What Determines the Maximum Angular Velocity for a Coin on a Turntable?

[cstryker02]

A 3.20g coin is placed 19.0cm from the center of a turntable. The coin has static and kinetic coefficients of friction with the turntable surface of (u)static=0.900 and (u)kinetic=0.400.

What is the maximum angular velocity with which the turntable can spin without the coin sliding?

 

[StatusX]

Do you know the equation for centripetal force? You can see it increases with angular velocity, so at a certain point it will be greater than the force of friction, and the coin will begin to move.

 

[wais]

To begin, we can use the formula for centripetal force to determine the maximum angular velocity. The force of friction between the coin and the turntable surface must be equal to the centripetal force in order to prevent the coin from sliding.

First, we can calculate the weight of the coin using the formula w = mg, where m is the mass of the coin and g is the acceleration due to gravity. Since the coin has a mass of 3.20g, its weight would be 3.20g * 9.8m/s^2 = 31.36m/s^2.

Next, we can calculate the centripetal force using the formula Fc = mv^2/r, where m is the mass of the coin, v is the velocity, and r is the radius (which in this case is 19.0cm or 0.19m).

Since we know the force of friction must be equal to the centripetal force, we can set up the equation Ff = (u)mg = mv^2/r. Plugging in the values we know, we get (u)mg = mv^2/r, where (u) is the coefficient of friction, m is the mass of the coin, g is the acceleration due to gravity, and r is the radius.

We can then solve for the maximum velocity, v, by rearranging the equation to v = sqrt((u)rg). Plugging in the values we know, we get v = sqrt((0.900)(0.19m)(31.36m/s^2)) = 3.68m/s.

Therefore, the maximum angular velocity with which the turntable can spin without the coin sliding is 3.68m/s. Any higher angular velocity would result in the coin sliding due to the force of friction being exceeded.