A student places an object with a mass of 100 g on a disk at a position 0.75 m from the center of the disk. The student starts rotating the disk. When the disk reaches a speed of 0.8 m/s, the object starts to slide off the disk. What is the coefficient of static friction between the object and the disk?
To find the coefficient of static friction between the object and the disk, we need to use the concept of centrifugal force and static friction.
First, let's calculate the centrifugal force acting on the object. The centrifugal force is given by the formula:
F = m * r * ω^2
Where:
F is the centrifugal force,
m is the mass of the object (100 g = 0.1 kg),
r is the distance of the object from the center of the disk (0.75 m), and
ω is the angular velocity of the disk.
Next, we need to determine the minimum static friction required to prevent the object from sliding off the disk. This force is equal to the centrifugal force acting on the object.
Now, let's equate the minimum static friction force to the maximum static friction force (F_s(max) = μ_s * N), where N is the normal force on the object, and μ_s is the coefficient of static friction. In this case, the normal force is equal to the weight of the object (N = m * g), where g is the acceleration due to gravity (9.8 m/s^2).
Now, we have two equations:
F = m * r * ω^2
F_s(max) = μ_s * N
From the equation F_s(max) = μ_s * N, substituting the value of N:
F_s(max) = μ_s * (m * g)
Since the minimum static friction is equal to the maximum static friction required, we can equate the two equations:
F = F_s(max)
m * r * ω^2 = μ_s * (m * g)
Now, we can solve for the coefficient of static friction (μ_s):
μ_s = (m * r * ω^2) / (m * g)
Substituting the given values:
μ_s = (0.1 kg * 0.75 m * (0.8 m/s)^2) / (0.1 kg * 9.8 m/s^2)
Simplifying the equation:
μ_s = (0.6 m^2/s^2) / (0.98 m^2/s^2)
μ_s ≈ 0.612
Therefore, the coefficient of static friction between the object and the disk is approximately 0.612.