Question

An airplane propeller is 2.28 m in length (from tip to tip) and has a mass of 132 kg . When the airplane's engine is first started, it applies a constant torque of 1880 N⋅m to the propeller, which starts from rest.

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

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

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

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

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

Answer 1

here,

the mass of propeller , m = 132 kg

the length of propeller , l = 2.28 m

the constant torque applied , T = 1880 N.m

a)

the angular acceleration of propeller , alpha = T /(m * l^2 /12)

alpha = 1880 /(132 * 2.28^2 /12) rad/s^2

alpha = 32.9 rad/s^2

b)

theta = 5 * 2pi rad

let the final angular speed be w

using third equation of motion

w^2 - w0^2 = 2 * alpha * theta

w^2 - 0 = 2 * 32.9 * (5 * 2 pi)

solving for w

w = 45.4 rad/s

the final angular velocity is 45.4 rad/s

c)

the work done , W = 0.5 * I * w^2

W = 0.5 * (m * l^2 /12) * 45.4^2 J

W = 0.5 * (132 * 2.28^2 /12) * 45.4^2 J

W = 5.9 * 10^4 J

d)

let the time taken be t

using first equation of motion

w = w0 + alpha * t

45.4 = 0 + 32.9 * t

solving for t

t = 1.38 s

the time taken is 1.38 s

the average power output of the engine during the first 5.00 rev , P = W/t

P = 5.9 * 10^4 /1.38 W

P = 4.28 * 10^4 W

e)

the instantaneous power output of the motor at the instant , P = T * w

P = 1880 * 45.4 W = 8.54 * 10^4 W