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 0.705, 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 that the crutch can have just before it begins to slip on the floor, we can use the concept of equilibrium.
Let's analyze the forces acting on the crutch. There are two main forces: the force exerted by the person on the crutch (directed upward) and the force exerted by the ground on the crutch (directed downward). These forces can be broken down into two components: the normal force perpendicular to the floor and the frictional force parallel to the floor.
1. Normal force (FN): The normal force equals the weight of the person, since the crutch is supporting their weight. Therefore, FN = mg, where m is the mass of the person and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Frictional force (Ff): The frictional force can be calculated using the coefficient of static friction (μs) and the normal force. Ff = μs * FN.
Now, let's consider the forces acting in the horizontal direction. The horizontal component of the normal force (FN_horizontal) must balance the horizontal component of the frictional force (Ff_horizontal) to maintain equilibrium.
Since the person is standing on crutches, the horizontal component of the normal force cancels out the horizontal component of the frictional force, preventing the crutch from sliding horizontally. Therefore, we can calculate the maximum angle by considering the ratio of these two forces.
The largest angle MAX can be determined using the following equation:
Tan(MAX) = Ff_horizontal / FN_horizontal
Note that the horizontal component of the normal force is equal to FN multiplied by the cosine of the angle MAX, and the horizontal component of the frictional force is equal to Ff multiplied by the sine of the angle MAX.
Tan(MAX) = (μs * FN * sin(MAX)) / (FN * cos(MAX))
Simplifying the equation:
Tan(MAX) = μs * sin(MAX) / cos(MAX)
Using trigonometric identities, we can rewrite the equation:
Tan(MAX) = μs * Tan(MAX)
Now, we can solve for the angle MAX:
MAX = arctan(μs)
Substituting the given coefficient of static friction μs = 0.705:
MAX = arctan(0.705)
Using a calculator, we find that the largest angle MAX is approximately 35.9 degrees.
To determine the largest angle MAX that the crutch can have just before it begins to slip on the floor, we can use the formula for the maximum angle of static friction.
The maximum angle of static friction can be calculated using the equation:
μ = tan(MAX)
Where μ is the coefficient of static friction.
Rearranging the equation, we have:
MAX = arctan(μ)
Given that the coefficient of static friction (μ) between the crutch and the ground is 0.705, we can calculate the largest angle MAX using the arctan function.
MAX = arctan(0.705)
Using a calculator, we find that MAX is approximately 35.89 degrees.
Therefore, the largest angle the crutch can have before it begins to slip on the floor is approximately 35.89 degrees.