A uniform ladder of mass, m, and length, l, leans at an angle, theta, against a frictionless wall.

Determine a formula for the minimum angle at which the ladder will not slip.

To determine the minimum angle at which the ladder will not slip, we need to consider the forces acting on the ladder. There are two main forces to consider: the weight of the ladder acting downwards and the normal force exerted by the wall on the ladder perpendicular to the wall.

Let's break it down step by step:

1. Draw a free body diagram: Start by drawing a diagram of the ladder leaning against the wall. Mark the weight of the ladder pointing downwards and the normal force exerted by the wall perpendicular to the wall.

2. Resolve the weight of the ladder: The weight of the ladder can be resolved into two components: one parallel to the wall (mg sinθ) and one perpendicular to the wall (mg cosθ), where θ is the angle at which the ladder is inclined.

3. Determine the friction force: Since the wall is frictionless, there is no friction force acting on the ladder parallel to the wall.

4. Use the condition for no slipping: In order for the ladder not to slip, the force of friction between the ladder and the ground (which is responsible for preventing slipping) must be greater than or equal to the parallel component of the weight of the ladder.

5. Express the condition mathematically: The condition for no slipping can be expressed as follows:
Friction force ≥ mg sinθ

6. Calculate the friction force: The friction force can be calculated as the product of the coefficient of static friction (μs) and the normal force (mg cosθ).

7. Substitute values and solve for the minimum angle: Substitute the expression for the friction force into the inequality condition for no slipping and solve for θ.

μs × mg cosθ ≥ mg sinθ

Simplifying the equation:
μs ≥ tanθ

Taking inverse tangent on both sides:
θ ≥ arctan(μs)

Therefore, the minimum angle at which the ladder will not slip is given by:
θmin = arctan(μs)

where μs is the coefficient of static friction between the ladder and the ground.

By using this formula, you can determine the minimum angle at which the ladder will not slip.

To determine the minimum angle at which the ladder will not slip, we need to consider the forces acting on the ladder.

Let's analyze the forces involved:

1. Weight (W): The weight of the ladder acts vertically downwards from its center of mass. The magnitude of the weight can be calculated as W = mg, where m is the mass of the ladder and g is the acceleration due to gravity (approximately 9.8 m/s²).

2. Normal Force (N): The force exerted by the wall on the ladder perpendicular to the wall is the normal force. At the minimum angle where the ladder does not slip, the vertical component of the normal force should balance the weight. So, N = W = mg.

3. Frictional Force (F): The frictional force opposes the tendency of the ladder to slip. It acts parallel to the wall and points downward.

Now let's consider the forces causing rotation:

4. Torque due to weight: The weight of the ladder creates a torque that tends to make it slip. The torque (τ) can be calculated as τ = W × d, where d is the distance from the center of mass of the ladder to the point of contact with the ground.

5. Torque due to friction: The frictional force also creates a torque that tends to rotate the ladder clockwise. The torque (τ') can be calculated as τ' = F × d, where F is the frictional force and d is the distance from the center of mass to the point of contact with the ground.

To prevent slipping, the frictional force (F) must counteract the torque due to the weight (τ). Therefore, F × d = W × d.

Substituting the values of W and N, we have:

F × d = W × d
F = W

Since N = W, the normal force equals the weight. Hence, there must be a minimum angle at which the normal force does not exceed the maximum static frictional force. Mathematically, this condition can be expressed as:

N ≤ μs × N

Where μs is the coefficient of static friction.

Substituting N = W = mg:

mg ≤ μs × mg

Canceling out mass and g:

1 ≤ μs

So, the minimum angle at which the ladder will not slip is such that the coefficient of static friction (μs) is greater than or equal to 1. In other words, at any angle where the coefficient of static friction is greater than or equal to 1, the ladder will not slip.