I can't give precise ratios because I don't know precisely how the weight is distributed in the flashlight. So I'll work out a simpler problem which should get all the key ideas across.
Say you have a uniformly dense rectangular block, with height r times the width. The tipping point occurs when the center of mas is straight up from the contact point. Draw the picture and you'll see that if you call the gravitational potential energy (mgh) of the flat block 1 unit, the tipping point has energy sqrt(1+r2
) and the upright energy is r. So the ratio of the work done to tip the block from the flat to the upright position to the reverse process is (sqrt(1+r2
)-r). If, for example, r=5, that comes out around 40.
The ratio of the maximum force needed is given by the ratio of the torques needed to counteract the gravitational torque around the pivot point. That's even simpler, since the gravitational torque just depends on how far the center of mass is displaced from the vertical line through the pivot. That ratio is just r.
(published on 04/20/2011)