What's the minimum coefficient of friction between the ladder (of length d) and the ground will keep it from slipping? (consider the wall frictionless) (θ=56.5 degrees)

I think the formula is supposed to be mg L/2 sin y, but it does not provide a length, nor a mass. How do I solve this problem?

To solve this problem, you can use the concept of rotational equilibrium. When the ladder is on the verge of slipping, the net torque acting on it will be zero.

First, let's determine the forces acting on the ladder. There are three forces:
1. The weight of the ladder acting downward, which can be represented as mg (where m is the mass of the ladder and g is the acceleration due to gravity).
2. The normal force acting perpendicular to the ground, which counteracts the weight and is equal to mg.
3. The friction force acting parallel to the ground, which prevents the ladder from slipping.

Since the wall is frictionless, the friction force between the ladder and the wall is zero. Thus, the only force that can prevent the ladder from slipping is the friction between the ladder and the ground.

Now, let's analyze the torques acting on the ladder. The torque due to the weight of the ladder is zero because it acts at the center of mass. Therefore, the only torque is due to the friction force.

The formula you mentioned, mgd/2 sin(theta), is the torque about the center of mass of the ladder due to the weight (which is not relevant in this case since it does not cause any rotation). Instead, we need to consider the torque about the bottom of the ladder.

The torque due to the friction force is given by the formula r * F * sin(theta), where r is the distance from the bottom of the ladder to the point where the friction force is applied (which is L/2, where L is the length of the ladder) and F is the friction force.

Since the net torque is zero, we can write the equation as r * F * sin(theta) = 0. Therefore, to prevent the ladder from slipping, F must be greater than or equal to zero.

Since the minimum coefficient of friction between the ladder and the ground is required, we can rewrite this as mu * N * L/2 * sin(theta) >= 0, where mu is the coefficient of friction and N is the normal force (equal to mg).

Now, set the inequality equal to zero to find the minimum coefficient of friction:
mu * mg * L/2 * sin(theta) >= 0

Dividing by mgL/2 sin(theta) (which is positive), we get:
mu >= 0 / mgL/2sin(theta)
mu >= 0

Therefore, the minimum coefficient of friction between the ladder and the ground to prevent it from slipping is zero.