What is a restoring force constant?

Question

A simple harmonic oscillator of mass 7 kg has a period of 50 seconds. What is its restoring force constant?

Answer 1

From the solution of the differential equation of simple harmonic motion,

k = m*4π²/T² = 7*4π²/50² = .11054 Newtons per meter

Answer 2

If by this you mean "What is the coefficient k of x in the differential equation" governing the small amplitude pendulum:

d^2 x / dt^2 + k x = 0,

it's k = (omega)^2 = (2 pi / P)^2 = (2 pi / 50) sec^(-2) =

0.01579... sec^(-2), to 4 significant figures.

The mass in the equation has simply divided out, here, so that the equation itself is actually about ACCELERATION, but not about FORCE. So, ironically, the "restoring force constant" doesn't have the dimensions of a force. If you felt more comfortable, expressing it in "force" terms, you'd have to at least multiply it by the mass, 7 kg (and maybe something else, since 'x' in the D.E, a SIDEWAYS displacement, could be a length, but it could also be an angle instead --- things having different dimensions [angles have none])! But that is rarely done; people know what they mean, despite the inconsistency between their use of the term "restoring force constant" and the dimensions of a force.

Live long and prosper