Question

An airplane propeller is 2.68 m in length (from tip to tip) and has a mass of 137 kg . When the airplane's engine is first started, it applies a constant torque of 1990 N⋅m to the propeller, which starts from rest.

What is the angular acceleration of the propeller? Treat the propeller as a slender rod.

What is the propeller's angular speed after making 5.00 rev ?

How much work is done by the engine during the first 5.00 rev ?

What is the average power output of the engine during the first 5.00 rev ?

What is the instantaneous power output of the motor at the instant that the propeller has turned through 5.00 rev ?

Answer 1

I= mL² /12 = 137*2.68^2/12=81.9990667

τ=I* α
α= τ/I = 1990/(81.9990667) =  24.2685689 rad/s²

What is the propeller's angular speed after making 5.00 rev ?

θ= 2pi*5 = 10pi

ωf^2 = ωi^2 +2*αθ = 0+2*αθ

ωf = sqrt(2*24.2685689*10*pi)=   39.0491889 rad/s

How much work is done by the engine during the first 5.00 rev

W =τ*θ = 1990*10*pi =  62517.6938 J

What is the average power output of the engine during the first 5.00 rev ?

ωf = ωi +α*t = 0+α*t

t = ωf/α = 39.0491889/24.2685689 =1.60904374 sec

P = w/t = 62517.6938 /1.60904374 = 38853.943 Watt

What is the instantaneous power output of the motor at the instant that the propeller has turned through 5.00 rev

he expression of instantaneous power

P = τ*ωf = 1990*39.0491889 = 77707.8859 Watt