A ceiling fan consists of five identical slender blades, each of length 0.60m and mass 0.40kg, rigidly attached to a hub of negligible mass. The blades are mounted so that they rotate in a horizontal plane about a fixed vertical axis passing through the hub. Each blade can be modeled as a thin rod pivoted about one end.
The fan operates at two speed settings:
• Low speed: ωL=1.5revs−1
• High speed: ωH=3.0revs−1
Assume the transition from low to high speed occurs under a constant net torque provided by the motor over a time interval of 4.0s, and ignore air resistance.
(a) Calculate the total kinetic energy of the fan at the low-speed setting.
(b) Determine the additional work the motor must do to bring the fan from low speed to high speed.
(c) Compute the magnitude of the constant net torque exerted by the motor during the 4.0s speed-up interval.
(d) After prolonged operation at high speed, the motor is turned off and a braking pad is gently applied to the rim of one blade at a radial distance of 0.55m from the hub. If the coefficient of kinetic friction between the pad and the blade is 0.25 and the normal force applied by the pad is 3.0N, estimate the time required for the fan to come to rest. Neglect all other torques.
(e) Explain qualitatively why the energy required to speed up the fan depends on the square of the final angular speed, whereas the work done by the braking pad depends linearly on the number of revolutions completed during the stopping process.