A coin rests 15 cm away from the center of a turntable. The turntable starts to rotate at an angular acceleration of .8 rad/s^2 and after 7 seconds it slides off. What is the coefficient of static friction between the coin and the surface?
Just give me step by step on how to solve this problem.
It starts sliding when the static friction force reaches its maximum value. Its maximum value is Us*M*g.
Us is the static coefficeint you want to determine
After 7 seconds, the angular velocity is 5.6 rad/s and the centripetal acceleration is
Rw^2 = -.15*(5.6)^2 = 4.7 m/s^2.
There is also continuing tangential acceleration of R*alpha = 0.12 m/s^2 The vector-sum acceleration is
4.701 m/s^2, so you can ignore the tangential acceleration effect.
Us*M*g = M R w^2
Us = R w^2/g = 4.7/9.8 = 0.48
To solve this problem, you can follow these steps:
Step 1: Calculate the angular velocity of the turntable at the point when the coin slides off.
- Use the formula: ω = ω0 + αt, where ω0 represents the initial angular velocity (which is 0 in this case), α is the angular acceleration (given as 0.8 rad/s^2), and t is the time (7 seconds in this case).
Step 2: Determine the radial acceleration of the coin.
- Use the formula: ar = rω^2, where ar is the radial acceleration, r is the distance of the coin from the center of the turntable (15 cm or 0.15 m), and ω is the angular velocity of the turntable at the point when the coin slides off (calculated in Step 1).
Step 3: Calculate the net force exerted on the coin in the radial direction.
- Since the coin is sliding off, the net force in the radial direction is equal to the friction force between the coin and the turntable.
- Use the formula: Fnet = mar, where Fnet represents the net force, ma is the mass of the coin multiplied by the radial acceleration (calculated in Step 2), and r is the distance of the coin from the center of the turntable (0.15 m).
Step 4: Determine the normal force.
- The normal force is the force exerted by the turntable on the coin perpendicular to the surface.
- Use the formula: Fn = mg, where Fn is the normal force, m is the mass of the coin (assuming it is known), and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Step 5: Calculate the coefficient of static friction.
- The coefficient of static friction (μs) is the ratio of the maximum frictional force to the normal force.
- Use the formula: μs = Ff / Fn, where μs is the coefficient of static friction, and Ff is the magnitude of the frictional force (calculated as the net force in Step 3).
By following these steps, you should be able to solve the problem and find the coefficient of static friction between the coin and the surface.