Question

A uniform ladder of length l and mass m1 rests against a frictionless wall. the ladder makes an angle θ with the horizontal.(a) find the horizontal and vertical forces the ground exerts on the base of the ladder when a firefighter of mass m2 is a distance x from the bottom. (answer using m1, m2, θ, gravity g, l, and x as necessary.) horizontal horizontal force= vertical force= (b) if the ladder is just on the verge of slipping when the firefighter is a distance d from the bottom, what is the coefficient of static friction between ladder and ground? μ =

Answer 1

Balancing horizontal and vertical forces,

N = (m1 + m2)g and

F = N'
To find F, taking moments of forces about B,

N'Lsinθ = m1g*(L/2)cosθ + m2g * xcotθ

=> N' = [(1/2)m1 + (x/L)m2 cosecθ] gcotθ

=> F = N' = [(1/2)m1 + (x/L)m2 cosecθ] gcotθ
When the ladder is on the verge of sliding,

x = d and F = μN = μ(m1 + m2)g

=> μ(m1 + m2)g = [(1/2)m1 + (d/L)m2 cosecθ] gcotθ

=> μ = [(1/2)m1 + (d/L)m2 cosecθ] cotθ / (m1 + m2).