Question

A ladder leans against a frictionless wall. It has a mass of M and is L meters long. The angle it makes with the ground is θ. In terms of the length, mass, angle, normal forces and the coefficient of static friction, write three equations for this static equilibrium case.

Answer 1

In static equilibrium, a ladder resting against a frictionless wall has three equilibrium conditions: vertical forces must sum to zero, horizontal forces must sum to zero, and total torque around any axis must be zero.

Step-by-step explanation:

When a ladder rests against a frictionless wall in static equilibrium, we apply Newton's laws to describe the forces acting on it. There are three crucial equations that come into play here:

1. The sum of forces in the vertical direction (y-axis) must be zero, which is written as ΣFy = N - M*g = 0, where N is the normal force exerted by the ground, M is the mass of the ladder, and g is the acceleration due to gravity.
2. Similarly, the sum of forces in the horizontal direction (x-axis) should also be zero: ΣFx = F - f = 0, where F is the normal force exerted by the wall, and f is the static friction force at the base of the ladder.
3. The third equation comes from the torque (τ) consideration, stating that the sum of torques around any axis must be zero: Στ = (N * L/2 * cos(θ)) - (M * g * L/2 * sin(θ)) - (f * L * sin(θ)) = 0. This takes into account that the center of mass of the ladder is midway along its length, and the force of gravity acts downward at this center of mass. For the ladder not to slip, the static friction must be enough to balance the torques.

The coefficient of static friction (μs) plays a vital role, as it determines the amount of frictional force that can be exerted without slipping, and it is defined by the equation f = μs * N.