physic problem help

Question

A 50 kg ladder leans against a frictionless at a 60° angle. If the ladder is 2.5 m long what is the force exerted by the wall, in Newtons, on the ladder?

Answer 1

This is a problem that is solved using the concept of Torques. Think of the ladder as a lever that is fixed at where it contacts the floor as a pivot point, and is acted upon by three forces:

F₁ = the force of gravity that acts at the Center of Mass of the ladder, which is 1.25 m from the pivot point

F₂ = the force that the wall applies at the top of the ladder, which is 2.5 m from the pivot point.

F₃ = the force that the floor applies against the ladder by the floor.

For each force, we must determine the angle it makes with the ladder. We do this because the torque formula is τ = F ⋅ D ⋅ sin(θ), where θ is the angle between the direction of the force and the direction of the ladder. The solution to this problem will require that each torque is calculated. The sum will be zero because the ladder is not rotating:

Σ τ = F₁ ⋅ D₁ ⋅ sin(θ₁) + F₂ ⋅ D₂ ⋅ sin(θ₂) + F₃ ⋅ D₃ ⋅ sin(θ₃) = 0

The force of gravity is directed straight down at the center of the ladder, so θ₁ is 60^o, D₁ is 1.25m, and F₁ is mg = 50g = 50*9.81 = 490.5. This force acts clockwise, so we assign a negative value to it.

The ladder is said to rest against the wall at a 60^o angle, but it is not clear to me what the angle is measured against. I assume it means that the angle between the wall and the ladder is 60^o, so the angle θ₂ between the force applied by the wall against the ladder is 30^o. D₂ is 2.5m and F₂ is the value we wish to calculate. Additionally, this force acts counter-clockwise, so we assign it a positive value.

The floor applies a force on the ladder at the pivot point, so D₃ = 0, so its torque is 0. In this case, the angle θ₃ does not matter, and nor does the value of F₃.

We have all the info that we need:

Σ τ = F₁ ⋅ D₁ ⋅ sin(θ₁) + F₂ ⋅ D₂ ⋅ sin(θ₂) + F₃ ⋅ D₃ ⋅ sin(θ₃)

0 = -(490.5) (1.25) sin(60^o) + F₂ (2.5) sin(30^o) + 0

Solving for F₂ leads to

F₂ = [ 490.5 * 1.25 * sqrt(2) / 2 ] / [ 2.5 * 1/2] = 347 N