A 9.60 g coin is placed 10.7 cm from the center of a turntable. The coin has a coefficient of static friction of 0.80 and a coefficient of kinetic friction of 0.60 with the turntable. The turntable's angular velocity slowly speeds up until the coin starts to slide off. Assuming that the local acceleration due to gravity is -9.80 m/s2?, calculate the angular speed (in rev/min) where the coin slides off.
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4.5
To calculate the angular speed at which the coin slides off, we need to determine the point where the friction force between the coin and the turntable reaches its maximum value, which is equal to the maximum static friction.
First, we need to calculate the maximum static friction force acting on the coin. The formula for static friction force is given by:
Fs = μs * N
Where:
Fs is the static friction force
μs is the coefficient of static friction
N is the normal force, which is the weight of the coin in this case
The weight of the coin can be calculated using the formula:
W = m * g
Where:
W is the weight of the coin
m is the mass of the coin
g is the acceleration due to gravity
Substituting the given values:
m = 9.60 g = 0.00960 kg
g = -9.80 m/s² (negative sign since it is acceleration due to gravity)
W = 0.00960 kg * (-9.80 m/s²)
W = -0.09408 N
The normal force is equal to the weight of the coin since it is not accelerating vertically or horizontally. Therefore, N = -0.09408 N.
Now, substituting the given coefficient of static friction into the formula:
Fs = μs * N
Fs = 0.80 * (-0.09408 N)
Fs = -0.075264 N
The negative sign indicates that the static friction force is acting in the opposite direction of the impending motion.
Next, we need to calculate the radial acceleration, which is given by:
ar = r * ω²
Where:
ar is the radial acceleration
r is the radius from the center of the turntable to the coin
ω is the angular velocity
Substituting the given values:
r = 10.7 cm = 0.107 m
Now, the net force acting on the coin is equal to the radial acceleration multiplied by the mass of the coin:
Fnet = m * ar
Fnet = (0.00960 kg) * (0.107 m) * ω²
Since the net force is equal to the friction force when the coin slides off, we can equate the two:
Fnet = Fs
0.00960 kg * 0.107 m * ω² = -0.075264 N
Simplifying the equation:
0.0010272 ω² = -0.075264
Divide both sides by 0.0010272:
ω² = -73.372
Take the square root of both sides to solve for ω:
ω ≈ ±8.56 rad/s
Since the angular speed is usually expressed in rev/min, we convert ω to rev/min. There are 2π radians in one revolution, and 60 seconds in one minute:
ω ≈ (8.56 rad/s) * (1 rev / 2π rad) * (60 s / 1 min)
ω ≈ 258.6 rev/min
Therefore, the angular speed at which the coin slides off the turntable is approximately 258.6 rev/min.