A coin rests 18.0 cm from the center of a turntable. The coefficient of static friction between the coin and turntable surface is 0.420. The turntable starts from rest at t = 0 and rotates with a constant angular acceleration of 0.650 rad/s2. (a) After 3.00 s, what is the angular velocity of the turntable? b) (b) At what speed will the coin start to slip? c) After what period of time will the coin start to slip on the turntable?
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To solve these problems, we need to understand the relationship between angular velocity, angular acceleration, and linear acceleration.
a) To find the angular velocity of the turntable after 3 seconds, we can use the kinematic equation for rotational motion:
ω = ω0 + αt
where ω is the final angular velocity, ω0 is the initial angular velocity, α is the angular acceleration, and t is the time.
Given:
ω0 = 0 rad/s (since the turntable starts from rest)
α = 0.650 rad/s^2
t = 3.00 s
Plugging in these values, we can calculate the angular velocity:
ω = 0 + (0.650 rad/s^2) * (3.00 s)
= 1.95 rad/s
So, after 3 seconds, the angular velocity of the turntable is 1.95 rad/s.
b) The coin will start to slip when the maximum static friction force between the coin and the turntable is exceeded. The maximum static friction force is given by:
f_max = μ_s * N
where μ_s is the coefficient of static friction and N is the normal force.
In this case, the normal force is equal to the weight of the coin since the coin is resting on the turntable. The weight of the coin is calculated using the formula:
Weight = mass * acceleration due to gravity
Given:
Coefficient of static friction (μ_s) = 0.420
Distance of the coin from the center of the turntable (r) = 18.0 cm = 0.18 m
Mass of the coin = m (unknown)
To calculate the mass of the coin, we need to know its weight. Assuming the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight:
Weight = m * 9.8 m/s^2
Now, we can calculate the normal force:
N = Weight = m * 9.8 m/s^2
Substituting this value into the expression for the maximum static friction force, we have:
f_max = μ_s * N
= μ_s * (m * 9.8 m/s^2)
The speed at which the coin will start to slip can be related to the angular velocity by:
v = r * ω
where v is the linear velocity, r is the distance from the center of the turntable, and ω is the angular velocity.
When the coin starts to slip, the static friction force is equal to the kinetic friction force:
f_kinetic = μ_k * N
where μ_k is the coefficient of kinetic friction. We can solve for this by setting the two expressions for the friction force equal to each other:
f_max = f_kinetic
Now, we can substitute the expressions for the friction forces:
μ_s * (m * 9.8 m/s^2) = μ_k * (m * 9.8 m/s^2)
Simplifying the equation, we find:
μ_s = μ_k
Therefore, the speed at which the coin starts to slip is not dependent on the mass of the coin.
c) To find the period of time after which the coin starts to slip, we can relate the angular acceleration to the speed at which the coin starts to slip:
α = (v^2 - v_0^2) / (2r)
where v is the linear velocity and v_0 is the initial linear velocity.
Given:
Coefficient of static friction (μ_s) = 0.420
Distance of the coin from the center of the turntable (r) = 18.0 cm = 0.18 m
Angular acceleration (α) = 0.650 rad/s^2
We can calculate the initial linear velocity using the relationship between linear velocity and angular velocity:
v_0 = r * ω_0
= 0.18 m * 0 rad/s
= 0 m/s
Plugging in these values, we can rearrange the equation to solve for the period of time:
α = (v^2 - v_0^2) / (2r)
0.650 rad/s^2 = (v^2 - 0^2) / (2 * 0.18 m)
Simplifying the equation, we find:
v^2 = 0.650 rad/s^2 * 2 * 0.18 m
= 0.234 rad/s
Taking the square root of both sides:
v = sqrt(0.234 rad/s)
Therefore, the coin will start to slip on the turntable after a period of time when its speed is equal to approximately 0.484 m/s.