Romeo takes a uniform 13.7-m ladder and leans it against the smooth (frictionless) wall of the Capulet residence. The ladder's mass is 23.2 kg and the bottom rests on the ground 3.43 m from the wall. When Romeo, whose mass is 70 kg, gets 88.3 percent of the way to the top, the ladder begins to slip. What is the coefficient of static friction between the ground and the ladder?

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To find the coefficient of static friction between the ground and the ladder, we can apply Newton's second law of motion and use the concept of torque.

First, let's determine the forces acting on the ladder. There are two forces:

1. The weight of the ladder: This force acts downward and can be calculated as the product of the mass of the ladder (23.2 kg) and the acceleration due to gravity (9.8 m/s^2). So, the weight force is W_ladder = (23.2 kg) * (9.8 m/s^2).

2. The normal force exerted by the ground: This force acts perpendicular to the ground and balances the weight of the ladder. It can be calculated as the product of the mass of Romeo (70 kg) and the acceleration due to gravity (9.8 m/s^2), plus the weight force of the ladder. Therefore, the normal force is N = (70 kg) * (9.8 m/s^2) + W_ladder.

Now, let's evaluate the torque acting on the ladder. The torque is caused by the weight of Romeo, the ladder, and any frictional force acting at the base. Since the ladder is about to slip, the frictional force reaches the maximum static friction force (F_friction = μ_s * N, where μ_s is the coefficient of static friction). The torque is given by:

Torque = Frictional force * distance from the base to point of rotation

In this case, the distance from the base to the point where Romeo is standing is (88.3% of ladder height) * (13.7 m) because Romeo is 88.3% of the way up the ladder. So, the torque equation becomes:

Torque = (μ_s * N) * (0.883 * 13.7 m)

Since the ladder is about to slip, the torque must be equal to or greater than zero. Therefore, we can set up the inequality:

Torque ≥ 0

Substituting the values into the torque equation:

(μ_s * N) * (0.883 * 13.7 m) ≥ 0

Simplifying the equation:

(μ_s * N) ≥ 0

Now, we can substitute the value of N in terms of the mass and acceleration due to gravity:

(μ_s * [(70 kg) * (9.8 m/s^2) + W_ladder]) ≥ 0
(μ_s * [(70 kg) * (9.8 m/s^2) + (23.2 kg) * (9.8 m/s^2)]) ≥ 0

Simplifying further:

μ_s * (70 kg) * (9.8 m/s^2 + 23.2 kg * 9.8 m/s^2) ≥ 0

Now, we have an inequality in terms of the coefficient of static friction, μ_s:

μ_s * (70 kg) * (9.8 m/s^2 + 23.2 kg * 9.8 m/s^2) ≥ 0

Solving for μ_s:

μ_s ≥ 0 / [(70 kg) * (9.8 m/s^2 + 23.2 kg * 9.8 m/s^2)]

Calculating the right-hand side of the inequality:

μ_s ≥ 0 / [(70 kg) * (9.8 m/s^2 + 23.2 kg * 9.8 m/s^2)]

μ_s ≥ 0 / [(70 kg) * (9.8 m/s^2 + (23.2 kg) * (9.8 m/s^2))]

Finally, we can calculate the value of μ_s and solve the inequality to obtain the coefficient of static friction between the ground and the ladder.