Question

A ladder 7.50 m long is leaning against a smooth (frictionless) wall at a point 7.0 m above the ground. The ladder weighs 200 N and an 800-N painter stands two thirds of the way up the ladder. What frictional force must act on the bottom of the ladder to prevent it from slipping for the painter to be safe?

Answer 1

$$FL\sin{\theta}=\frac{1}{2}WL\cos{\theta}+\frac{2}{3}W_pL\cos{\theta}$$

$$\theta=\arccos{\frac{7}{7.5}}=21\degree$$

$$F\tan{21}=\frac{1}{2}W+\frac{2}{3}W_p\\ F\tan{21}=\frac{1}{2}200+\frac{2}{3}800\\F=1650\ N$$