A uniform ladder of length L rests against a smooth, vertical wall as shown in the figure below. If the mass of the ladder is m and the coefficient of static friction between the ladder and the ground is muks=0.4, find the minimum angle theta at which the ladder does not slip.
To find the minimum angle theta at which the ladder does not slip, we need to consider the forces acting on the ladder.
Let's break down the forces:
1. Weight of the ladder: The weight of the ladder acts downward and can be calculated as W = m * g, where m is the mass of the ladder and g is the acceleration due to gravity.
2. Normal force: The normal force acts perpendicular to the surface of contact between the ladder and the ground. In this case, since the ladder is against a smooth vertical wall, the normal force is acting perpendicular to the ground.
3. Friction force: The friction force acts parallel to the ground and opposes the tendency of the ladder to slip. The maximum static friction force can be calculated as F_friction = muks * Normal force, where muks is the coefficient of static friction.
4. Horizontal component of the weight: This component acts parallel to the ground and tends to make the ladder slide downward. This force can be calculated as F_horizontal = W * sin(theta).
To find the minimum angle theta at which the ladder does not slip, we need to find the point where the friction force equals the horizontal component of the weight.
We have:
F_friction = F_horizontal
Substituting the values, we get:
muks * Normal force = W * sin(theta)
Since Normal force is equal to the weight of the ladder (Normal force = W), we can rewrite the equation as:
muks * W = W * sin(theta)
Dividing both sides by W, we get:
muks = sin(theta)
Now, we need to find the inverse sine (also known as arcsine) of muks to calculate theta. Using the following equation:
theta = arcsin(muks)
Substituting the value of muks as 0.4, we can calculate the minimum angle theta at which the ladder does not slip:
theta = arcsin(0.4)
Using a calculator, we find that theta ≈ 23.58 degrees.
Therefore, the minimum angle theta at which the ladder does not slip is approximately 23.58 degrees.