Equilibrium and stability of bevel gauges: the role of arm angle and length

Abstract We investigate the equilibrium and stability of bevel gauges resting on an idealised punctual support, assuming identical thickness and mass density for both arms and perfect contact. Extending our previous analysis of L-squares, we generalise the model to variable inter-arm angles and examine how equilibrium orientation and stability depend on arm lengths, arm thickness and inter-arm angle. Using the principle of virtual work, we derive a theoretical expression for the equilibrium angle, including its sign, and complement it with a stability analysis that identifies the conditions for stable or unstable equilibria. This leads to the definition of a critical thickness that separates two distinct regimes in the evolution of equilibrium configurations as the longer arm length increases, and below which all equilibrium positions remain stable. Experiments conducted with laser-cut bevel gauges confirm the theoretical predictions. Beyond its mechanical interest, this simple system offers an accessible framework for illustrating how geometry controls equilibrium and stability, and can be rapidly adapted for classroom demonstrations.