A woman is standing on the middle of a ladder that leans up against a building, with an angle of 65 degrees from the ground. The wall is very slippery and the ladder is very light. What is the minimum coefficient of friction needed between the ground and the ladder so that the ladder doesn't fall down?
I bet summing moments around any point will solve this.
To determine the minimum coefficient of friction needed between the ground and the ladder so that it doesn't fall down, we need to analyze the forces acting on the ladder.
First, let's draw a diagram to visualize the situation.
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Ladder |_
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| point of contact with ground
In this diagram, the vertical line represents the wall, the horizontal line represents the ground, and the diagonal line represents the ladder.
Now, let's identify the forces acting on the ladder.
1. Weight (W): This force acts vertically downwards from the center of gravity of the ladder. The weight 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 (N): This force acts perpendicular to the ground and opposes the weight of the ladder. It can be calculated as N = W * cos(θ), where θ is the angle between the ladder and the ground.
3. Friction force (f): This force acts parallel to the ground and opposes the tendency of the ladder to slide down. It can be calculated as f = μ * N, where μ is the coefficient of friction.
For the ladder to remain in equilibrium and not slide down, the friction force (f) must be greater than or equal to the component of the weight force (W) parallel to the ground. This component can be calculated as W_parallel = W * sin(θ).
So, the inequality becomes:
f >= W_parallel
Substituting the values, we get:
μ * N >= W_parallel
μ * W * cos(θ) >= W * sin(θ) (substituting N = W * cos(θ))
μ * cos(θ) >= sin(θ)
μ >= tan(θ)
Now, substituting the given angle (θ) of 65 degrees:
μ >= tan(65 degrees)
Calculating this value, we get:
μ >= 2.14
Therefore, the minimum coefficient of friction needed between the ground and the ladder is approximately 2.14.