Question

a light ladder is supported on a rough floor and leans against a rough wall.how far up the ladder can a man climb without slipping taking place?

Answer 1

A light ladder is supported on a rough floor and leans against a rough wall. How far up the ladder can a man climb without slipping taking place?

Solution.

The necessary conditions for mechanical equilibrium for a system of particles.

The vector sum of all external forces is zero:

Projection on OX:

$$F _ {f 1} = N _ {2};$$

Projection on OY:

$$m g = N _ {1} + F _ {f 2};$$

The forces of friction:

$$F _ {f 1} = \mu_ {1} N _ {1};$$ $$F _ {f 2} = \mu_ {2} N _ {2};$$

$\mu_{1}$ is the coefficient of friction between the ladder and the floor;

$\mu_{2}$ is the coefficient of friction between the ladder and the wall.

$$\mu_ {1} N _ {1} = N _ {2};$$ $$N _ {1} = \frac {N _ {2}}{\mu_ {1}};$$ $$m g = N _ {1} + \mu_ {2} N _ {2};$$ $$m g = \frac {N _ {2}}{\mu_ {1}} + \mu_ {2} N _ {2};$$ $$N _ {2} = \frac {\mu_ {1} m g}{1 + \mu_ {1} \mu_ {2}}.$$

The sum of the moments of all external forces about line O is zero:

$$N _ {2} \cos \alpha l + F _ {f 2} \sin \alpha l - m g \cos \alpha x = 0;$$ $$m g \cos \alpha x = N _ {2} \cos \alpha l + F _ {f 2} \sin \alpha l;$$ $$m g \cos \alpha x = N _ {2} \cos \alpha l + \mu_ {2} N _ {2} \sin \alpha l;$$ $$m g \cos \alpha x = N _ {2} l (\cos \alpha + \mu_ {2} \sin \alpha);$$ $$m g \cos \alpha x = \frac {\mu_ {1} m g l}{1 + \mu_ {1} \mu_ {2}} (\cos \alpha + \mu_ {2} \sin \alpha);$$ $$x = \frac {\mu_ {1} m g l}{1 + \mu_ {1} \mu_ {2}} \cdot \frac {(c o s \alpha + \mu_ {2} s i n \alpha)}{m g c o s \alpha};$$ $$x = \frac {\mu_ {1} l}{1 + \mu_ {1} \mu_ {2}} \cdot \frac {(c o s \alpha + \mu_ {2} s i n \alpha)}{c o s \alpha};$$ $$h = x \cos \alpha ;$$ $$h = \frac {\mu_ {1} l}{1 + \mu_ {1} \mu_ {2}} \cdot \frac {(c o s \alpha + \mu_ {2} s i n \alpha)}{c o s \alpha} c o s \alpha ;$$ $$h = \frac {\mu_ {1} l (c o s \alpha + \mu_ {2} s i n \alpha)}{1 + \mu_ {1} \mu_ {2}}.$$

Answer:

$$h = \frac {\mu_ {1} l (c o s \alpha + \mu_ {2} s i n \alpha)}{1 + \mu_ {1} \mu_ {2}}.$$