An improved calculation of the mass for the resonant spring pendulum

M. G.

Olsson

, “,” (), – ().

H. M.

Lai

, “,” (), – ().

W. R.

Mellen

, “,” , – ().

D. E.

Holzwarth

and

J.

Malone

, “,” , ().

Equation (25) in Ref. 2 for the period with which the energy oscillates between these modes involves elliptical integrals of the first kind, even for the case of a massless spring as a simple pendulum.

E. E.

Galloni

and

M.

Kohen

, “,” (), – ().

S. Y.

 

Mak

, “,”   (), – ().

See also

P. K.

Glanz

, “,” (), – () for a discussion of the effect of pre-loading a spring on the conservation of energy.

T. C.

Heard

and

N. D.

Newby

, Jr., “,” (), – (). See especially Eq. (20).

J. T.

Cushing

, “,” (), – (). See footnote 16.

R.

 

Weinstock

, “,”   (), – ().

See also

M.

Parkinson

, “,” (), – () for the same result from iterative methods rather than differential equations.

R. J. Stephenson, Mechanics and Properties of Matter (Wiley, New York, 1952), pp. 113–114.

See also

L.

Ruby

, “,” , – () for the same result from iterative methods to avoid the calculus for an algebra-based course. Ruby also attempts a simplified nonuniform approximation, but incorrectly relates Mak’s (Ref. 7) static correction to his dynamic correction.

R.

Weinstock

, “,” (), – ().

F. A.

McDonald

, “,” (), – ().

J. M.

Bowen

, “,” (), – ().

As mentioned in this article, a mass hanging in equilibrium from a spring will satisfy $kz=mg.$ A graph of displacement versus added-mass then gives a slope of $g/k.$ As described in Appendix B, a mass that is hung from a spring and set to oscillating has a characteristic period. A plot of $[T/2π]2$ vs $m$ then gives a line with slope $1/k.$ In principle, $g/k$ divided by $1/k$ will give a calculation of the local gravitational field. In a paper in preparation, these values are compared to the acceleration due to gravity measured by a spark-machine free-fall experiment and the gravitational field measured by a swinging pendulum. The free-fall experiment gave repeatable results consistently lower than expected. The springs and pendula independently gave repeatable results that were higher than expected and consistent with each other. The systematic uncertainties are still being investigated.

Philip R. Bevington and D. Keith Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw–Hill, New York, 1992), 2nd ed., Chaps. 6, 11.