## Vertical Spring and Hanging Mass

Our

prototypefor SHM has been ahorizontalspring attached to a mass,But it is often easier for us to set up a

verticalspring with a hanging mass. Now the force ofgravitycomes into play. Does this change what we expect for the period of this simple harmonic oscillator?Consider a vertical spring on which we hang a mass m; it will stretch a distance x because of the weight of the mass,

That stretch is given by x = m g / k. k is the spring constant of the spring.

Now pull the mass down an additional distance x',

The spring is now exerting a force of

F _{spring}= - k xF

_{spring}= - k (x' + x)F

_{spring}= - k x' - k xF

_{spring}= - k x' - mgWhen we add in the force of gravity, we have

F _{net}= F_{spring}+ mgF

_{net}= - k x' - mg + mgF

_{net}= - k x'Now this looks

exactlylike our prototypical equation with the displacement x' now being measuredfrom the new equilibriumposition. This means that everything we have learned about our prototypicalhorizontalSHO is entirely applicable to this more usualverticalSHO.(c) Doug Davis, 2001; all rights reserved

UCMSimple PendulumReturn to ToC, Simple Harmonic Motion