Static equilibrium problem I am designing a balanced-arm lamp structure for a robotics project and I obviously want the lamp to stay on its base without falling when the arm is extended. For that, I applied the conditions for static equilibrium and I come to an equality (relation between the masses and the distances of the base, head, arm, etc. of the lamp) that ensures that the angular acceleration is 0. So far I am sure that my reasoning is correct, but I think that the result I got is not what I am looking for. Experiencing with a real lamp or with every similar object in general you can apply a certain weight to the arm before the lamp starts falling. So experience and common sense tells me I should arrive to an inequality, but I don't know how to get there. Can somebody offer some light to the subject?
 A: Balanced-arm lamps typically have a heavy and wide base. If you take moments about the edge of the base (which would be the fulcrum if the lamp tipped) then the idea is that the large weight at the centre of the base multiplied by its small moment arm is greater than the weight of the lamp holder multiplied by its large moment arm. To find the approximate maximum horizontal distance that the lamp holder can be extended, divide the weight of the base by the weight of the lamp holder, and multiply by the radius of the base. For a more accurate calculation you will need to take into account the weight of each arm, which you can assume acts through its mid-point.
An alternative design is to use a counterweight. Again, the idea is that a heavy weight with a smaller moment arm counterbalances a smaller weight with a larger moment arm. But this time the counterweight can be extended beyond the edge of the base. A mechanical linkage can move the counterweight out by an appropriate distance as the lamp holder arm is extended.
