Unfortunately, a ladder is most likely to slip when the climber is at the top. This is because the moment of the climber in the direction in which the ladder would slip is at its maximum when the climber is at the top. We can analyse the forces on a ladder as it leans against the wall to find any one of

The maximum distance up the ladder the climber can ascend

The minimum value of the coefficient of friction which will stop the ladder from slipping when the climber is at the top

The least horizontal force which must be exerted at the base of the ladder to stop it from slipping

The least angle that the ladder can make with the horizontal while not slipping

To analyse the first two, Suppose a ladder of mass 30 kg and length 10m leans against a smooth wall (so the reaction force at the wall is normal to the wall). The ladder makes an anglewith the horizontal, and a man of mass 70 kg climbs the ladder. If the coefficient of friction between the ladder and the ground isfind the maximum distance up the ladder the man can climb.

When the ladder is just about to slip the force of friction at the ground is equal towhereis the normal reaction force.

Resolving vertically for the ladder:

The force of limiting friction is then

Resolving horizontally for the ladder gives

Now take moments about B. For the ladder not to slip,

so

Ifthen the climber can get to the top so