Question

The propeller of a plane has a rotational inertia of 890 kg*m². Each blade has a length of 3.3 m. The propellor starts from rest and reaches a final angular speed of 16 rad/s in only 8 seconds.

a) What was the angular acceleration of the propeller?

b) What was the net torque exerted on the propeller while it was accelerating?

c) When t = 4 s, what was ω for the propeller?

d) When t = 4 s, through what angle θ had the propeller rotated? Your answer should be entered in radians.

e) When t = 4 s, determine a_c for the tip of a blade.

f) When t = 4 s, determine a_tan for the tip of a blade.

g) When t = 4 s, determine a_tot for the tip of a blade.

Thank you for your help!

Answer 1

Rotational inertia of the propeller of the plane = I = 890 kg.m²

Length of each blade = L = 3.3 m

Initial angular speed of the propeller = ₁ = 0 rad/s (At rest)

Angular speed of the propeller after 8 sec = ₂ = 16 rad/s

Time period = T₁ = 8 sec

Angular acceleration of the propeller =

₂ = ₁ + T₁

16 = 0 + (8)

= 2 rad/s²

Net torque on the propeller = _net

I = _net

(890)(2) = _net

_net = 1780 N.m

T₂ = 4 sec

Angular speed of the propeller at 4 sec = ₄

₄ = ₁ + T₂

₄ = 0 + (2)(4)

₄ = 8 rad/s

Angle through which the propeller has rotated in 4 sec =

= ₁T₂ + T₂²/2

= (0)(4) + (2)(4)²/2

= 16 rad

Centripetal acceleration of for the tip of the blade at 4 sec = a_c

a_c = ₄²L

a_c = (8)²(3.3)

a_c = 211.2 m/s²

Tangential acceleration of the tip of blade at 4 sec = a_tan

a_tan = L

a_tan = (2)(3.3)

a_tan = 6.6 m/s²

Total acceleration of the tip of blade at 4 sec = a_tot

a_tot = 211.3 m/s²

a) Angular acceleration of the propeller = 2 rad/s²

b) Net torque exerted on the propeller while accelerating = 1780 N.m

c) Angular speed of the propeller at 4 sec = 8 rad/s

d) Angle through which the propeller has turned in 4 sec = 16 rad

e) Centripetal acceleration of the tip of the blade at 4 sec = 211.2 m/s²

f) Tangential acceleration of the tip of the blade at 4 sec = 6.6 m/s²

g) Total acceleration of the tip of the blade at 4 sec = 211.3 m/s²