To determine the maximum angular speed for which the coin stays in place, we can use the concept of centripetal force.

First, let's convert the mass of the coin from grams to kilograms. Since 1 gram is equal to 0.001 kilograms, the mass of the coin is 8.6 g * 0.001 kg/g = 0.0086 kg.

Next, we need to calculate the force of static friction between the coin and the turntable. The formula for static friction is given by fs = μs * N, where μs 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, which is given by N = m * g, where m is the mass of the coin and g is the acceleration due to gravity (approximately 9.8 m/s^2).

So, N = 0.0086 kg * 9.8 m/s^2 = 0.08428 N.

Now, we can calculate the force of static friction using the coefficient of static friction. fs = 0.52 * 0.08428 N = 0.0438 N.

The force of static friction provides the centripetal force required to keep the coin in place as the turntable rotates. The centripetal force is given by Fc = m * r * ω^2, where m is the mass of the coin, r is the distance from the center of the turntable to the coin, and ω is the angular speed.

In this case, m = 0.0086 kg, r = 0.105 m (since the distance is given in centimeters, we convert it to meters by dividing by 100), and Fc = fs = 0.0438 N.

Substituting these values into the equation, we have 0.0438 N = 0.0086 kg * 0.105 m * ω^2.

Simplifying the equation, we get ω^2 = 0.0438 N / (0.0086 kg * 0.105 m).

Calculating the right side of the equation, we have ω^2 = 49.4118 rad^2/s^2.

Finally, taking the square root of both sides, we find ω = √(49.4118 rad^2/s^2) = 7.03 rad/s.

Therefore, the maximum angular speed for which the coin stays in place is approximately 7.03 rad/s.

Question

To determine the maximum angular speed for which the coin stays in place, we can use the concept of centripetal force.

First, let's convert the mass of the coin from grams to kilograms. Since 1 gram is equal to 0.001 kilograms, the mass of the coin is 8.6 g * 0.001 kg/g = 0.0086 kg.

Next, we need to calculate the force of static friction between the coin and the turntable. The formula for static friction is given by fs = μs * N, where μs 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, which is given by N = m * g, where m is the mass of the coin and g is the acceleration due to gravity (approximately 9.8 m/s^2).

So, N = 0.0086 kg * 9.8 m/s^2 = 0.08428 N.

Now, we can calculate the force of static friction using the coefficient of static friction. fs = 0.52 * 0.08428 N = 0.0438 N.

The force of static friction provides the centripetal force required to keep the coin in place as the turntable rotates. The centripetal force is given by Fc = m * r * ω^2, where m is the mass of the coin, r is the distance from the center of the turntable to the coin, and ω is the angular speed.

In this case, m = 0.0086 kg, r = 0.105 m (since the distance is given in centimeters, we convert it to meters by dividing by 100), and Fc = fs = 0.0438 N.

Substituting these values into the equation, we have 0.0438 N = 0.0086 kg * 0.105 m * ω^2.

Simplifying the equation, we get ω^2 = 0.0438 N / (0.0086 kg * 0.105 m).

Calculating the right side of the equation, we have ω^2 = 49.4118 rad^2/s^2.

Finally, taking the square root of both sides, we find ω = √(49.4118 rad^2/s^2) = 7.03 rad/s.

Therefore, the maximum angular speed for which the coin stays in place is approximately 7.03 rad/s.

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