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 0.937, determine the largest angle MAX that the crutch can have just before it begins to slip on the floor.
F along crutch
angle T from vertical (I assume)
F normal = F cos T
Fmax friction = .937 *Fnormal
= .937 F cos T
F along floor = F sin T
so for max
F sin T = .937 F cos T
Tan T = .937
T = 43.1 degrees from vertical
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 static friction.
Static friction is the force that prevents an object from moving when a force is applied to it and it has not yet overcome this force. It can be calculated using the equation:
fs ≤ μs * N
where fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force between the object and the surface it is in contact with.
In this scenario, the force of static friction acts along the crutch, perpendicular to the ground. This force prevents the crutch from slipping on the floor. The force exerted by the person on the crutches can be decomposed into two components: one along the crutch and one perpendicular to it.
Since the force exerted on the crutch by the ground is directed along the crutch, the vertical component of the force exerted by the person is equal to the normal force (N), which is given by:
N = mg
where m is the mass of the person and g is the acceleration due to gravity.
To determine the largest angle MAX, we need to find the angle where the force of static friction is at its maximum. This occurs when the force exerted along the crutch by the person is equal in magnitude and opposite in direction to the force of static friction.
Now, let's break down the components of the force exerted by the person along and perpendicular to the crutch.
The force along the crutch is given by:
F_parallel = F * cosθ
where F is the total force exerted by the person and θ is the angle between the force and the crutch.
The force perpendicular to the crutch is given by:
F_perpendicular = F * sinθ
To reach the maximum angle MAX, the force of static friction (fs) must be at its maximum value, which is fs = μs * N.
Now, let's set up the equation using the components of the force exerted by the person:
fs = F_parallel
μs * N = F * cosθ
Substituting N = mg and F_parallel = F * cosθ into the equation, we have:
μs * mg = F * cosθ
Solving for θ, we get:
θ = cos^(-1)(μs * g)
Substituting the given value of the coefficient of static friction (μs = 0.937) and the acceleration due to gravity (g = 9.8 m/s^2), we can calculate the largest angle MAX:
θ = cos^(-1)(0.937 * 9.8)
Evaluating this expression, we find:
θ ≈ 21.3 degrees
Therefore, the largest angle MAX that the crutch can have just before it begins to slip on the floor is approximately 21.3 degrees.