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 last 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 is 0.1 and the man must be able to climb to the top, find the minimum horizontal force which must be exerted at the base to prevent the ladder slipping.
When the ladder is just about to slip the force of friction at the ground is equal towhereis the normal reaction force. An extra horizontal force X must be exerted at the base to prevent slipping.
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
If no horizontal force is available,must be increased for the climber to get to the top without the ladder slipping.
Resolving vertically,
When the ladder is about to slip, friction is limiting and equal toResolving horizontally,
Take moments about B