufficiently large to keep the coils open.