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 0.94, determine the largest angle èMAX that the crutch can have just before it begins to slip on the floor.
To determine the largest angle èMAX, we need to consider the conditions for an object just about to slip or move. In this case, the conditions for an object just about to slip involve the maximum static friction force that can act on it.
Let's break down the forces acting on the crutch:
1. The weight of the person acting vertically downward.
2. The normal force exerted by the ground, perpendicular to the ground.
3. The static friction force acting parallel to the ground, opposing the tendency of the crutch to slip.
Since the crutch is about to slip, the maximum static friction force (Ff_max) is at its limit and can be calculated using the equation:
Ff_max = μs * N
where μs is the coefficient of static friction between the crutch and the ground, and N is the magnitude of the normal force.
From the given information, we know that the coefficient of static friction is 0.94, which means μs = 0.94.
To find the normal force, we need to consider the forces acting perpendicular to the ground. Since there are no other vertical forces mentioned, the normal force (N) is equal in magnitude to the weight of the person.
To calculate N, we need to know the weight of the person (W). Assuming a mass of m kg, the weight can be determined using the formula:
W = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Once we have W, we can substitute it in for N:
N = W
Now that we have both the μs and N, we can find Ff_max.
Ff_max = 0.94 * N
Next, we need to analyze the forces acting on the crutch along the inclined direction (vector perpendicular to the angle èMAX).
Let's consider the sum of forces acting in the perpendicular direction (y-direction):
N * cos(èMAX) - W = 0
Simplifying this equation, we get:
N * cos(èMAX) = W
Since we already know that N = W, we can substitute it in:
W * cos(èMAX) = W
Dividing the equation by W:
cos(èMAX) = 1
We know that the maximum value of cos(èMAX) is 1.
Therefore,
cos(èMAX) = 1
Taking the inverse cosine of both sides, we find:
èMAX = cos^(-1)(1)
Since the cosine of 0 degrees is 1, we have:
èMAX = 0 degrees
So, the largest angle èMAX that the crutch can have just before it begins to slip on the floor is 0 degrees, which means the crutch needs to be perfectly horizontal to avoid slipping.