Final answer:
To solve for the forces and coefficient of static friction for a ladder against a wall, we analyze the horizontal and vertical components, ensuring they balance due to the system being in equilibrium, and use right-angle trigonometry tied to the ladder's inclination angle.
Step-by-step explanation:
When a ladder rests against a wall, we deal with the equilibrium of forces and the principle of static friction. To find the reaction forces and friction coefficient, we start by identifying the forces acting on the ladder: the weight of the ladder, the normal force from the ground, the frictional force at the base, and the force from the wall. The normal reaction at the base is perpendicular to the ground, while the frictional force is parallel to it. Since the wall is slippery, we can assume there's no frictional force at the top.
To solve for the forces, we begin by analyzing the vertical and horizontal components of the forces. The sum of the forces in the vertical direction must equal zero since the ladder is in a state of equilibrium. This gives us the equation: N = W + w, where W is the weight of the ladder, and w is the weight of the person. Equally, in the horizontal direction, we know that the force from the wall (Fw) is equal to the frictional force (f) at the base of the ladder. This can be expressed by Fw = f.
The static friction force can be determined using the coefficient of static friction (μs) and the normal force (N) using the equation f = μs * N. To find the coefficient of static friction that prevents slipping, we can use μs = f / N once we have f and N from the equilibrium equations. For the angles involved, we use trigonometric ratios. For example, the cosine of the angle between the ladder and the ground helps to figure out the horizontal component of forces while sine helps with the vertical component.