A coin rests 18.0 cm from the center of a turntable. The coefficient of static friction between the coin and turntable surface is 0.390. The turntable starts from rest at t = 0 and rotates with a constant angular acceleration of 0.850 rad/s2. (b) At what angular speed (ω) will the coin start to slip?
To determine the angular speed at which the coin will start to slip, we need to consider the forces acting on the coin and analyze the conditions for slipping to occur.
First, let's calculate the static frictional force between the coin and the turntable surface. We can use the formula:
Static frictional force (fs) = coefficient of static friction (μ) * normal force (N)
The normal force (N) on the coin is equal to its weight, which can be calculated as:
N = mass (m) * acceleration due to gravity (g)
Now, we need to express the mass of the coin in terms of its position from the center of the turntable. Since the coin is resting at a distance of 18.0 cm from the center, we can use the formula:
mass (m) = (distance from the center / radius of the turntable) * total mass of the turntable
Given that the coefficient of static friction (μ) is 0.390, the acceleration due to gravity (g) is approximately 9.8 m/s², and the total mass of the turntable is unknown, we can set up the following equations:
fs = μ * N
N = m * g
m = (distance from the center / radius) * total mass of the turntable
Substituting the equations, we have:
fs = μ * m * g
Now, the torque (τ) acting on the coin is given by:
τ = fs * distance from the center
The torque τ can also be expressed as:
τ = moment of inertia (I) * angular acceleration (α)
For a point mass like a coin, the moment of inertia (I) is given by:
I = m * (distance from the center)²
Equating the two expressions for torque, we get:
m * (distance from the center)² * α = μ * m * g * distance from the center
Simplifying,
(α * distance from the center) = μ * g
Now, we can solve for the angular acceleration (α) at which the coin will start to slip:
α = μ * g / distance from the center
Given that the angular acceleration is 0.850 rad/s², the distance from the center is 18.0 cm (or 0.18 m), and the acceleration due to gravity (g) is approximately 9.8 m/s², we can plug in the values:
0.850 rad/s² = (0.390 * 9.8 m/s²) / 0.18 m
Simplifying the expression, we find:
0.850 rad/s² = 2.13 rad/s²
Thus, the angular speed (ω) at which the coin will start to slip is approximately 2.13 rad/s.