Statics of a ladder leaning against a rough wall

Abstract

A ladder leaning against a wall is an example of a statically indeterminate structure for which the equations of equilibrium are insufficient to determine the reaction forces. In such problems the reaction forces also depend on the elasticity of the system and the way in which the system was set up. Introductory physics texts usually make the ladder problem tractable by assuming that the wall is frictionless. This paper argues that the correct condition is that the frictional force at the top of the ladder is equal to the coefficient of static friction times the normal force, i.e., the maximum possible static friction at the wall. The stability of the ladder against falling depends on the frictional force at the bottom of the ladder which would normally be less than its maximum possible value independently of conditions at the top of the ladder.