Randall D. Knight — Dynamics II: Motion in a Plane: Problem 63

For safety, elevators have a rotational governor, a device that is attached to and rotates with one of the elevator’s pulleys. The governor, shown in FIGURE P8.63, is a disk with two hollow channels holding springs with metal blocks of mass \( m \) attached to their free ends. The faster the governor spins, the more the springs stretch. At a critical angular velocity \( \omega_c \), the metal blocks contact the housing, which completes a circuit and activates an emergency brake. The spring force on a mass, which we will explore more thoroughly in Chapter 9, is \( F_{Sp} = k(r - L) \), where \( k \) is the spring constant measured in N/m, and \( L \) is the relaxed (unstretched) length of the spring. Suppose a rotational governor has \( L = 0.75R \), and the emergency brake activates when the metal blocks reach \( r = R \). What is the critical angular velocity in rpm if \( R = 16 \text{ cm} \), \( k = 18 \text{ N/m} \), and \( m = 35 \text{ g} \)? Ignore gravity.