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 1.21, determine the largest angle θMAX that the crutch can have just before it begins to slip on the floor.
A steel ball is dropped from a diving platform (with an initial velocity of zero). Use the approximate value of g = 10 m/s2.
(a) Through what distance does the ball fall in the first 2.0 s of its flight?
To determine the largest angle θMAX that the crutch can have just before it begins to slip on the floor, we can use the concept of equilibrium and the coefficient of static friction.
Let's break down the forces acting on the crutch:
1. Weight (mg): The force exerted by gravity on the crutch, acting vertically downward. Its magnitude can be calculated as the product of the mass (m) of the person and the acceleration due to gravity (g).
2. Normal force (N): The force exerted by the ground on the crutch, acting perpendicular to the surface. Since the person is standing on crutches, the normal force cancels out the weight of the person's body, resulting in equilibrium.
3. Friction force (f): The force exerted by the ground on the crutch, acting parallel to the surface. The magnitude of this force depends on the coefficient of static friction (μs) and the normal force (N).
Now, let's analyze the forces in terms of their components:
1. Vertical forces: The weight (mg) and the normal force (N) have equal magnitudes but opposite directions. Therefore, their vertical components cancel each other out.
2. Horizontal forces: The friction force (f) opposes any tendency of the crutch to slip. It acts parallel to the surface, opposite to the horizontal component of the weight (mg sinθ), where θ is the angle between the crutch and the ground.
Using Newton's second law, we can write the equation of equilibrium in the horizontal direction:
f - mg sinθ = 0
Rearranging the equation:
f = mg sinθ
The maximum value of static friction is given by:
fMAX = μsN
Since the crutch is on the verge of slipping, the maximum friction force fMAX is equal to the force of friction f.
Therefore, we have:
mg sinθMAX = μsN
The normal force N is equal to the vertical component of the weight:
N = mg cosθ
Substituting the expression for N, we get:
mg sinθMAX = μs(mg cosθ)
Canceling out the mass (m) on both sides, we have:
sinθMAX = μs cosθ
Now, divide both sides by cosθ:
tanθMAX = μs
Finally, take the inverse tangent (arctan) of both sides to find the largest angle θMAX:
θMAX = arctan(μs)
Substituting the given coefficient of static friction (μs = 1.21) into the equation, we can calculate the largest angle:
θMAX = arctan(1.21)
Using a scientific calculator or mathematical software, we can find:
θMAX ≈ 50.67 degrees
Therefore, the largest angle θMAX that the crutch can have just before it begins to slip on the floor is approximately 50.67 degrees.