The person in the drawing is standing on crutches. Assume that the force exerted on each crutch by the ground is directed along the crutch, as the force vectors in the drawing indicate. If the coefficient of static friction between a crutch and the ground is 0.793, determine the largest angle MAX that the crutch can have just before it begins to slip on the floor.
To determine the largest angle MAX at which the crutch can begin to slip on the floor, we need to analyze the forces acting on the crutch.
Let's consider the forces involved:
1. Weight (W): The force exerted by the person's body on the crutch due to gravity.
2. Normal force (N): The force exerted by the ground on the crutch perpendicular to the ground.
3. Friction force (Ff): The force exerted parallel to the ground by the static friction between the crutch and the ground.
Based on the information given, the force exerted on each crutch by the ground is directed along the crutch. Therefore, we can resolve the forces into components along and perpendicular to the crutch.
Let's break down the forces into their components:
1. Weight (W): The weight of the person can be divided into two components:
- W ⊥: The component of weight perpendicular to the crutch.
- W ||: The component of weight parallel to the crutch.
2. Normal force (N): The normal force is divided into two components:
- N ⊥: The component of normal force perpendicular to the crutch.
- N ||: The component of normal force parallel to the crutch.
3. Friction force (Ff): The friction force can also be divided into two components:
- Ff ⊥: The component of friction force perpendicular to the crutch.
- Ff ||: The component of friction force parallel to the crutch.
Now let's analyze the forces acting parallel to the crutch:
Equilibrium in the parallel direction:
Ff || = W ||
Since the crutch is on the verge of slipping, the maximum force of static friction (Ff_max) can be determined using the equation:
Ff_max = μs * N ||
where μs is the coefficient of static friction.
To find N ||, we can analyze the forces acting perpendicular to the crutch:
Equilibrium in the perpendicular direction:
N ⊥ = W ⊥
To determine N ||, we can use the following relationship:
N || = N ⊥ * tan(MAX)
where MAX is the largest angle the crutch can have just before slipping.
By substituting N || into the equation for Ff_max, we get:
Ff_max = μs * N ⊥ * tan(MAX)
Since N ⊥ = W ⊥, we can rewrite the equation as:
Ff_max = μs * W ⊥ * tan(MAX)
Now, rearranging the equation to solve for MAX, we have:
tan(MAX) = Ff_max / (μs * W ⊥)
Finally, taking the inverse tangent of both sides, the largest angle MAX can be found.
Note: To find W ⊥, we need to know the mass (m) of the person and the acceleration due to gravity (g).
W ⊥ = m * g * cos(θ)
where θ is the angle the crutch makes with the vertical.
After calculating W ⊥, you can use the given coefficient of static friction (μs) and substitute the values into the equation to find the largest angle MAX.