Final answer:
To determine the forces on a ladder leaned against a wall, apply static equilibrium principles to balance the ladder's and person's weights with the reaction forces at the base and top of the ladder. Use torque formulas and trigonometric relationships involving the center of mass, normal force, and friction to perform the calculations.
Step-by-step explanation:
To calculate the magnitudes of the forces on an extension ladder leaned against a wall, one can apply the principles of static equilibrium. The ladder's weight and the weight of the person on it create forces that must be countered by the reaction forces at the base and the point where the ladder touches the wall.
Firstly, we need to understand the forces acting on the ladder: the ladder's weight (WL), the person's weight (WP), the normal force at the base (N), and the friction force at the base (f). The ladder makes contact with the frictionless rain gutter only as a normal force (no horizontal force due to frictionlessness), so the force at the top would be a vertical force only.
The ladder is in a state of static equilibrium, meaning that the sum of all forces and the sum of all torques (moments) around any point is zero. This results in two primary conditions: ΣFx = 0 and ΣFy = 0 for forces, and Στ = 0 for torques. Considering that the ladder and the floor form a right triangle, we can use trigonometric relationships to solve for these forces.
In practice, these calculations can be quite involved, requiring an understanding of physics principles such as the center of mass, torque, normal force, and friction. The formula for torque (τ) is the force multiplied by the distance from the pivot point, and it is applied perpendicularly.