Chapter 8.8, Problem 43E

David and Goliath The force F (in newtons) required to maintain an object in a circular path varies jointly with the mass m (in kilograms) of the object and the square of its speed (measured in meters per second) and inversely with the radius r (in meters) of the circular path. Suppose that David has a rope that is 3 meters long. On the end of the rope he has attached a pouch that holds a 0.5-kilogram stone. Suppose that David is able to spin the rope in a circular motion at the rate of 50 revolutions per minute.

1. a. The spinning rate of 50 revolutions per minute can be converted into an angular velocity ω (lowercase Greek letter omega) using the formula ω = 2π (revolutions per minute). Write the spinning rate as an angular velocity rounded to two decimal places.
2. b. Use the fact that v = ωr, where r is the radius of the circle, to find the linear velocity (in meters per minute) of the stone if it were released. Now convert the linear velocity to meters per second.
3. c. Suppose that the force on the rope required to keep the rock in a circular motion is 2.3 newtons. Use the result from part (b) to find the constant of proportionality k.
4. d. David is fairly certain that he will require more force than this to beat Goliath in a battle, so he increases the circular motion to 80 revolutions per minute and increases the length of the rope to 4 meters. What is the force required to keep the stone in a circular motion?