A small object is placed 13.0 cm from the center of a phonograph turntable. It is observed to r

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A small object is placed 13.0 cm from the center of a...

A small object is placed $13.0 \mathrm{~cm}$ from the center of a phonograph turntable. It is observed to remain on the table when it rotates at $33 \frac{1}{3}$ rev/min but slides off when it rotates at $45.0$ rev/min. Between what limits must the coefficient of static friction between the object and the surface of the turntable lie?

Key Concepts

Force Equilibrium in Rotating Systems

For a rotating object to remain in equilibrium without sliding, the inward (centripetal) force required must not exceed the maximum static frictional force available. By comparing the centripetal force demand at different angular speeds with the frictional force limit, one can establish bounds on the coefficient of static friction necessary to prevent slipping.

Static Friction

Static friction is the force that resists the onset of sliding between two surfaces in contact. For an object in uniform circular motion on a rotating table, static friction must provide the necessary centripetal force to keep the object in its circular path. The coefficient of static friction relates the maximum possible frictional force to the normal force, and its value determines whether the object will remain in place or begin to slip.

Angular Velocity Conversion

Angular velocity conversion is the process of converting rotational speed from units such as revolutions per minute (rpm) to radians per second (rad/s). This conversion is essential for using standard formulas in circular motion, such as computing centripetal acceleration with the relation a = r?².

Uniform Circular Motion

Uniform circular motion describes an object moving in a circle at a constant speed. Although the speed remains constant, the direction changes continuously, which requires an inward (centripetal) acceleration. This concept is key when analyzing objects on rotating platforms, where the inward force is provided by friction.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path. It is proportional to the mass of the object, the square of its tangential speed (or angular speed), and inversely proportional to the radius of the circle. In rotating systems, this force must be supplied by another force, such as static friction, to prevent the object from sliding outward.

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