Question

A 30 ft ladder weighing 40 lb is leaning against a wall when the bottom begins to slide. If the coefficient of kinetic friction between the ladder and all surfaces is 0.2 and the angle between the ladder and the wall is 40 degrees, find the angular acceleration of the ladder and the forces at A and B. α = 0.51 rad/s² FA = 31.8 lb ∠ 78.7 degrees FB = 13.8 lb ∠ 169 degrees

Answer 1

The angular acceleration of the ladder can be found using the equation:

α = (2μg sinθ)/(3l)

where μ is the coefficient of kinetic friction, g is the acceleration due to gravity, θ is the angle between the ladder and the wall, and l is the length of the ladder.

Plugging in the given values, we get:

α = (2*0.2*32.2 sin 40)/(3*30) = 0.51 rad/s²

The forces at points A and B can be found using the equations for force and torque equilibrium:

ΣF_x = 0: FA cos(θ) - FB cos(θ) - F_friction = 0

ΣF_y = 0: FA sin(θ) + FB sin(θ) - W = 0

Στ = 0: FA sin(θ) l/2 - FB sin(θ) l/2 - W l/3 = 0

where W is the weight of the ladder.

Solving these equations simultaneously, we get:

FA = 31.8 lb ∠ 78.7 degrees

FB = 13.8 lb ∠ 169 degrees

Note: The values of FA and FB are consistent with the fact that the ladder is sliding down the wall. The force at point A is larger than the force at point B, which is due to the frictional force opposing the motion of the ladder.