a penny is placed at the outer edge of a disk(radius =0.150m) that rotates about its axis perpendicular to the plane of the disk at its center. the period of rotation is 1.80s. find the minimum coefficient friction necessary to allow the penny to rotate along with the disk
To find the minimum coefficient of friction necessary for the penny to rotate along with the disk, we need to analyze the forces acting on the penny.
The only force acting horizontally is the friction force between the penny and the disk. The maximum value of the friction force can be expressed as:
friction force (F) = coefficient of friction (μ) * normal force (N)
In this case, the normal force between the penny and the disk is equal to the weight of the penny. This is because the penny is in equilibrium, meaning that the net force acting on it is zero. The weight of the penny (W) can be calculated using the formula:
W = mass (m) * gravitational acceleration (g)
Let's calculate the weight of the penny first.
The mass of a penny is approximately 2.5 grams, or 0.0025 kg (1 gram = 0.001 kg).
The gravitational acceleration is approximately 9.8 m/s^2.
W = 0.0025 kg * 9.8 m/s^2
W ≈ 0.0245 N
Now, to find the minimum coefficient of friction (μ_min), we need to consider the centripetal force required to keep the penny moving along with the disk.
The centripetal force is given by the formula:
centripetal force (Fc) = mass (m) * angular velocity (ω)^2 * radius (r)
The angular velocity (ω) can be calculated using the formula:
ω = 2π / T
where T is the period of rotation.
Let's calculate the angular velocity first.
ω = 2π / 1.80 s
ω ≈ 3.487 rad/s
Now, we can calculate the centripetal force.
Fc = 0.0025 kg * (3.487 rad/s)^2 * 0.150 m
Fc ≈ 0.0182 N
Since the friction force (F) must be equal to or greater than the centripetal force (Fc) for the penny to rotate along with the disk, we can set up the following equation:
F = μ * N ≥ Fc
Substituting the values we calculated, we get:
μ * 0.0245 N ≥ 0.0182 N
Finally, solving for the minimum coefficient of friction (μ_min):
μ_min ≥ 0.0182 N / 0.0245 N
μ_min ≥ 0.744
Therefore, the minimum coefficient of friction necessary to allow the penny to rotate along with the disk is approximately 0.744.