A ladder ABC of length 20m is leaning against a wall at an angle from the horizontal. There is a roller at the top of the ladder that rests against the vertical wall. The coefficient of friction between the ladder and the horizontal surface at A is 0.5. Determine:

(a) the reactions at A and C;
(b) the smallest angle that the ladder alone can make with the wall without slipping if the ladder has a mass of 18kg.

(a) sum of torques = zero so mg(10 cos(theta) = Fc(20 sin(theta))

(mass is assumed at center)

sum of forces = zero so Fay = mg (really the normal component of Fa).
and Fc = Fax = (mu)Fay

So Fc = (mu)mg (if you assume the 18kg given in part b = .5*18*9.8)
Fa has x and y components, Fax = Fc =(mu)mg, Fay = mg. By pythagorean Fa = sqrt(mg^2 + (mu)mg^2)
(b) For no slipping the torques must be zero:
mg(10cos(theta))= (mu) mg (20 sin(theta))
(oddly the mass is not needed for part b)
tan(theta) = 10/(.5 * 20)
theta = 45o

To solve this problem, we can use the concepts of equilibrium and the conditions for a ladder not to slip. Let's break down the steps to determine the reactions at points A and C and find the smallest angle that the ladder can make with the wall without slipping.

(a) Reactions at A and C:

1. Draw a free-body diagram of the ladder, showing all the forces acting on it.

- Label the weight of the ladder, W, acting downward at its center of mass.
- Draw the normal force, N, acting vertically upward at point A.
- Show the friction force, F, acting horizontally to the right at point A.
- Finally, indicate the reaction force, R, acting vertically downward at point C.

2. Apply the conditions for equilibrium:

- The sum of all forces in the x-direction (horizontal) should equal zero.
F - friction force = 0
F = friction force

- The sum of all forces in the y-direction (vertical) should equal zero.
N - W - R = 0
N - R = W
N = R + W

3. Substitute the given information into the equations:

- The coefficient of friction, μ, is given as 0.5.
- The weight of the ladder, W, is calculated by multiplying its mass (18 kg) by the acceleration due to gravity (9.8 m/s^2).
W = 18 kg * 9.8 m/s^2

4. Solve the equations simultaneously using the substitution method or any other preferred method. This will give you the values of the reactions at points A and C.

(b) Smallest angle without slipping:

1. The condition for a ladder not to slip is when the force of friction at point A is equal to or greater than the force component parallel to the surface.

The force component parallel to the surface is given by F_parallel = N * sin(θ),
where θ is the angle between the ladder and the horizontal surface.

2. Determine the maximum force of friction using F_max = μ * N.

3. Set up the inequality:

F_parallel ≤ F_max
N * sin(θ) ≤ μ * N

4. Simplify the inequality:

sin(θ) ≤ μ

5. Substitute the given coefficient of friction, μ, and solve for the smallest angle θ that satisfies the inequality.

θ = sin^(-1)(μ)

Note: The angle θ in this case represents the angle between the ladder and the horizontal surface, not the angle of elevation.

By following these steps, you should be able to determine the reactions at points A and C and find the smallest angle that the ladder can make with the wall without slipping.