Question #120972

A fan blade is initially rotating an angular speed of 4.6 rpm. It slows down and eventually comes to rest in a time of 32 seconds after turning through a total of 8.8 revolution. Find a.) The angular speed in rad/s. b.) The average angular velocity ang c.) The average angular acceleration.

Expert's answer

Initial angular velocity is "\\omega_o=4.6\\times \\frac{2\\pi}{60}=0.48\\:rad\/s"

Final angular velocity is "\\omega_f=0\\:rad\/s"

Final time "t=t_f=32s" and angle traversed is "\\theta=8.8\\times2\\pi=55.29 \\:rad"

(a).As we have not provided at which time we need to find the angular speed, thus

we assume time t is arbitrary, hence

"\\omega=\\omega_o+(-\\alpha) t\\implies \\omega=\\omega_o-\\alpha t\\:rad\/s"As angular acceleration is decreasing.

(b). Since, average angular velocity is given by

"\\omega_{av}=\\frac{\\Delta \\theta}{\\Delta t}"Thus, on plugin the value we get,

"\\omega_{av}=\\frac{55.29-0}{32-0}=1.72 \\: rad\/s"

(c). Since,

"\\alpha_{av}=\\frac{\\Delta \\omega}{\\Delta t}=\\frac{\\omega_f-\\omega_o}{t_f-t_o}"Hence, on plugin the value we get

"\\alpha_{av}=\\frac{0-0.48}{32-0}=-0.015 \\:rad\/s^2"

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