Suppose you swing a ball in a vertical circle on a 1.0...

Suppose you swing a ball in a vertical circle on a 1.0 -m-long string. As you probably know from experience, there is a minimum angular velocity $\omega_{\min }$ you must maintain if you want the ball to complete the full circle. If you swing the ball at $\omega<\omega_{\text {min }},$ then the string goes slack before the ball reaches the top of the circle. What is $\omega_{\min } ?$ Give your answer in $\mathrm{rpm}$.

Key Concepts

Circular Motion

This concept describes the movement of an object along a curved path with a fixed radius. In problems involving vertical circular motion, the object requires a centripetal force to maintain its curved trajectory, which is especially important when external forces like gravity act on the object.

Minimum Speed Condition for Complete Circular Motion

In a vertical circle, there is a critical speed or angular velocity below which the string or rod would go slack because it would not provide the necessary centripetal force. At the top of the circle, the minimum condition is met when the gravitational force exactly supplies this centripetal force, meaning that the tension in the string is zero.

Centripetal Force Balance with Gravity

At the top of the vertical circle, the necessary centripetal force is provided entirely by gravity at the minimum angular speed. This balance is described by the equation mg = m(v²/r), where g is the acceleration due to gravity, v is the tangential speed, and r is the radius of the circle.

Angular Velocity and Unit Conversion

Angular velocity is typically expressed in radians per second (rad/s) but can also be converted to revolutions per minute (rpm). Converting between these units involves understanding that one revolution is 2? radians and that time conversion factors are required, making it possible to express theoretical speeds in practical terms.

Transcript

Please give Ace some feedback