The first part of the document below shows how I do an SHM analysis for a pendulum. The restorative force pulling it back to equilibrium acts along the radius of the motion so perpendicular to the string. This means that no component of tension is involved, The resultant force turns out to be the sine component of the weight mg. But sin(theta) = x/L. For small angles x is approximately equal to the true displacement.
The balloon analysis takes up the bottom two thirds of the page. It works in much the same way except that two forces act on the balloon vertically - the upthrust and the weight. I have turned them into a resultant F. Hence the intertial mass m does not cancel this time although the solution as the same form. I got a theoretical answer of 1,1 seconds having measured it as 7 seconds. I will post later about how I got estimates for the upthrust and mass.
